Boris
Hasselblatt
Tufts University
Mathematical billiards
Oberseminar Nonlinear Dynamics 📅
Usual time
Tuesday at 3:15 p.m.
Usual venue
Free University Berlin, Arnimallee 7 (rear building), room 140
Number of talks
324
June 2025
January 2025
Jia-Yuan
Dai
National Tsing Hua University
Hybrid bifurcations: Periodicity from eliminating a line of equilibria
Abstract.
We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations, which consists of a bifurcation without parameters and a classical bifurcation. Our main result classifies the hybrid bifurcation when a line of equilibria with an exchange point of normal stability vanishes. We showcase the efficacy of our approach by proving stable periodic coexistent solutions in an ecosystem of two competing predators with Holling's type II functional response. This is a joint work with Alejandro López Nieto, Phillipo Lappicy, Nicola Vassena, and Hannes Stuke. Reference: A. López Nieto, P. Lappicy, N. Vassena, H. Stuke, and J.-Y. Dai. Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria. https://arxiv.org/pdf/2310.19604
Nicola
Vassena
Universität Leipzig
Takens-Bogdanov points and Global Hopf bifurcation in reaction networks
Abstract.
In this talk, I will explore two distinct yet complementary approaches for identifying periodic solutions in ODE systems arising from reaction networks. The first method focuses on locating an equilibrium where the Jacobian has an algebraically double eigenvalue zero (a Takens-Bogdanov point). The second method examines changes in stability at an equilibrium with an invertible Jacobian, leading to global Hopf bifurcation. I will discuss how -- perhaps not so surprisingly?- the sufficient conditions for these two bifurcations overlap.
July 2024
Hans
Engler
Georgetown University
The Lorenz System of 1996
Abstract.
The meteorologist and applied mathematician Edward Lorenz is famous for discovering chaotic behavior in dynamical systems in 1963. In 1996, Lorenz introduced a dynamical system that describes very simple 'weather' on a cartoon planet: a scalar quantity evolves on a circular array of N sites, undergoing radiative forcing, dissipation, and nonlinear advection. Lorenz proposed this system as a test bed for numerical weather prediction. Since then, it has found much use as a test case in data assimilation. Related systems have been studied by other authors earlier. Mathematically, this is an nonlinear N-dimensional dynamical system that is invariant under rotating the sites. There is a single parameter, namely forcing strength. For small forcing strength, there is no 'weather': the only possible stable solutions are constant in space and time. As the strength of the forcing increases, periodic wave patterns appear that move around the circle of sites. These periodic patterns are not unique - the same forcing strength may be associated with stable patterns that are qualitatively different, depending on the system's initial state. For even larger forcing, the motion becomes chaotic. Regular wave patterns are replaced by moving irregular wave trains that are short-lived, similar to changing weather systems that move across a landscape. The talk will introduce the main properties of this and of related systems. The appearance of periodic solutions can be explained with bifurcation theory for all these versions. By using the discrete Fourier transform and explicit computation of normal form coefficients, the stability of bifurcating periodic solutions and the coexistence of multiple such solutions for the same radiative forcing can be understood. The system also shows delayed transition to instability as the forcing parameter increases slowly. This is joint work with John Kerin, a former Georgetown University undergraduate student.
Sören
von der Gracht
Universität Paderborn
Exploring exotic symmetries to explain exotic behavior of network dynamical systems
Abstract.
Many dynamical systems models of real world processes exhibit the structure of a network consisting of nodes with connections between them. The specific interaction structure of a network can produce remarkable dynamics beyond that of the individual nodes. Prominent examples include synchronization and highly complex branching behavior in bifurcations, phenomena that are not found in dynamical systems without the structure of a network. Network dynamical systems are not well understood mathematically, which makes it hard to quantify and control their behavior. The reason is that most of the established machinery of dynamical systems theory fails to distinguish between networks and general dynamical systems. Several mathematical tools that are tailor-made for network problems have been proposed recently. Strikingly, they have one thing in common: they exploit the algebraic nature of networks. In this talk, I will give an overview over some recent results regarding the question which dynamical behavior and generic bifurcations are dictated by the network structure of a system. In particular, I will illustrate how structural and algebraic properties culminate in symmetries of the governing equations and how these can be exploited for (partial) answers. This includes classical symmetries but also more exotic concepts such as monoid and quiver representations.
Phillipo
Lappicy
Universidad Complutense de Madrid, Spain
An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension
Abstract.
Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation. This is a joint work with E. Beatriz.
June 2023
Jia-Yuan
Dai
National Chung Hsing University, Taiwan
Exploring a new mechanism of periodic orbits: Dynamics of two predators competing for a prey
Abstract.
We prove the existence of stable periodic orbits of an ODE system that describes the dynamics of two predators competing for the same prey. The prey population grows logistically in the absence of predation and the predators feed on the prey with Holling's type-II nonlinearity. We explain how stable periodic orbits, located far away from the boundary planes, are triggered by perturbing an elliptic Hopf bifurcation point without parameters. To this end, we prove that the ODE system admits an elliptic Hopf bifurcation and the existence and stability of periodic orbits are ensured by the averaging method. This is a joint work with A. López Nieto, P. Lappicy, H. Stuke, and N. Vassena.
Phillipo
Lappicy
Universidad Complutense de Madrid, Spain
On the exceptional Bianchi models
Abstract.
Most (if not all) rigorous results regarding spatially homogeneous and anisotropic cosmological singularities are based on the Bianchi types VIII and IX models, which constitute four-dimensional dynamical systems. However, there is one exceptional model which has the same dimensionality, but with almost no rigorous results: the Bianchi type VI_{-1/9} models. These exceptional models should play a distinguished role in the generic asymptotic dynamics towards cosmological singularities, especially when small spatial inhomogeneities are considered. We will discuss some early explorations/findings which are based on joint work with C. Uggla.
November 2022
Yulia
Petrova
Instituto de Matematica Pura e Aplicada
and
Sergey
Tikhomirov
University of Duisburg-Essen
Two tube model of miscible displacement: travelling waves and normal hyperbolicity
Abstract.
We study the motion of miscible liquids in porous media with the speed determined by Darcy's law. The two basic examples are the displacement of viscous liquids and the motion induced by gravity. Such motion often is unstable and creates patterns called viscous fingers (picture attached). We concentrate on important for applications property of viscous fingers - speed of their propagation. The work is inspired by the results of F. Otto and G. Menon for a simplified model, called transverse flow equilibrium (TFE). In this work a rigorous upper bound was proved using the comparison principle. At the same time numerical experiments suggest that the actual speeds are better than Otto-Menon estimates. We consider a two-tubes model -- the simplest model we were able to construct which includes transverse liquid flow. For this model for the gravitational fingers we were able to find families of travelling waves and found the relation between original model and TFE simplification. The main tool in the proof in normal hyperbolicity. For viscous liquid it seems that the phenomenon is the same but up to now it is work in progress. This is a joint work with Yalchin Efendiev.
June 2022
Babette
de Wolff
VU Amsterdam
Delayed feedback stabilization & unconventional symmetries
Abstract.
In 1992, the physicist Pyragas proposed a time delayed feedback scheme to stabilize periodic solutions of ordinary differential equations. The feedback scheme (now known as `Pyragas control`) has been adapted to symmetric systems whose symmetries can be described by groups. In this talk, we explore how Pyragas control can be adapted to systems with `unconventional symmetries`, i.e. symmetries that cannot be described by groups. In the first part of the talk, we review a fundamental observation that gives insight in Pyragas control without symmetry. In the second part of the talk we give an example of a system of three coupled oscillators with `unconventional symmetriess`. For this example, we discuss what a Pyragas-like control scheme looks like and what the stabilization properties of this control scheme are. The first part of the talk based on joint work with Isabelle Schneider (FU Berlin/Universitäst Rostock); the second part of the talk is based on joint work with Bob Rink (VU Amsterdam).
May 2022
Oleksandr A.
Burylko
Institute of Mathematics NAS of Ukraine and Potsdam Institute for Climate Impact Research
Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators
Abstract.
Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.
Leonhard
Schülen
Technische Universität Berlin
The solitary route to chimera states
Abstract.
We show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit we demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. We demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators.
Yulia
Petrova
Instituto de Matemática Pura e Aplicada, Rio de Janeiro
On the impact of dissipation ratio on vanishing viscosity solutions of Riemann problems for chemical flooding models
Abstract.
We are interested in solutions of the Riemann problem arising in chemical flooding models for enhanced oil recovery (EOR). To distinguish physically meaningful weak solutions we use vanishing viscosity admissibility criterion. We demonstrate that when the flow function depends non-monotonically on the chemical agent concentration (which corresponds to the surfactant flooding), non-classical undercompressive shocks appear. They correspond to the saddle-saddle connections for the traveling wave dynamical system and are sensitive to precise form of the dissipation terms. In particular we prove the monotonic dependence of the shock velocity on the ratio of dissipative coefficients. The talk is based on joint work with F. Bakharev, A. Enin and N. Rastegaev (arxiv: 2111.15001).
Sergey
Tikhomirov
St.Petersburg State University and Instituto de Matemática Pura e Aplicada, Rio de Janeiro
Mixing zone in miscible displacement: application in polymer flooding and theoretical attempts of improving
Abstract.
Injection of less viscous fluid to a more viscous one generates instabilities, which are often called 'viscous fingers'. In case of miscible displacement it generates a mixing zone, where both fluids are presented. This phenomenon has a negative impact on flooding using chemical slugs in oil fields. We study the mathematical model describing the behavior on the rear front of a polymer slug. It consists of conservation of mass, incompressibility condition and Darcy law. This model often is called the Peaceman model. We study the size of the mixing zone appearing on the rear end of the polymer slug, provide pessimistic estimates and its numerical validation and apply estimates to the graded viscosity banks technology (GVB or tapering) to reduce the volume of used polymer without loss of effectiveness. Further optimization is possible with more delicate estimates of the mixing zone. In current work in progress with Yu. Petrova and Ya. Efendiev we consider a simplified model replacing multidimensional space with two tubes interacting with each other. In this system we numerically observe two traveling waves with different speeds. This behavior mimics the mixing zone of the multidimensional Pieceman model. We will speak on our progress in this direction.
Marius
Yamakou
Friedrich-Alexander-Universität Erlangen-Nürnberg
Transitions between weak-noise-induced resonance phenomena in a multiple timescales neural system
Abstract.
We consider a stochastic slow-fast nonlinear dynamical system derived from a computational neuroscience model. Independently, we uncover the mechanisms that underlie two forms of weak-noise-induced resonance mechanisms, namely, self-induced stochastic resonance (SISR) and inverse stochastic resonance (ISR) in the system. We then show that SISR and ISR are related through the relative geometric positioning (and stability) of the fixed point and the generic folded singularity of the system's critical manifold. This result could explain the experimental observation in which real biological neurons with identical physiological features and stochastic synaptic inputs sometimes encode different information.
November 2021
Jaroslav
Hlinka
Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic
Perturbations both trigger and delay seizures due to generic properties of slow-fast relaxation oscillators
Abstract.
The mechanisms underlying the emergence of seizures are one of the most important unresolved issues in epilepsy research. In this work, we analyze white and how perturbations, exogenous or endogenous, may promote or delay seizure emergence [1], following previous observations in vivo, in vitro and in silico [2]. To this aim, due to the increasingly adopted view of epileptic dynamics in terms of slow-fast systems, we perform a theoretical analysis of the phase response of a generic relaxation oscillator. As relaxation oscillators are effectively bistable systems at the fast time scale, it is intuitive that perturbations of the non-seizing state with a suitable direction and amplitude may cause an immediate transition to seizure. By contrast, and perhaps less intuitively, smaller amplitude perturbations have been found to delay the spontaneous seizure initiation. By studying the isochrons of relaxation oscillators, we show that this is a generic phenomenon, with the size of such delay depending on the slow flow component. Therefore, depending on perturbation amplitudes, frequency and timing, a train of perturbations causes an occurrence increase, decrease or complete suppression of seizures. This dependence lends itself to analysis and mechanistic understanding through methods outlined in this paper. We illustrate this methodology by computing the isochrons, phase response curves and the response to perturbations in several epileptic models possessing different slow vector fields. While our theoretical results are applicable to any planar relaxation oscillator, in the motivating context of epilepsy they elucidate mechanisms of triggering and abating seizures, thus suggesting stimulation strategies with effects ranging from mere delaying to full suppression of seizures.
July 2021
Messoud
Efendiyev
Helmholtz Zentrum München
Mathematical modeling of biofilms and their long-time dynamics
Abstract.
In this talk we deal with mathematical modelling of biofilms, that show biofilm performance is non-uniform. Moreover, our model leads to a new class of degenerate PDEs. Effect of degeneracy to large time behavior of solutions will also be considered.
Carsten
Conradi
Hochschule für Technik und Wirtschaft Berlin
Monomial parameterizations in the analysis of biochemical reaction networks
Abstract.
The dynamics of biochemical reaction networks can be described by ODEs with polynomial right hand side. In this presentation networks are considered where the steady state variety can be parameterized by monomials. I present two applications of these monomial parameterizations in the analysis of reaction networks: (i) deciding multistationarity and (ii) establishing Hopf bifurcations. Here multistationarity refers to the existence of at least two positive solutions to the polynomial steady state equations. And if a monomial parameterization exists, then this question is equivalent to the feasibility of at least one linear inequality system (out of many). The results presented here can be used to determine parameter values where multistationarity or Hopf bifurcations occur.
June 2021
Sebastian
Wieczorek
University College Cork
Rate-Induced Tipping Points
Abstract.
Many systems are subject to external disturbances or changing external conditions. For a system near a stable state (an attractor) we might expect that, as external conditions change with time, the stable state will change too. In many cases the system may adapt to changing external conditions and track the moving stable state. However, tracking may not always be possible owing to nonlinearities and feedbacks in the system. So far, the focus has been on critical levels of external conditions (dangerous autonomous bifurcation points) where the stable state turns unstable or disappears, causing the system to suddenly move to a different and often undesired state. We describe this phenomenon as bifurcation-induced tipping or B-tipping. However, critical levels are not the only critical factor for tipping. Some systems can be particularly sensitive to how fast the external conditions change and have critical rates: they suddenly and unexpectedly move to a different state if the external input changes too fast. This happens even though the moving stable state never loses stability in the classical autonomous sense! We describe this phenomenon as rate-induced tipping or R-tipping. Being a genuine non-autonomous bifurcation, R-tipping is not captured by the classical bifurcation theory and requires an alternative framework. In the first part of the talk, we demonstrate R-tipping in a simple ecosystem model where environmental changes are represented by time-varying parameters. We then introduce the concept of basin instability and show how to complement the classical bifurcation diagram with information on nonautonomous R-tipping that cannot be captured by the classical bifurcation analysis. In the second part of the talk, we develop a general mathematical framework for R-tipping with decaying inputs based on the concepts of thresholds, edge states and special compactification of the nonautonomous system. This allows us to transform the R-tipping problem into a connecting heteroclinic orbit problem in the compactified system, which greatly simplifies the analysis. We explain the key concept of threshold instability and give rigorous testable criteria for R-tipping to occur in arbitrary dimension. In the third part of the talk, we discuss the so-called "compost-bomb instability", which is an example of R-tipping without an obvious threshold. We use geometric singular perturbation theory and desingularisation to reveal non-obvious R-tipping thresholds and edge states.
Bhumika
Thakur
Jacobs University, Bremen
Data driven identification of nonlinear dynamics using sparse regression with applications in plasma physics
Abstract.
Data driven techniques are increasingly finding applications in physical sciences and plasma physics is no exception. Many plasma processes are highly complex and nonlinear and often the exact form of the equations governing their dynamics is not known. If we can construct these equations from the experimental data, then we can further our understanding of these processes and use techniques such as model reduction to isolate dominant physical mechanisms. A large number of regression techniques are available for identification of system dynamics from data, with varying degrees of generality and complexity. Sparse identification of nonlinear dynamics (SINDy) algorithm is one such technique that can be used to find parsimonious models. I will talk about this algorithm and discuss some examples where it is being applied in plasma physics with a focus on our ongoing attempt at finding the model equations for anode glow oscillations observed in a glow discharge plasma device.
Philipp
Lorenz-Spreen
MPI for Human Development, Berlin
Modeling radicalization dynamics and polarization in temporal networks
Abstract.
Echo chambers and opinion polarization have been recently quantified in several sociopolitical contexts, across different social media, raising concerns for the potential impact on the spread of misinformation and the openness of debates. Despite increasing efforts, the dynamics leading to the emergence of these phenomena remain unclear. Here, we propose a model that introduces the phenomenon of radicalization, as a reinforcing mechanism driving the evolution to extreme opinions from moderate initial conditions. Empirically inspired by the dynamics of social interaction, we consider agents characterized by heterogeneous activities and homophily. We analytically characterize the transition from a global consensus to an emerging radicalization that depends on parameters, which can be interpreted as the controversialness of a topic and the strength of social influence people exert on each other. Finally, we offer a definition of echo-chambers via our model and contrast the model's behavior against empirical data of polarized debates on Twitter, qualitatively reproducing the observed relation between users' engagement and opinions, as well as opinion segregation based on the interaction network. Our findings shed light on the dynamics that may lie at the core of the emergence of echo chambers and polarization in social media.
Chunming
Zheng
MPI for Physics of Complex Systems, Dresden
Transition to Synchrony in the Three-Dimensional Noisy Kuramoto Model
Abstract.
We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found, that alignment of the velocities occurs for arbitrary small attractive coupling. Adding noise to the system resolves this singular limit.
Georgi
Medvedev
Drexel University
Unfolding chimeras: Where Turing meets Penrose
Abstract.
The coexistence of coherence and incoherence is arguably the most interesting effect in the theory of synchronization and possibly in nonlinear science in general discovered in the past two decades. Despite intense research chimera states still present many challenging questions to the nonlinear science community. There is no consensus on how to define chimera states. Further, the theory is only available for chimera states lying in the Ott-Antonsen manifold, which is an elegant but a very special case. In this work, we suggest a new way for studying chimera states based on the combination of the linear stability analysis of mixing and a beautiful method of Penrose for Vlasov equation in plasma physics. This approach yields a new qualitative description of chimera states and provides very accurate quantitative estimates. Our results are universal in the sense that the structure and bifurcations of chimera states are explained in terms of the qualitative properties of the distribution of intrinsic frequencies and network topology, and, thus, are relevant for interacting particle systems of all scales from neuronal networks, to power grids, to astrophysics. This talk is based on the joint work with Hayato Chiba (Tohoku University) and Matthew Mizuhara (The College of New Jersey).
May 2021
Alicia
Dickenstein
University of Buenos Aires
Algebra and geometry in the study of enzymatic networks?
Abstract.
I will try to show in my lecture that the question in the title has a positive answer, summarizing recent mathematical results about signaling networks in cells obtained with algebro-geometric tools.
Maximilian
Engel
FU Berlin
Lyapunov exponents in random dynamical systems and how to find and use them
Abstract.
This talk aims to give an overview on various notions of Lyapunov exponents (LEs) in random dynamical systems, that is, systems whose evolution in time is governed by laws exhibiting randomness: from finite-time LEs to classical asymptotic LEs and corresponding spectra up to LEs for processes conditioned on staying in bounded domains. We demonstrate how these notions, especially of a first, dominant LE, become relevant in the context of stochastic bifurcations, in finite and infinite dimensions.
April 2021
Eddie
Nijholt
University of Illinois
Exotic symmetry in networks
Abstract.
Network dynamical systems appear all throughout science and engineering. Despite this prevalence, it remains unclear precisely how network structure impacts the dynamics. One very successful approach in answering this question is by identifying symmetry. Of course, there are many networks that do not have any form of symmetry, yet which still show remarkable dynamical behavior. Instead a wide array of other network features (such as node-dependency, synchrony spaces, and so forth) are known to impact the dynamics. We will see that most of these features can still be captured as symmetry, provided one widens the definition. That is, instead of considering classical group symmetry, one has to allow for more "exotic structures", such as semigroups, categories and quivers. In many cases the network topology itself can even be seen as such a symmetry. An important consequence is that network structure can therefore be preserved in most reduction techniques, which in turn makes it possible to analyse bifurcations in such systems. In order to best explain these notions, l do not assume any familiarity with group symmetry -or their exotic counterparts- on the part of the audience.
Alan David
Rendall
Johannes Gutenberg-Universität Mainz
Bogdanov-Takens bifurcations and the regulation of enzymatic activity by autophosphorylation
Abstract.
An important mechanism of information storage in molecular biology is the binding of phosphate groups to proteins. In this talk we consider the case of autophosphorylation, where the protein is an enzyme and the substrate to which it catalyses the binding of a phosphate group is that enzyme itself. It turns out that this often leads to more complicated dynamics than those seen in the case where enzyme and substrate are distinct. We focus on the example of the enzyme Lck (lymphocyte-associated tyrosine kinase) which is of central importance in the function of immune cells. We study a model for the activation of Lck due to Kaimachnikov and Kholodenko and give a rigorous proof that it admits periodic solutions. We do so by showing that it exhibits a generic Bogdanov-Takens bifurcation. This is an example where this approach gives a simpler proof of the existence of periodic solutions than ones using more elementary techniques. Joint work with Lisa Kreusser.
February 2021
Alexandre
Nolasco de Carvalho
University of São Paulo at São Carlos
Gradient Non-autonomous Dynamics
Abstract.
In this lecture we present our approach for the study of the asymptotics of autonomous and non-autonomous dynamical systems. This unified approach shows how some different notions of attractors play a role in the description of the dynamics and enable us to address the gradient structure within non-autonomous attractors. As an application we characterize the gradient structure within the uniform attractor for a non-autonomous Chafee-Infante problem. A few, in our view, interesting and challenging problems, for future studies, are presented at the end of the lecture.
Svetlana
Gurevich
Westfälische Wilhelms-Universität Münster
Dynamics of temporal and spatio-temporal localized states in time-delayed systems
Abstract.
Time-delayed systems describe a large number of phenomena and exhibit a wealth of interesting dynamical regimes such as e.g., fronts, localized structures or chimera states. They naturally appear in situations where distant, pointwise, nonlinear nodes exchange information that propagates at a finite speed. In this talk, we review our recent theoretical results regarding the existence and the dynamics of temporal, spatial and spatio-temporal localized structures in the output of semiconductor mode-locked lasers. In particular, we discuss dispersive effects which are known to play a leading role in pattern formation. We show that they can appear naturally in delayed systems [1] and we exemplify our result by studying the influence of high order dispersion in a system composed of coupled optical microcavities.
David
Müller-Bender
Chemnitz University of Technology
Laminar chaos in systems with time-varying delay
Abstract.
For many systems arising in nature and engineering, the influence of time-delays cannot be neglected. Due to environmental fluctuations that affect the delay generating processes such as transport processes, the delays are in general not constant but rather time-varying. Although it is known that variable delays can lead to interesting phenomena, their effect on the dynamics of systems is not completely understood. In this talk, it is demonstrated that a temporal delay variation can change the dynamics of a time-delay system drastically. A recently discovered type of chaos called laminar chaos is introduced, which can only be observed for a certain type of time-varying delays. It is characterized by nearly constant laminar phases with periodic duration, where the intensity of the laminar phases varies chaotically from phase to phase. In contrast to the high-dimensional turbulent chaos, which is also observed for constant delay, laminar chaos is low-dimensional. Furthermore, it is shown experimentally and theoretically that laminar chaos is a robust phenomenon. A time-series analysis toolbox for its detection is provided, which is benchmarked by experimental data and by time-series of a nonlinear delayed Langevin equation.
Bernold
Fiedler
Freie Universtität Berlin, Germany
Global attractors of scalar parabolic equations: the Thom-Smale complex
Abstract.
We consider the global attractors of scalar parabolic equations u<sub>t</sub>=u<sub>xx</sub>+f(x,u,u<sub>x</sub>) under Neumann boundary conditions. The Thom-Smale complex decomposes a global attractor into the unstable manifolds of its (hyperbolic) equilibria u<sub>t</sub>=0. In general this is not a topological cell complex -- not even in the presence of a gradient structure. For the above PDEs, however, the Thom-Smale complex turns out to be a signed regular topological cell complex: the boundaries of the unstable manifolds are topological spheres, each with a signed hemisphere decomposition. On the other hand, the equilibria u<sub>t</sub>=0 are characterized by a meander curve, which arises from a shooting approach to the ODE boundary value problem u<sub>xx</sub>+f(x,u,u<sub>x</sub>) =0. We explore a minimax characterization of boundary neighbors, along the meander. Specifically, we identify the precise geometric locations of these boundary neighbors in the signed Thom-Smale complex. This opens an approach to the construction of global attractors with prescribed Thom-Smale complex. Many examples will illustrate this joint work with Carlos Rocha, dedicated to the memory of Geneviève Raugel. See arxiv:1811.04206 and doi: 10.1007/s10884-020-09836-5.
Carlos
Rocha
CAMGSD, Instituto Superior Técnico, Universidade de Lisboa, Portugal
A minimax property for global attractors of scalar parabolic equations
Abstract.
We overview recent results on the geometric and combinatorial characterization of global attractors of semiflows generated by scalar semilinear partial parabolic differential equations under Neumann boundary conditions. In special, we discuss a minimax property of the boundary neighbors of any specific unstable equilibrium. This property allows the identification of the equilibria on the cell boundary of any chosen equilibrium. This is a joint work with B. Fiedler.
January 2021
Tibor
Krisztin
University of Szeged, Hungary
Periodic and connecting orbits for the Mackey-Glass equation
Abstract.
We consider the Mackey-Glass equation x' (t)=-ax(t)+b (x^k(t-1))/(1+x^n (t-1)) with positive parameters a,b,k,n. First, for the discontinuous limiting (n → ∞) case, an orbitally asymptotically stable periodic orbit is constructed for some fixed parameter values a,b,k. Then it is shown that for large values of n and with the same parameters a,b,k, the Mackey-Glass equation also has an orbitally asymptotically stable periodic orbit near to the periodic orbit of the discontinuous equation. The shape of the obtained stable periodic solutions can be complicated. The existence of connecting orbits will be discussed as well.
Michal
Hadrava
Institute of Computer Science of the Czech Academy of Sciences, Prague
A Dynamical Systems Approach to Spectral Music: Modeling the Role of Roughness and Inharmonicity in Perception of Musical Tension
Abstract.
Tension-resolution patterns seem to play a dominant role in shaping our emotional experience of music. Whereas in traditional Western music, these patterns are mainly expressed through harmony and melody, many contemporary musical compositions (e.g. so-called 'spectral music') employ sound materials lacking any perceivable pitch structure, rendering these two compositional devices useless. Motivated by recent advances in music-theoretical and neuroscientific research into the related phenomenon of dissonance, we propose a neurodynamical model of musical tension based on a spectral representation of sound and hence applicable to any kind of sound material, pitched or non-pitched.
Serhiy
Yanchuk
Technical University Berlin
Delay systems and machine learning applications
Abstract.
A single dynamical system with time-delayed feedback (DDE) can emulate networks. This property of delay systems made them extremely useful tools for Machine Learning applications. Here we describe several possible setups. The first setup is the reservoir computing where the DDE plays the role of a high-dimensional reservoir that performs specific computational tasks. We discuss which dynamical properties of such a reservoir are important. These properties include the conditional Lyapunov exponents and the eigenvalue spectrum of the linearized DDE. The second setup is the Deep Neural Network, which can be emulated with a DDE. We present a method for folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops. This single-neuron deep neural network consists of only a single nonlinearity and appropriately adjusted modulations of the feedback signals. The connection weights are determined via a modified back-propagation algorithm that we have developed for such networks.
December 2020
Elisenda
Feliu
University of Copenhagen
Understanding of bistability and Hopf bifurcations in biochemical reaction networks
Abstract.
In the context of (bio)chemical reaction networks, the dynamics of the concentrations of the chemical species over time are often modelled by a system of parameter-dependent ordinary differential equations, which are typically polynomial or described by rational functions. The polynomial structure of the system allows the use of techniques from algebra to study properties of the system around steady states, for arbitrary parameter values. In this talk I will present the formalism of the theory of reaction networks, and how applied algebra plays a role in the study of three main questions: determination of bistability, determination of Hopf bifurcations, and parameter regions for multistationarity. I will present new results tackling these questions by using a ubiquitous and challenging network from cell signaling (the dual futile cycle) as a case example. For this network, which is relatively small, several basic questions, such as the existence of oscillations, the parameter region of multistationarity, and whether multistationarity implies bistability, remain unresolved. The results I will present arise from different joint works involving Conradi, Kaihnsa, Mincheva, Sadeghimanesh, Torres, Yüürük, Wiuf and de Wolff.
Felix
Kemeth
Johns Hopkins University
2-Cluster Fixed-Point Analysis of Mean-Coupled Stuart-Landau Oscillators in the Center Manifold
Abstract.
Clustering in phase oscillator systems with long-range interactions has been subject to theoretical investigations for many years. By mapping the dynamics of mean-coupled Stuart-Landau oscillators onto the center manifold of the Benjamin-Feir instability, we aim to add to the theoretical understanding of clustering in systems beyond phase oscillator ensembles. In particular, we discuss the formation of 2-cluster states in this lower dimensional manifold and outline the resulting implications for the dynamics of the coupled oscillator ensemble. Joint work with Bernold Fiedler, Katharina Krischer and Sindre Haugland.
November 2020
Giulia
Ruzzene
Universitat Pompeu Fabra, Barcelona and IFISC, Palma de Mallorca
Control of chimera states in multilayer networks
Abstract.
Chimera states are one of the most intriguing and studied types of partial synchronization. In small systems, which are the most relevant for experimental situations, chimera states present various instabilities. Therefore, it is natural to investigate methods to control them. We propose a control mechanism based on the idea of a pacemaker oscillator, which allows to control the position of a chimera state within a network and to prevent its collapse to the fully synchronous state. We show how this mechanism developed for ring networks of phase oscillators can be applied to multilayer networks with more complex node dynamics, such as FitzHugh-Nagumo oscillator. In particular, we show that it allows to remotely control a chimera state in one layer via a pacemaker in the other layer.
Angela
Stevens
Universität Münster
Mathematical modeling of regeneration phenomena in biology
Abstract.
Some organisms can regenerate from nearly any kind of severe injury. Regeneration does not function this way in humans. Understanding the underlying mechanisms in model organisms like flatworms is therefore of strong interest. In our mathematical model differences between bulk and tissue surface dynamics play an important role and will be discussed in detail. Joint work with Arnd Scheel.
July 2020
Rico
Berner
TU Berlin
From coherence to incoherence: Stability islands in adaptive neuronal networks
Maria
Barbarossa
Frankfurt Institute of Advanced Studies
Modeling and simulating the early dynamics of COVID-19 in Germany
June 2020
Nicolas
Perkowski
FU Berlin
Infinite regularization by noise
Abstract.
It is a very classical yet still surprising result that noise can have a regularizing effect on differential equations. For example, adding a Brownian motion to an ODE with bounded and measurable vector field leads to a well posed equation with Lipschitz continuous flow. While the equation without noise may have none or many solutions. Classical proofs of such results are based on stochastic analysis and on the close link between the Brownian motion and the heat equation. In that derivation it is not obvious which property of the noise gives the regularization. A more recent approach by Catellier and Gubinelli leads to pathwise conditions under which regularization occurs. I will present a simplified version of their approach and use it to construct (very irregular) paths which are infinitely regularizing: after adding them to an ODE we have a unique solution and an infinitely smooth flow - even if the vector field is only a tempered distribution. This is joint work with Fabian Harang.
Sindre W.
Haugland
TU München
A hierarchy of symmetries: Coexistence patterns of four (and more) Stuart-Landau oscillators with nonlinear global coupling
Abstract.
The symmetry of a system of globally coupled identical units is high: For any solution, interchanging the behavior of any two elements still guarantees a solution. A full N! permutations are possible in total, either not changing the current solution or producing a mirror-image equivalent in phase space. Together, these N! permutations form a symmetry group with many subgroups. Thus, a large number of different partially symmetric solutions might in principle exist. Here, we study N=4 Stuart-Landau oscillators with a particularly intriguing form of global coupling. Its solutions indeed exhibit symmetries corresponding to many of the subgroups of the group of all permutations. Moreover, the transitions between these solutions, together with additional bifurcations more subtly influencing the symmetry, form an intriguing interconnected hierarchy of differentiated oscillator dynamics. We will also take a brief look at how this hierarchy develops if the ensemble size is scaled up.
Peter
Koltai
FU Berlin
Reaction coordinates (order parameters) for metastable systems
Abstract.
We consider non-deterministic dynamical systems showing complex metastable behavior but no local separation in fast and slow coordinates. We raise the question whether and when such systems exhibit a low-dimensional parametrization supporting their effective statistical dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates (or order parameters), such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus less prone to the curse of dimension with respect to the state space of the dynamics than global methods. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as −but not only −molecular dynamics.
May 2020
Eckehard
Schöll
TU Berlin
Partial synchronization patterns in the brain
Bernold
Fiedler
FU Berlin
Reaction, diffusion, and advection in one space dimension - an introduction
Abstract.
Online talk in the Lisbon WADE - Webinar in Analysis and Differential Equations, 17:00 s.t. Berlin time.
Kostya
Blyuss
University of Sussex, England
Mathematical models of epidemics: Insight for control of COVID-19
Carlos
Rocha
Instituto Superior Tecnico, Lisboa, Portugal
Meanders, zero numbers and the cell structure of Sturm global attractors
Abstract.
We overview recent results on the geometric and combinatorial characterization of global attractors of semiflows generated by scalar semilinear partial parabolic differential equations under Neumann boundary conditions. This is a joint work with B. Fiedler.
April 2020
Markus
Kantner
WIAS Berlin
Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions
January 2020
Martin
Brokate
TU München
Sensitivity results for rate independent evolutions
Abstract.
Rate independent evolutions are evolutions whose solution operators commute with time transformations. They are inherently nonlinear and nonsmooth. We present some introduction, outlining various mathematical approaches as well as relations to other fields. We then address sensitivity, in particular the question whether the solution operators possess weak differentiability properties.
Frits
Veerman
University of Heidelberg
Toll roads and freeways: Defects in bilayer interfaces in the multi-component functionalised Cahn-Hilliard equation
Abstract.
We study a multi-component extension of the functionalised Cahn-Hilliard equation, which provides a framework for the formation of patterns in fluid systems with multiple amphiphilic molecules. The assumption of a length scale dichotomy between two amphiphilic molecules allows the application of geometric techniques for the analysis of patterns in singularly perturbed reaction-diffusion systems. For a generic two-component system, we show that solutions to the four-dimensional connection problem provide the leading order approximation for solutions to the full eight-dimensional barrier problem, which can be obtained through a perturbative expansion in the layer width. Moreover, we show that a saddle-node bifurcation of bilayer solutions in the four-dimensional connection problem acts as a source of so-called defect solutions, i.e. solutions to the barrier problem that are not also solutions to the connection problem. The analysis combines geometric singular perturbation theory with centre manifold theory in an infinite-dimensional context.
November 2019
Jia-Yuan
Dai
National Center for Theoretical Sciences
Hyperbolicity of Ginzburg-Landau vortex solutions
Abstract.
We prove that each equilibrium of the Ginzburg-Landau equation restricted on the invariant subspace of vortex solutions is hyperbolic, that is, its associated linearization possesses nonzero eigenvalues. This result completely describes the global attractor of vortex solutions, and also yields the Ginzburg-Landau spiral waves of nodal type. This result is a joint work with Dr. Lappicy.
Ralf
Toenjes
University of Potsdam
The Constructive Role of Noise in The Dynamics on Network Hubs for Network Synchronization
Abstract.
We describe and analyze a coherence resonance phenomenon for synchronization in bipartite networks of well connected hubs and followers when the hubs are subjected to noise. Using the Ott-Antonsen ansatz for globally coupled phase oscillators the dynamics of the mean fields is described by a low-dimensional system of Langevin equations. Averaging over the fast stochastic dynamics of the hubs yields ordinary differential equations which predict the coherence resonance reasonably well.
Nicolas
Perkowski
From hopping particles to stochastic PDEs
Abstract.
I will try to give an overview of some of my research interests, focusing on certain a priori ill posed stochastic PDEs and their derivation. An important task in stochastics is to find and construct 'universal' models that describe a given phenomenon. For example, any random variable that is given by the superposition of many small independent influences is approximately Gaussian, independently of the concrete nature of the small influences, and therefore we call the Gaussian distribution universal. When trying to derive universal models for phenomena that evolve in space and time, formal calculations often suggest that we should consider nonlinear stochastic PDEs driven by space-time white noise. This is a problem because due to the irregularity of the noise the solution might be too irregular to make sense of the nonlinearities in the equation. But in recent years we found new ways of overcoming these problems, making sense of the equations, and proving their universality in some cases.
Phillipo
Lappicy
Universidade de Sao Paulo, Freie Universität Berlin
Horava-Lifshitz Gravity: Bifurcations and Chaos
Abstract.
Lorentzian causal structure, general covariance, and scale invariance are first principles that play a key role in the nature of generic spacelike singularities in general relativity. To bring a new perspective on the contributions of these first principles regarding the chaotic aspects of generic singularities, we consider the initial singularity in spatially homogeneous Bianchi type VIII and IX models in Horava-Lifshitz gravity, which replaces relativistic first principles with anisotropic scalings of Lifshitz type. To describe the nature of the initial singularity in these models, we make use of mathematical tools that include symbolic dynamics and chaos. For the present class of models, it is shown that general relativity is a critical case that corresponds to a bifurcation where chaos becomes generic. For different models nearby the general relativistic critical case, Cantor sets and iterate function systems are important for describing the chaotic aspects of generic singularities. This is joint work with Juliette Hell and Claes Uggla.
October 2019
Pavel
Gurevich
Freie Universität Berlin
and
Hannes
Stuke
Freie Universität Berlin
Reinforcement learning
July 2019
Toshiyuki
Ogawa
Meiji University, Tokyo
Drift bifurcation of traveling waves in a reaction-diffusion system with 3 competing species
June 2019
Stefan
Ruschel
TU Berlin
Pulsing dynamics of a laser system with delayed feedback
Abstract.
We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber in the presence of delayed optical self-feedback. We present some recent results regarding the stability and bifurcations of periodic light pulses that are sustained in the external cavity feedback loop, so called external cavity pulse trains.
Matthias
Wolfrum
WIAS Berlin
Temporal dissipative solitons in systems with time delay
May 2019
Sebastian
Eydam
WIAS Berlin
Mode-locked solutions and coherence echoes in systems of globally-coupled phase oscillators
Abstract.
In systems of Kuramoto type globally-coupled phase oscillators with equidistant natural frequencies, one can observe an interesting collective phenomenon below the synchronization threshold. The collective behavior is characterized by sharp pulses in the mean-field amplitude and therefore appropriately called mode locked. We discuss the emergence of this particular type of solution as well as the typical bifurcation scenarios that are found along their stability boundaries. In large ensembles, where the natural frequencies follow from a multi-modal distribution, mode-locked solutions are observed and the breakdown of the pulsation due to the broadening of the frequency width is explored. Echo phenomena have a long history and are found in various systems with one of the most prominent examples being the appearance of spin-echoes. We consider an echo-type response phenomenon that occurs after multiple external stimuli in ensembles of phase oscillators below the synchronization threshold. An intuitive explanation of the coherence echo phenomenon in the language of mode locking is given and it is shown that the nonlinear global interaction that supports mode-locked solutions can also enhance the echo response significantly.
April 2019
Irina
Kmit
Humboldt-Universität zu Berlin
Classical bounded and almost periodic solutions to quasilinear first-order hyperbolic systems
Abstract.
We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the L2-generalized solutions to the initial-boundary value problems become eventually C2-smooth for any initial L2-data. We investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on R. We prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system. Assuming that the coefficients of the hyperbolic system are almost periodic, we prove that the bounded solution is almost periodic also.
Oleh
Omel'chenko
Universität Potsdam
Traveling chimera states
Abstract.
We overview recent results about the traveling chimera states observed in a ring of nonlocally coupled phase oscillators with broken reflection symmetry of the coupling kernel. These states manifest themselves as coherence-incoherence patterns moving along the ring. As the coupling asymmetry grows they undergo a sequence of transformations, which can be adequately explained using the continuum limit Ott-Antonsen equation. In the context of this equation the chimera states correspond to its smooth traveling wave solutions. Using the mathematical methods from the PDE and integral equations theory we carry out asymptotic analysis of these traveling waves, describe an algorithm for their numerical continuation and explore the spectrum of the corresponding linearized equation. We show that traveling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated traveling chimera states.
February 2019
Alejandro
Kocsard
Universidade Federal Fluminense
Random walks, synchronization and existence of invariant measures II
Abstract.
In this second talk we will review some results we discussed in the first one, but from a 'dynamical systems' point of view. Finally we shall discuss some applications of these results to the so called Burnside problem on manifolds.
January 2019
Alejandro
Kocsard
Universidade Federal Fluminense
Random walks, synchronization and existence of invariant measures
Abstract.
In this talk we start discussing some classical results due to Furstenberg [Fur63] about random product of matrices and how a certain form of synchronization naturally appears in this context. Then we shall consider some extensions of these results for random walks on compact smooth manifolds [Led84, Ant84, Cra90]. Finally we will see how these ideas can be used to study the existence of invariant (probability) measures for smooth actions on manifolds by some groups which are a priori non-amenable.
Edgard
Pimentel
Pontifical Catholic University of Rio de Janeiro
Geometric methods in regularity theory for nonlinear PDEs
Abstract.
In this talk we examine the regularity theory of the solutions to a few examples of nonlinear PDEs. Arguing through a genuinely geometrical method, we produce regularity results in Sobolev and Hölder spaces, including some borderline cases. Our techniques relate a problem of interest to a further one - for which a richer theory is available - by means of a geometric structure, e.g., a path. Ideally, information is transported along such a path, giving access to finer properties of the original equation. In the first part of the talk, we cover examples including elliptic and parabolic fully nonlinear problems, the Isaacs equation and a double divergence model. Then we proceed to the setting of state-dependent degenerate problems and report recent (optimal) results. We close the talk with a discussion on open problems and further directions of work.
Willie
Hsia
National Taiwan University
On the mathematical analysis of the synchronization theory with time-delayed effect
Abstract.
In this lecture, we will introduce a newly developed mathematical theory on the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase synchronization and frequency synchronization are in view. This is joint work with Bongsuk Kwon, Chang-Yeol Jung and Yoshihiro Ueda.
November 2018
Hannes
Stuke
FU Berlin
A dynamical systems approach to outlier robust machine learning
Abstract.
We consider a typical problem of machine learning - the reconstruction of probability distributions of observed data. We introduce the so-called gradient conjugate prior (GCP) update and study the induced dynamical system. We will explain the dynamics of the parameters and show how one can use insights from the dynamical behavior to recover the ground truth distribution in a way that is more robust against outliers. The developed approach also carries over to neural networks.
Lutz
Schimansky-Geier
Humboldt-University Berlin
Celestial mechanics of fruit flies or a theory for mushroomers
Abstract.
The topic of global search in complex environments have been often investigated. But a search can also be local in the sense that it is centered at a given home position. In the latter case, the searcher does not only look for a new target but is also required to regularly return to the home position. Such behavior is typical for many insects and achieves technical importance for self-navigating robotic systems. We propose a stochastic nonlinear model for local search which does not distinguish between the two aims. The dynamics bases on an unique pursuit and escape behavior of the heading from the position vector realizing thereby optimal exploration of space and the return to the home. We discuss the mechanics of the searcher and inspect the role of noise. Such randomness is present in the decision making rule of selecting the new heading direction. We consider Levy noises with different degree of discontinuity and report about steady spatial densities for the searchers. Also we report about an optimal noise intensity that a searcher finds a target at nearby places. For this noise value the required time for finding the target becomes minimal which appears to be the consequence of different relaxation processes in the spatial and the angular dynamics. Further extensions of the model are discussed during the lecture.
October 2018
Jan
Sieber
University of Exeter
A nonsmooth saddle-node in a non-autonomous dynamical system
July 2018
Anne
Shiu
Harnessing steady-state parametrizations arising from reaction networks
Abstract.
Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. This talk focuses on certain biological signaling networks, namely, multisite phosphorylation networks. Many of these systems exhibit 'toric steady states' (that is, the ODEs generate a binomial ideal), and more generally the set of steady states admits a rational parametrization, that is, the set is the image of a map with rational-function coordinates. We describe how this parametrization allows us to investigate the dynamics of two multisite phosphorylation networks: the emergence of bistability in a network underlying ERK regulation, and the capacity for oscillations in a mixed processive/distributive phosphorylation network. This is joint work with Carsten Conradi and Maya Mincheva.
Nils
Berglund
Institut Denis Poisson, Universite d'Orleans
Metastable dynamics of Allen-Cahn SPDEs on the torus
June 2018
Jens
Starke
Universität Rostock
Multiscale analysis of collective behaviour in particle systems
May 2018
Murad
Banaji
Middlesex University, London
Inheritance of behaviours in bio/chemical networks
Abstract.
Consider the following imprecise questions about dynamical systems with some network structure: If a small network is embedded in a larger one, then must the larger one 'inherit' some behaviours of the smaller one? Can the presence of certain 'motifs' be sensibly used to make predictions about dynamical behaviours of a network? Once such questions are phrased precisely, the answers turn out to depend on the class of networks studied, the dynamical behaviour of interest and, crucially, the chosen notion of 'subnetwork'. My talk will focus on some recent results on chemical reaction networks (CRNs), where the behaviours in question are multistationarity and oscillation. Network modifications such as adding or deleting reactions, adding or deleting species from reactions, and inserting intermediates into a reaction are considered for their effects on multistationarity and oscillation. For example, it can be shown that both behaviours are inherited in the induced subnetwork partial ordering on fully open CRNs, but the same does not hold for general CRNs. Under certain mild assumptions, growing a CRN by inserting intermediates into reactions preserves the capacity for multistationarity in a CRN. The theorems are proved using essentially local techniques -- the implicit function theorem and regular and singular perturbation theory -- and hold for various kinetics including (but not exclusively) mass action kinetics. The results on multisationarity are joint with Casian Pantea.
Felix
von Oppen
Free University Berlin
Elements of a topological quantum computer
Abstract.
This talk will discuss elements of a topological quantum computer with an emphasis on introducing two paradigmatic models, namely the Kitaev chain and the toric (or surface) code.
Alexander
Linke
WIAS Berlin
On the role of the Helmholtz-Leray projector for a novel pressure-robust discretization theory for the incompressible Navier-Stokes equations
Abstract.
The talk discusses several physical regimes of the incompressible Navier--Stokes equations with respect to the role of the pressure and the role of the Helmholtz--Leray projector in the Navier--Stokes momentum balance. It is emphasized that not the forces in the momentum balance themselves matter for the complexity of the flow field, but their Helmholtz--Leray projector. Since the Helmholtz--Leray projector vanishes for arbitrary gradient fields, a semi norm and a corresponding equivalence class of forces play naturally a major role for the evolution of incompressible flows. Novel pressure-robust mixed finite element methods are designed for an appropriate discrete treatment of this equivalence class of forces. On the contrary, classical, (only) inf-sup stable mixed methods do not care about the existence of such an equivalence class. In order to deliver accurate simulation results, they have to resort to expensive high order ansatz spaces. Finally, the talk will indicate suitable applications for efficient and accurate low-order pressure-robust mixed methods like flows with electrolytes and flows with stagnation points, and it will discuss possible extensions to compressible low-Mach number flows.
Marc
Timme
Technische Universität Dresden
Network dynamics as an inverse problem
April 2018
Misha
Zaks
Universität Potsdam
Collective dynamics of globally coupled excitable units
February 2018
Marek
Fila
Comenius University, Bratislava
A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation
Abstract.
We discuss a Gagliardo-Nirenberg-type inequality for functions with fast decay. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense. The talk will consist of two parts - an introduction of one hour and a presentation of proofs of also one hour.
January 2018
Jia-Yuan
Dai
Freie Universität Berlin
Existence of local solutions of Gowdy spacetimes
Abstract.
We consider a class of Gowdy spacetimes that reduces the Einstein's field equation to a system of two semilinear wave equations, by assuming a universe without matter, in which the gravitational wave fronts repeat in space and are mutually parallel. To prove the existence of local solutions of the system, we add a linear perturbation and seek periodic-in-space solutions. The idea of the proof is to apply the Lyapunov-Schmidt reduction. We will solve a related small-divisor problem and discuss how to design the correct functional setting that fits to the nonlinearity. This ongoing research is a joint work with Dr. Hannes Stuke.
November 2017
Nikita
Begun
Saint Petersburg State University
Chaos for the saw map
Abstract.
We consider dynamics of a scalar piecewise linear "saw map" with infinitely many linear segments. In particular, such a map occurs as the Poincare map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as systems with a feedback loop containing the hysteretic stop operator. We analyze chaotic sets and attractors of the ''saw map'' depending on its parameters.
October 2017
Carlos
Rocha
Instituto Superior Técnico of Lisbon
Global Attractors for Non-autonomous Systems
Abstract.
We survey the notions of global attractors for non-autonomous systems, and consider small non-autonomous perturbations of dynamical systems to discuss the resulting changes on the global attractors. We review the notion of Morse-Smale dynamical system and extend this notion to the non-autonomous framework, based on a recent joint work with R. Czaja and W. Oliva.
Jan
Sieber
University of Exeter
Smooth center manifolds for delay-differential equations
Abstract.
Delay-differential equations (DDEs) with state-dependent delays are, as far as is known, at best continuously differentiable once as dynamical systems. That is, the time-t map does not depend on its argument with a higher degree of smoothness than 1. However, as I will show, center manifolds near equilibria are still as smooth as expected from the spectral gap and from the smoothness of coefficients. In particular, I will review what precisely "smoothness of coefficients" means.
July 2017
Jia-Yuan
Dai
Freie Universität Berlin
Spiral Waves in Circular and Spherical Geometries. The Ginzburg-Landau Paradigm
Abstract.
We prove the existence of spiral waves for the complex Ginzburg-Landau equation in the circular and spherical geometries. Instead of applying the shooting method from the literature, we establish a functional approach and adopt global bifurcation analysis to solve the spiral wave elliptic equation. Moreover, we prove the existence of two new patterns: frozen spirals in the circular and spherical geometries and spiral-pairs, that is, spirals with two tips, in the spherical geometry.
Augusto
Visintin
Università di Trento
Compactness and structural stability of nonlinear flows
Abstract
June 2017
Igor
Franović
Institute of Physics Belgrade, University of Belgrade
Bistability, rate oscillations and slow rate fluctuations in networks of noisy neurons with coupling delay
Abstract.
Spontaneous activity of cortical neurons is typically characterized as a doubly-stochastic process, underlying two distinct forms of variability. While the local spike-train variability is reflected on the fast timescale, the variability associated with much longer timescales involves macroscopic irregular fluctuations of the firing rate. The latter fluctuations apparently emerge by coherent switching of neurons between the “up” and “down” states of membrane potential, and are believed to play important functional roles. In order to gain qualitative insight into the mechanisms behind such switching phenomena, we consider a random network of rate-based neurons influenced by external and internal noise, as well as the coupling delay. The network behavior is analyzed by deriving the second-order stochastic mean-field model, which describes the network dynamics in terms of the mean-rate and the associated variance. The mean-field model is used to study the stability and bifurcations in the thermodynamic limit, as well as the fluctuations due to the finite-size effect. For the thermodynamic limit, it is established that (i) the network may exhibit coexistence between two stationary levels in a wide range of parameters, whereby the two types of noise affect the levels in a fundamentally different fashion, and (ii) coupling delay may give rise to oscillations of the mean-rate. The slow rate fluctuations are demonstrated to emerge via two distinct scenarios. In the delay-free case, the leading mechanism can be seen as noise-induced transitions between two metastable states, quite reminiscent to fluctuations of a particle in a double-well potential. In the second scenario, which involves the cooperative action of noise and delay, the fluctuations can be interpreted as stochastic mixing between two different oscillatory regimes.
May 2017
Phillipo
Lappicy
Freie Universität Berlin
Einstein constraints: a dynamical approach.
Abstract.
The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar solutions of Schwarzschild type, which describe the space of initial data of certain black holes, for example. The first main result gives a detailed study of the axially symmetric solutions, since the domain is now one dimensional and nodal properties can be used to describe certain asymptotics of the rescaled self-similar solutions. Such asymptotics describe the possible metrics arising at an event horizon of a black hole, depending on the metric inside the horizon. Those are described by Sturm attractors. In particular, we compute an example for a prescribed scalar curvature. The second main result states a symmetrization property of certain metrics in the event horizon, namely, how the symmetry of the spherical domain can influence the symmetry of solutions.
Pavel
Gurevich
Freie Universität Berlin
A short introduction to machine learning: towards (un)certainty quantification.
Abstract.
Machine learning is a rapidly developing field that deals with searching for and generating patterns in data. It is nowadays a very broad field encompassing many tasks and methods. In my talk, I will give a short overview, rapidly narrowing my focus towards a particular topic, namely, neural networks and their ability not only to find patterns but also quantify how certain they are in doing so. If time permits, I will conclude with a discussion of our joint work with Hannes Stuke on certainty quantification.
Hannes
Stuke
Freie Universität Berlin
Parabolic blow-up in complex time.
Abstract.
In my talk I will give an overview of my research on parabolic differential equations and blow-up. Blow-up denotes the phenomenon, when solutions diverge to infinity. By considering complex time we try to establish a connection between two opposed worlds: on the one hand bounded solutions, in particular equilibria and heteroclinic orbits and on the other hand blow-up in equations with entire nonlinearities.
April 2017
Juliana
Pimentel
UFABC, Brazil
Permutation characterization for slowly non-dissipative equations.
Abstract.
We study the longtime behavior of slowly non-dissipative scalar reaction-diffusion equations. By extending known results, we are able to obtain a complete decomposition of the non-compact global attractor and still manage to determine the heteroclinic connections based on the Sturm permutation method.
February 2017
Alessandro
Torcini
Université de Cergy-Pontoise, Laboratoire de Physique Théorique et Modélisation
Death and rebirth of neural activity in sparse inhibitory networks
Abstract.
In this presentation, we clarify the mechanisms underlying a general phenomenon present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of the neural activity, as expected, but it can also promote neural reactivation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neurons' death). However, the random pruning of the connections is able to reverse the action of inhibition, i.e. in a sparse network a sufficiently strong synaptic strength can surprisingly promote, rather than depress, the activity of the neurons (neurons' rebirth). Thus the number of firing neurons reveals a minimum at some intermediate synaptic strength. We show that this minimum signals a transition from a regime dominated by the neurons with higher firing activity to a phase where all neurons are effectively sub-threshold and their irregular firing is driven by current fluctuations. We explain the origin of the transition by deriving an analytic mean field formulation of the problem able to provide the fraction of active neurons as well as the first two moments of their firing statistics. The introduction of a synaptic time scale does not modify the main aspects of the reported phenomenon. However, for sufficiently slow synapses the transition becomes dramatic, the system passes from a perfectly regular evolution to an irregular bursting dynamics. In this latter regime the model provides predictions consistent with experimental findings for a specific class of neurons, namely the medium spiny neurons in the striatum.
Shalva
Amiranashvili
WIAS Berlin
Scattering of waves at solitons
January 2017
Matthias
Wolfrum
WIAS Berlin
Interfaces in a chain of coupled bistable elements
Sergey
Tikhomirov
Saint-Petersburg State University, Russia
Depinning bifurcation in quasiperiodic media
Christian
Bick
Somerville College, Oxford, UK/University of Exeter, UK
From Weak Chimeras to Switching Dynamics of Localized Frequency Synchronization Patterns
Abstract.
We review some recent results on weak chimeras, a mathematically rigorous description for chimera states-solutions with coexisting synchronization and incoherence-in networks of identical oscillators. Moreover, we give some preliminary results on how to construct networks of identical phase oscillators with heteroclinic connections between weak chimeras of saddle type. These networks exhibit dynamic switching of localized synchronization which could for example encode information in neural networks.
December 2016
Jan
Sieber
University of Exeter, UK
Convergence of equation-free methods in the case of finite time scale separation
Abstract.
A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods originally proposed by IG Kevrekidis): first, a lifting operator maps a set of low dimensional coordinates into the high-dimensional space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for sufficiently large healing time (which is a method parameter). More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. We demonstrate for an example from Barkley et al (SIAM J. Appl. Dyn. Sys. 5(3) 2006) that the ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic differential equations, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically small. In the example, the ratio between the decay rates of fast and slow variables is 2.
November 2016
Tobias
Jäger
Friedrich-Schiller-Universität Jena
Mode-locking phenomena in low-dimensional dynamics
Jens
Griepentrog
Humboldt Universität Berlin
Solution regularity of parabolic variational inequalities
Abstract.
You are kindly invited to attend the special event on the occasion of the 65th birthday of Lutz Recke on Monday, November 21st, at 15:15 within the Research Seminar 'Applied Analysis' of the Institute of Mathematics of the Humboldt University. You are kindly invited to the seminar as well as to cake, coffee and wine afterwards. Guests are always welcome!
Martin
Väth
Freie Universität Berlin
Abstract Volterra Equations
Abstract.
A definition of abstract Volterra operators/equations will be proposed which contains renewal and delay type equations. The relation between these problems will be illustrated and as a sample application a result about the essential spectral radius for linear abstract Volterra operators will be presented. Similar techniques apply also for the nonlinear case.
Sebastian
van Strien
Imperial College London, UK
Dynamics on heterogenous networks
Abstract.
Networks in which some nodes are highly connected and others have low connectivity are ubiquitous (they are used to model the brain, the internet, cities etc). In this talk I will consider coupled dynamics on networks of this type. It turns out that weakly connected and highly connected nodes in this network often develop rather different kinds of dynamics, and that one can predict the behaviour for a time window which grows exponentially with the size of the network. These results are proved using ergodic theory and invariant manifolds. This work is joint with Matteo Tanzi and Tiago Pereira.
Bernhard
Brehm
Freie Universität Berlin
Particle Horizons in the Mixmaster Universe
Abstract.
The Mixmaster Universe has been proposed by Misner (1969) as a model for a chaotic big bang cosmological singularity. This cosmological model describes Bianchi IX spatially homogeneous, anisotropic vacuum space-times. In 1970, Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that particle horizons form towards the big bang. In other words, backwards light-cones remain spatially bounded, and spatially separate regions causally decouple. We prove this BKL conjecture, for almost every solution. More specifically, the answer to this question depends on the convergence speed towards the Mixmaster attractor. Ringström (2001) showed that this convergence occurs at all. We introduce a novel expanding measure in order to prove that the convergence is fast enough to guarantee the formation of particle horizons for Lebesgue almost every solution. The talk is addressed at a nonspecialist audience.
June 2016
Mike
Field
Marie Curie Fellow, Imperial College; Research Professor, Rice University
Functional Asynchronous Networks: Factorization of Dynamics and Function
Abstract.
We describe the theory of functional asynchronous networks and one of the main results, the Modularization of Dynamics Theorem, which for a large class of functional asynchronous networks gives a description of dynamics and function in terms of properties of constituent subnetworks. For these networks we can give a complete description of the network function in terms of the function of the events comprising the network and thereby answer a question originally raised by Alon in the context of biological networks: “Ideally, we would like to understand the dynamics of the entire network based on the dynamics of the individual building blocks.” (page 27, An Introduction to Systems Biology, Uri Alon.)
Yuri
Maistrenko
TU Berlin & National Academy of Sciences of Ukraine
Solitary states in coupled oscillators
Abstract.
Ensembles of identical oscillators can display remarkable spatial or spatiotemporal patterns, so-called solitary states, in which one or a few oscillators split off and behave differently than the others synchronized. At further variations of a control parameter, more and more oscillators leave the coherent cluster manifesting eventually the phenomenon of spatial chaos. Due to these peculiar properties, solitary states can signal to a new scenario for coherence-incoherence transition different from the chimera states. In the talk, the solitary state appearance is reported for Kuramoto model with attractive and repulsive interactions [1], for non-locally coupled Kuramoto model with inertia [2-3] and Hansel-Mato- Maunier model, as well as for a nonlinear delayed-feedback system ([4]).
Alan
Rendall
Universität Mainz
Dynamical systems arising from the Calvin cycle
Abstract.
I will talk about various aspects of the dynamics of ODE models of the Calvin cycle of photosynthesis, based on work done together with Juan Velazquez, Dorothea Möhring and Stefan Disselnkötter. Issues discussed include boundedness, persistence (whether concentrations can tend to zero asymptotically) and existence and stability of steady states. The modelling of this biological system is presented both because of its intrinsic interest and because of the more general insights it provides for the understanding of biochemical processes.
May 2016
Björn
de Rijk
Universiteit Leiden, The Nederlands
Factorization of the Evans function via the Riccati transformation
Abstract.
In the spectral stability analysis of pattern solutions, the presence of a small parameter can reduce the complexity of the linear stability problem. The spectrum of the linearization about certain type of patterns is given by the zero set of an analytic function, the so-called Evans function. Our reduction method yields a factorization of the Evans function in accordance with the scale separation induced by the small parameter. For some specific equations this product structure has yet been established by geometric arguments. Our analytic method formalizes and generalizes the factorization procedure. The main tool for the reduction is the Riccati transformation. We employ our techniques to study the stability of stationary, spatially periodic pulse patterns to a general class of singularly perturbed reaction-diffusion systems. Our Evans-function analysis complemented with a careful analysis of the spectral curve attached to the origin leads to explicit conditions for nonlinear diffusive stability.
Martin
Väth
Free University Berlin
A Primer on Nonlinear Scalar Conservation Laws
Abstract.
The talk is an introduction and survey on basic properties of scalar conservation laws with emphasis on Burgers' equation. It covers an explanation for the formation of shocks, the introduction of weak solutions and the associated “jump condition”, a discussion of rarefactions and how to overcome them with an entropy condition.
January 2016
Fredi
Tröltzsch
Technische Universität Berlin, Germany
Optimization of nonlocal distributed feedback controllers with time delay for the Schlögl model
Abstract.
A class of Pyragas type nonlocal feedback controllers is investigated for the 1D Schlögl model, a semilinear parabolic equation that is also known as Nagumo equation. The main goal is to find an optimal kernel in the controller such that the solution of the controlled equation is close to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optimal control problem with the kernel taken as control function. The well-posedness of the problem and necessary optimality conditions are discussed. Special emphasis is laid on time-periodic target functions. A particular issue is the periodic behavior of solutions to the feedback equation. This is joint work with P. Nestler and E. Schöll.
Hans-Otto
Walther
Justus Liebig University Giessen, Germany
Differential Equations, Delays, and Frechet Manifolds
Abstract.
For differential equations with bounded state-dependent delay the initial value problem is well-posed on a submanifold in a Banach space of maps on a compact interval, with all solution operators differentiable, under mild smoothness hypotheses on the functional defining the differential equation. In the general case, with the delay not necessarily bounded, a similar result holds provided the delay is locally bounded. Such results, however, miss solutions whose histories do not belong to the ambient Banach space of maps on the negative halfline. This suggests to study the initial value problem for data in the Frechet space of all continuously differentiable maps on the negative halfline. The lecture presents a result which provides a continuous semiflow on a submanifold of the Frechet space whose solution operators are continuously differentiable in an appropriate sense.
Edgar
Knobloch
University of California at Berkeley, CA/US
Spatially localized structures in dissipative systems
Abstract.
Spatially localized structures arise frequently in nature. In this lecture I will describe a number of examples from different physical systems, followed by a discussion of the basic ideas behind the phenomenon of nonlinear self-localization that is responsible for their existence. I will illustrate these ideas using a simple phenomenological model and explain why the qualitative predictions of this model help us understand the properties of much more complicated systems exhibiting spatial localization.
December 2015
Helge
Dietert
University of Cambridge, UK
Stability and bifurcation for the Kuramoto model
Abstract.
The Kuramoto model is a prototype for synchronisation behaviour of heterogeneous oscillators due to a global coupling. In this model the totally unsynchronised state appears to be stable in simulations with a large number of oscillators. In order to understand this stability, I will first recall the mean-field (thermodynamic) limit, which recasts the problem to a PDE on the density of the oscillators. By a careful study of the PDE in Fourier space, we can then understand the apparent stability through the heterogeneity of the oscillators. With this we can show perturbative and global stability results and derive a center manifold reduction to determine the bifurcation behaviour. The mathematical structure is similar to the Vlasov system, where the stability is called Landau damping.
November 2015
Serhiy
Yanchuk
WIAS, Germany
Dynamic jittering and spiking solutions in oscillators with pulsatile delayed feedback
Abstract.
Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with nonequal interspike intervals emerge. We show that the number of the emerging, so-called “jittering” regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the “multijitter” bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.
Michael
Herrmann
WWU Münster, Germany
Hysteresis in discrete forward-backward diffusion equations
Abstract.
We study the dynamics of phase interfaces in discrete diffusion equations with bistable nonlinearity. In the first part we identify a hysteretic free boundary problem for the parabolic scaling limit by combining heuristic arguments with numerical evidence. Afterwards we discuss the rigoros justification for the bilinear and the trilinear case. Joint work with Michael Helmers (University of Bonn).
Ivan
Ovsyannikov
University of Bremen, Germany
Criteria of birth of Lorenz attractors
Abstract.
The birth of Lorenz attractors is proved for a system which we call an extended Lorenz model. It appears as a normal form for some class of codimension-three (and codimension-four) bifurcations. Thus, the proof of the birth of the Lorenz attractor in this model will immediately prove the birth of such attractor in these bifurcations. With a transformation of coordinates and parameters the extended Lorenz model can be also transformed to the original Lorenz equations (for some parameter values it may require time-reversal). Thus, the proof of an attractor in the extended model gives immediately a Lorenz attractor or repeller in the Lorenz system. This is a joint work with D. Turaev, http://arxiv.org/abs/1508.07565/
July 2015
Georgi
Medvedev
Drexel University, Philadelphia, PA/US
Coupled Dynamical Systems on Large Graphs and Graph Limits
Abstract.
The continuum limit is an approximate procedure, by which coupled dynamical systems on large graphs are replaced by an evolution integral equation on a continuous spatial domain. This approach has been instrumental for studying dynamics of diverse networks throughout physics and biology. We use the combination of ideas and results from the theories of graph limits and nonlinear evolution equations to develop a rigorous justification for using the continuum limit in a variety of dynamical models on deterministic, random, and quasirandom graphs. As an application, we discuss stability of spatial patterns in the Kuramoto model on certain Caley and random graphs.
January 2015
Abderrahim
Azouani
Mohamed Premier University, National School of Applied Sciences, Al Hoceima, Morocco
Feedback control of nonlinear dissipative dynamical systems using general interpolant observables and continuous data assimilation
Abstract.
In this work we propose a new feedback control for controlling general dissipative evolution equations using any of the determining systems of parameters (modes, nodes, volume elements, etc...) without requiring the presence of separation in spatial scales, i.e. without assuming the existence of an inertial manifold. For more reaching applications, we present a continuous data assimilation algorithm based on our feedback control ideas in the context of the incompressible two-dimensional Navier-Stokes equations.
Sergey
Tikhomirov
MPI for Mathematics in the Science Leipzig, Germany
Shadowing and random walks
Abstract.
We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional analog of Hammel-Yorke-Grebogi's conjecture concerning the interval of shadowability for a typical pseudotrajectory is not correct. The main technique is reduction of the shadowing problem to the ruin problem for a simple random walk.
Oleksandr
Burylko
National Academy of Sciences of Ukraine
Weak chimeras in minimal networks of coupled phase oscillators
Abstract.
We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a 'weak chimera' as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks.
December 2014
Claes
Uggla
Karlstad University, Schweden
Scalar field cosmology and dynamical systems
Abstract.
In this talk I’ll discuss Einstein’s field equations and how Killing and homothetic Killing vectors in cosmology in combination with physical first principles, such as general covariance and scale invariance, induce a hierarchical solution space structure, where simpler models act as building blocks for more complicated ones (note that similar considerations are equally applicable when it comes to modified gravity theories). To illustrate the consequences of these quite general aspects, I will consider several examples that will furthermore shed light on a variety of heuristic concepts such as attractor and tracker solutions. I’ll focus on flat FLRW cosmology with a source that consists of: a scalar field representation of a modified Chaplygin gas; monomial scalar fields and perfect fluids; inverse power-law scalar fields and perfect fluids. I’ll show how physical principles can be used to obtain regular dynamical systems on compact state spaces with a hierarchy of invariant subsets and how local and global dynamical systems techniques then subsequently make it possible to obtain a global understanding of the associated solution spaces and their properties.
Hayato
Chiba
Kyushu University, Japan
A spectral theory of linear operators on a Gelfand triplet and its application to dynamics of coupled oscillators
Abstract.
The Kuramoto model is a system of ODEs (coupled oscillators) to describe synchronization phenomena. In this talk, an infinite dimensional Kuramoto model is considered, and Kuramoto's conjecture on a bifurcation diagram of the system will be proved. A linear operator obtained from the linearization of the Kuramoto model has the continuous spectrum on the imaginary axis, so that the dynamics of the system cannot be revealed via the usual spectrum theory. To handle such continuous spectra, a new spectral theory of linear operators based on Gelfand triplets is developed. In particular, a generalized eigenvalue is defined. It is proved that a generalized eigenvalue determines the stability and bifurcation of solutions.
October 2014
Elena
Tobisch
Johannes Kepler University Linz, Austria
Nonlinear resonances in bounded domains
Abstract.
Evolutionary dispersive nonlinear PDEs with periodic boundary conditions are frequently met among the famous equations of mathematical physics, e.g. Boussinesq eq., Hasegawa-Mima eq., Kadomtsev-Petviashvili eq., Schroedinger eq. and others. We regard a PDE of this class in two space dimensions and with small nonlinearity. This allows reducing the study of entire Fourier space of solutions to the study of resonantly interacting Fourier modes. As resonance condition in this case is equivalent to a Diophantine equation of 6 to 8 variables in high powers, according to the 10th Hilbert problem there exists no general algorithm for solving it. However, the use of a special form of the Diophantine equations appearing in physics in this context allows solving them by the method of q-class decomposition, [1]. Once the solutions of resonance conditions are found, their structure can be studied; its most interesting property turned out to be the decomposition of the Fourier space into non-intersecting subspaces with independent time evolution. Application of these results to the problem of fluid mechanics is briefly described (water waves and atmospheric planetary waves). References: [1]. Kartashova E. Nonlinear resonance analysis: Theory, computation, applications (Cambridge University Press, 2010).
July 2014
Hiroshi
Matano
University of Tokyo
Propagating terrace for semilinear diffusion equations
Abstract.
In this talk I will discuss the behavior of spreading fronts in semilinear diffusion equations on the entire space. Here, by a "spreading front", I mean the expanding level sets of a solution that starts from compactly supported (or rapidly decaying) nonnegative initial data. If the nonlinearity is multi-stable, the dynamics of a solution can no longer be described by a single front, but by what we call a "propagating terrace", which roughly means a layer of several fronts that expand to infinity at different speeds. I will first give a brief review of my earlier result for the one-dimensional case (joint work with Thomas Giletti and Arnaud Ducrot), in which the term "propagating terrace" was first introduced. I will then discuss the higher dimensional case, and show that every solution eventually behaves like what we call a "radially symmetric propagating terrace". This latter part is joint work with Yihong Du.
June 2014
Jens
Rademacher
Universität Bremen
Patterns in Landau-Lifschitz-Gilbert-Slonczewski equations for spintronic devices with aligned fields
Abstract.
The self-organized emergence of spatio-temporal patterns is a ubiquitous phenomenon in nonlinear processes on large homogeneous domains. In this talk a class of Landau-Lifshitz-Gilbert-Slonczewski equations is studied from this viewpoint, highlighting various aspects of the theory. The model describes magnetization dynamics in the presence of an applied field and a spin polarized current. Here we consider the case of axial symmetry and focus on coherent structure solutions that occur due to the symmetry in one space dimension. This is joint work with Christof Melcher (RWTH).
May 2014
Dmitry
Turaev
Imperial College London, UK
Exponential and super-exponential growth of the number of periodic orbits in iterated function systems
Marcus
Hauser
Otto-von-Guericke-Universität Magdeburg
Interaction of Scroll Waves
Abstract.
Scroll waves are the three-dimensional counterparts of spiral waves occurring in excitable reaction-diffusion systems. Single scroll waves may undergo various instabilities that play an important role in the formation of cardiac arrhythmias, like ventricular tachycardia and fibrillation. While a substantial effort has been devoted to the study of the dynamics of single scroll waves, experimental investigations of the interaction of scroll waves are rare. The interaction of a pair of scroll waves with each other was investigated. The scroll waves were created in a Belousov-Zhabotinsky reaction medium and observed by optical tomography. We studied scroll waves whose filaments were parallel to each other and oriented along the vertical axis of the reaction cylinder [1]. The dynamics of such a pair of scroll waves was found to depend on the distance between the filaments: When the distance d between the two filaments was shorter than the wavelength λ of the scroll wave (but larger than the extension of a spiral core), then the filaments displaced each other, until the inter-filament distance d became comparable to the wavelength (i.e. until d ≈ λ). Once d ≈ λ, the scroll waves rotated without being disturbed by each other. When the distance between the two filaments was shorter than the radius of the spiral core, then two behaviours were observed: Locally co-rotating scroll waves (i.e. scrolls that presented the same sense of rotation) always repelled each other. By contrast, locally counter-rotating scroll waves may suffer a “crossover collision” [2], leading to a rupture and a subsequent reconnection of the filaments. Each of the reconnected filaments consisted of parts that originated from the two original filaments. The conditions for rupture and reconnection of filaments will be discussed. [1] D. Kupitz, M. J. B. Hauser, J. Phys. Chem. A 117, 12711 (2013). [2] B. Fiedler and R. M. Mantel, Documenta Math., 5, 695 (2000).
April 2014
Boris
Hasselblatt
Tufts University, Medford, MA/US
Statistical properties of deterministic systems by elementary means
Abstract.
The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by Poincaré in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hopf argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders.
February 2014
Irina
Kmit
Humboldt University Berlin
Hopf bifurcation for one-dimensional hyperbolic PDEs
January 2014
Philipp
Hövel
Technical University Berlin
Dynamics of two neural oscillators in the presence of heterogeneous coupling delays
Abstract.
Investigations of nonlinear dynamics in coupled systems have seen a huge increase in interest during the last years. The size of considered systems ranges from a few coupled elements to complex networks, and collective dynamics may arise in various patterns, of which in-phase (or zero-lag) synchronization is just the most prominent one. The signal transmission between coupled elements is often not instantaneous. Thus, non-zero transmission times have to be taken into account as crucial quantities that influence the dynamics of network nodes to a large extent. In this presentation, I will discuss synchronization effects in the presence of delayed coupling for two excitable neural systems. In particular, I will investigate the effects of heterogeneous delays in the coupling [1]. Depending upon the coupling strengths and the time delays in the mutual and self-coupling, the compound system exhibits different types of synchronized oscillations of variable period. I will present an analysis of this behavior based on the interplay of the different time delays. The numerical results are supported by analytical findings. In addition, I elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function. [1] A. Panchuk, D. P. Rosin, P. Hövel, and E. Schöll: Synchronization of coupled neural oscillators with heterogeneous delays Int. J. Bif. Chaos 23, 1330039 (2013).
Josef
Ladenbauer
Technical University Berlin
Modeling populations of adaptive neurons: spike train properties and network dynamics
Abstract.
Many types of neurons exhibit spike rate adaptation, a gradual decrease in spiking activity following a sudden increase in stimulus intensity. This phenomenon is typically produced by slowly deactivating transmembrane potassium currents, which effectively inhibit neuronal responses and can be controlled by neuromodulators. In this talk I will present recent theoretical work on (networks of) model neurons, showing (i) how these adaptation currents change the relationship between fluctuating synaptic input, spike rate output and the spike train statistics of single neurons and (ii) how they contribute to spike synchronization as well as spike rate oscillations in recurrent networks of excitatory and inhibitory neurons.
December 2013
Jan
Sieber
University of Exeter, UK
Periodic orbits in equations with state-dependent delay
Atsushi
Mochizuki
RIKEN Advanced Science Institute, Japan
Rate sensitivity of biochemical reaction systems: A function-free approach
November 2013
Antonio
Politi
University of Aberdeen, UK
Coarsening processes in Discrete Nonlinear Schroedinger-type models
Abstract.
The Discrete Nonlinear Schroedinger equation (DNLSE) is a paradigmatic model used to describe both Bose-Einstein condensates and propagation in waveguides. The DNLSE behaviour is discussed from the point of view of statistical mechanics, with a particular emphasis given to the convergence to the equilibrium state in the regime of the so-called negative (absolute) temperatures. The observed coarsening is studied with the help of simplified models of exclusion-process type.
October 2013
Dmitrii
Turaev
Imperial College London, UK
Arnold diffusion in the a priori chaotic case
Abstract.
Let a real-analytic Hamiltonian system have a normally-hyperbolic cylinder such that the Poincare map on the cylinder has a twist property. Let the stable and unstable manifolds of the cylinder intersect transversely. The homoclinic channel is a small neighbourhood of the union of the cylinder and the homoclinic. We show that generically (in the real-analytic category) in the channel there always exist orbits which deviate from the initial condition unboundedly.
Alexander
Skubachevskii
Peoples' Friendship University of Russia, Russia
The Vlasov–Poisson equations in infinite cylinder and controlled plasma
Abstract.
We consider the first mixed problem for the Vlasov-Poisson equations in infinite cylinder, describing evolution of densities for ions and electrons in rarefied plasma with external magnetic field. We construct in explicate form a stationary solution with densities of charged particles in interior cylinder. In a neighborhood of stationary solution it is proved existence and uniqueness of classical solution with supports of densities of charged particles locating at some positive distance from cylindrical boundary. This work was supported by the RFBR (grant No. 12-01-00524).
Oleksandr
Burylko
National Academy of Sciences Kyiv, Ukraine
Analysis of bifurcations and the study of competition in phase oscillator networks with positive and negative coupling
Abstract.
A globally coupled phase oscillators of Kuramoto type with positive (conformist) and negative (contrarian) couplings are considered in (Hong & Strogatz (HS), 2011). Here we generalize the HS system to include a phase shift at the interaction function. The system is analyzed by using bifurcation theory as well as a detailed study of geometry of the invariant manifolds. Results include a rich repertoire of dynamical regimes (multistability, complex heteroclinic cycles, chaos, etc). Some of these interesting regimes are not possible in HS system.
June 2013
Atsushi
Mochizuki
RIKEN, Tokyo
Phenomenology of rate sensitivity in reaction networks
Abstract.
One experimentally possible procedure to understand the dynamics of chemical reaction systems (including metabolic networks) is based on perturbations of reaction rates. For example we may increase/decrease or knockout enzymes that mediate reactions in the systems. Frequently, however, the experimental response to perturbations seems difficult to understand. In this talk, we present a mathematical approach to determine the equilibrium response from the structure of the network. In particular, some perturbations may cause large deviations, while other perturbations may go unnoticed.
Hayato
Chiba
University of Kyushu
Renormalization group methods for ODEs/PDEs
Abstract.
It is known that the Swift-Hohenberg equation, which is a fourth-order PDE, can be reduced to the Ginzburg-Landau equation (amplitude equation), which is a second-order PDE, by means of a singular perturbation method. In this talk, a reduction of a certain class of (a system of ) nonlinear parabolic equations is proposed based on the renormalization group method, which is one of the perturbation methods applicable to a wide class of ODEs/PDEs.
May 2013
Haibo
Ruan
University of Hamburg
A degree theory for synchrony-breaking bifurcations in coupled cell systems
Abstract.
A network is a graphical entity consisting of nodes and links between the nodes. In the theory of coupled cell systems, nodes of the network are interpreted as individual dynamical systems whose mutual interactions are described by the coupling structure of the network. One important and most studied collective dynamics on networks is the synchronization: a set of cells is said to be synchronized, if their individual dynamics coincide over time. Fully synchronized states where all cells are in synchrony, are rare instances. The more common phenomenon is partial synchronization where communities or clusters of cells are synchronized. A synchrony-breaking bifurcation refers to a local bifurcation, where a fully synchronous equilibrium loses its stability and bifurcates to states of less synchrony. In this talk, we introduce a lattice degree which can be used to study synchrony-breaking bifurcations, both of steady states and of oscillating states, and to predict the existence of these bifurcating branches, together with their multiplicity, symmetry and synchrony types.
Martin
Väth
Free University Berlin
Darbo's Theorem, Essential Maps, and Essential Pairs
Abstract.
Darbo established ub 1955 a fixed point theorem in which the compactness hypothesis of Schauder's fixed point theorem was dramatically relaxed. Essential Maps, introduced by A. Granas and also by M. Furi, M. Martelli, and A. Vignoli, are an extension and homotopic analogue of degree theory. Both ideas were married, historically with very technical proofs, for applications in so-called nonlinear spectral theory. In the talk a stunningly simple combination of the two ideas is presented. This even applies to a new concept of so-called essential pairs which appears to be the natural extension and homotopic analogue of recently developing degree theories for function triples.
April 2013
Marek
Fila
Comenius University Bratislava
Extinction of solutions of the fast diffusion equation
Abstract.
We consider positive solutions of the Cauchy problem for the fast diffusion equation. Sufficient conditions for extinction of solutions in finite time are well known. We shall discuss results on the asymptotic behavior of solutions near the extinction time obtained in collaboration with John R. King, Juan Luis Vazquez, Michael Winkler and Eiji Yanagida. We shall pay particular attention to a critical case with slow asymptotics.
Serhiy
Yanchuk
Humboldt University of Berlin
Delay induced patterns in 2D neuronal lattices
Pavel
Gurevich
Free University Berlin
Hysteresis with diffusive thresholds
Abstract.
We describe a system with fluctuating hysteresis thresholds by means of diffusion equations, where the role of the spatial variable is played by hysteresis threshold. To illustrate our approach, we consider a prototype model that describes a population of two-phenotype individuals in a varying environment. The key features are as follows: The individuals can switch between the two phenotypes by hysteresis law (non-ideal relay). The thresholds for each individual can fluctuate. Under the assumption that this fluctuation obeys the Gaussian distribution, we arrive at a system where the density of the population obeys a reaction-diffusion equation with discontinuous hysteresis. The collective impact of the population on the environment is described in terms of the Preisach operator with a measure given by the population density. This measure becomes time dependent and is now a part of the solution. In the talk, we formulate a well-posedness theorem and discuss emerging spatial patterns.
January 2013
Frits
Veerman
Leiden University
Pulses in singularly perturbed reaction-diffusion systems
Abstract.
The existence of pulse solutions in a two component system of singularly perturbed reaction-diffusion equations is established using geometric singular perturbation theory. The stability of these pulse solutions can be analysed using Evans functions techniques, leading to explicit expressions, even in the general setting. A Gierer-Meinhardt system with an additional slow nonlinear term is used as an explicit example, showing new phenomena.
Shalva
Amiranashvili
WIAS Berlin
Extreme events in optical supercontinuum
Abstract.
Optical supercontinuum is a conversion of a short light pulse into a complicated filed pattern with an ultra-wide spectrum. Many interesting objects are hiding behind this pattern, including solitons, linear wave packets, and short-living sub-pulses with the extremely high intensities. Detailed insight into the supercontinuum is provided by a numerical solution of the corresponding propagation equation.
December 2012
Sergio
Oliva
University Sao Paulo, Brasil
Inflammatory phase dynamics in diabetic wounds: a PDE approach
Abstract.
The objective of the present paper is the modeling and analysis of the dynamics of macrophages and certain growth factors in the inflammatory phase, the first one of the wound healing process. It is the phase where there exists a major difference between diabetic and nondiabetic wound healing, an effect that we will consider in this talk. We will propose and analyze a partial differential equation as a model for the interaction of some of the crucial elements involved in this phase of the wound healing. This model will generalize a previous existing ODE model.The final model is a system of 3 PDE equations with nonlinear boundary conditions and we show that it is a well posed problem both from a mathematical and biological point of view. More concretely, we will prove there exists a bounded invariant set where all the solutions are global and positive, and show some numerical experiments. Work done in collaboration with Neus Consul and Marta Pellicer.
November 2012
Klaus
Böhmer
Philipps-Universität Marburg
Dew drops on spider webs: a symmetry breaking bifurcation for a parabolic differential-algebraic equation
Abstract.
Lines of dew drops on spider webs are frequently observed on cold mornings. In this lecture we present a model explaining their generation. Although dew is supposed to condense somehow evenly along the thread, only lines of drops are observed along the spider thread. What are the reasons for this difference? We try to give an explanation by concentrating on some essential aspects only. This every-day observation is an example of one of the fascinating scenarios of nonlinear problems, symmetry breaking bifurcation. Despite many simplifications the model still provides very interesting mathematical challenges. In fact the necessary mathematical model and the corresponding numerical methods for this problem are so complicated that in its full complexity it never has been studied before. We analyse and numerically study symmetry breaking bifurcations for a free boundary value problem of a degenerate parabolic differential-algebraic equation employing a combination of analytical and numerical tools.
Vasily
Denisov
Lomonosov Moscow State University
On stabilization problem for parabolic equations
Abstract.
We discuss necessary and sufficient conditions on the lower order coefficients of parabolic equations under which the solution of the Cauchy problem stabilizes (x-uniformly) to zero on every compact in Euclidean space. Bounded and growing initial functions are considered.
October 2012
Kristial Uldall
Kristiansen
Technical University of Denmark
On slow manifolds where the fast normal motion is oscillatory
Igor
Franovic
currently University Potsdam
Excitable systems under noise and delayed couplings: Self-organization and paradigm for hierarchies of mean-field approximations
June 2012
Atsushi
Mochizuki
RIKEN Advanced Science Institute, Japan
Pattern formation in biology
Abstract.
I introduce two mathematical studies for pattern formations in biology. The first topic is on network and spot patterns in plants. Various differentiation processes in plant development are regulated by a plant hormone, auxin. Leaf vascular networks and phyllotaxis patterns in meristem are generated from network-like and spot-like distributions of auxin, respectively. Inhomogeneous distribution of auxin is generated by anisotropic distribution of auxin efflux carrier protein called PINFORMED (PIN) in cells. We develop mathematical models for the dynamics of auxin and PIN distribution in plant organs, and discuss possible mechanisms for switching between two different patterns, network-like and spot-like patterns. The second topic is on a simplest system of cell differentiation and pattern formation by a species of bacteria. A multicellular filamentous cyanobacterium, Anabaena, produces specialized nitrogen-fixing cells named heterocysts that appear about every 10 cells. We develop a 1-dimensional cellular automaton model for the dynamics of cyanobacteria, which includes stochastic cell division and differentiation. We determined distribution of heterocyst interval analytically. From the comparison with experimental data we conclude that age dependency of division and differentiation is essential for the observed patterns. This is joint work with Yoshinori Hayakawa and Jun-ichi Ishihara.
Jens
Starke
Technical University of Denmark
Traveling waves and oscillations in particle models
Abstract.
The macroscopic behaviour of microscopically defined particle models is investigated by equation-free techniques where no explicitly given equations are available for the macroscopic quantities of interest. We investigate situations with an intermediate number of particles where the number of particles is too large for microscopic investigations of all particles and too small for analytical investigations using many-particle limits and density approximations. By developing and combining very robust numerical algorithms, it was possible to perform an equation-free numerical bifurcation analysis of macroscopic quantities describing the structure and pattern formation in particle models. The approach will be demonstrated with examples from traffic and pedestrian flow. The presented traffic flow on a single lane highway shows besides uniform flow solutions also traveling waves of high density regions. Bifurcations and co-existence of these two solution types are investigated. The approach is validated by a comparison with analytical findings. The pedestrian flow shows the emergence of an oscillatory pattern for two crowds passing a narrow door in opposite directions. The oscillatory solutions appear due to a Hopf bifurcation. This is detected numerically by an equation-free continuation of a stationary state of the system. Furthermore, an equation-free two-parameter continuation of the Hopf point has been performed to investigate the oscillatory behaviour in detail using the door width and relative velocity of the pedestrians in the two crowds as parameters. This is in parts joint work with Rainer Berkemer, Olivier Corradi, Yuri Gaididei, Poul Hjorth and Mads Peter Soerensen.
May 2012
Fengqi
Yi
Harbin Engineering University, China
Hopf bifurcations and steady state bifurcations of some semi-linear reaction diffusion equations
Abstract.
Some semi-linear reaction diffusion equations subject to Neumann boundary conditions on bounded domains are considered. Hopf and steady state bifurcation analysis are carried out in details. In particular, for the concrete problems, we show the existence of multiple spatially nonhomogeneous periodic orbits while the system parameters are all spatial homogenous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions of certain reaction-diffusion systems. These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.
Serhiy
Yanchuk
Humboldt University Berlin
Patterns in non-homogeneous rings of delay coupled systems
April 2012
Nadezhda M.
Ratiner
Voronezh State University
Some applications of the topological degree for Fredholm proper maps with positive index to differential equations
Abstract.
The talk will be devoted to some examples demonstrating applications of the topological degree for Fredholm proper maps with positive index. Nonlinear boundary value problem for system of ordinary differential equations, bifurcation problem for elliptic boundary value problem and oblique boundary value problem will be considered. The talk is partially based on a joint paper with V. G. Zviagin.
Bernhard
Brehm
Free University Berlin
Formal Kasner-map shadowing and qualitative results on the convergence of Bianchi 9 trajectories to heteroclinic chains
Abstract.
The Bianchi 9 cosmologies have a well known compact heteroclinic attractor. Heteroclinic chains on this attractor are described by orbits of the mixing, non-injective Kasner map (related to continued fraction expansions). It has been long conjectured, that the dynamics of the Kasner-map (i.e. the structure of the heteroclinic network) essentially describes the long-term dynamics of trajectories which limit to the attractor. Unfortunately, it is not clear at all what we should precisely mean by this heuristic. The choice of definitions is here certainly a matter of taste as well as a tradeoff between attainability, readability and strength (for particular applications) of results. Aside from global (e.g. statistical or global topological) properties, one important notion is what we should mean by convergence of two trajectories towards each other or towards a heteroclinic chain. I will introduce the notion of 'convergence in order', apply it to a classic example, and show some theorems pertaining to this choice of specification of the Kasner-map heuristic. Furthermore, I will argue that this notion of convergence sits at a good 'spectral gap' for Bianchi 9 cosmologies: The definition is weak enough to allow for relatively easy results, while being strong enough that these results are still meaningful. Unfortunately the notion of convergence is too weak to answer many important questions; however, any strengthening of the notion of convergence seems to make similar results much harder to prove or outright wrong. (well, there is a reason for some conjectures on Bianchi cosmologies still being unresolved after fifty years) The notion of convergence in order will be used to show that every (non-Taub) trajectory in Bianchi 9 converges 'in order' to a (possibly non-unique) formal heteroclinic chain of the Kasner-map. Furthermore, we will show (if time permits) that for every infinite heteroclinic chain, there exists a codimension one set of initial conditions converging to it (by degree-type arguments). We will close by explaining why this result is not at all as spectacular as it might sound at first glance.
February 2012
Nina
Uraltseva
St.Petersburg State University
Two-Phase Parabolic Obstacle Problem: Regularity Properties of the Free Boundary
Abstract.
In this talk we describe the methods, developed in the last decade, for studying the regularity of the free boundary in the vicinity of branch points. These methods are based on the use of various monotonicity formulas, blow-up technique and some observations of geometric nature.
Daria
Apushkinskaya
Saarland University
Two-Phase Parabolic Obstacle Problems: L∞-estimates for Derivatives of Solutions
Abstract.
Consider the two-phase parabolic obstacle problem with non-trivial Dirichlet condition Δu − ∂tu = λ+χ{u>0} − λ−χ{u<0} in Q=Ω×(0;T), u = φ on ∂pQ. Here T<+∞, Ω ⊂ R^n is a given domain, ∂pQ denotes the parabolic boundary of Q, and λ± are non-negative constants satisfying λ++λ−>0. The problem arises as limiting case in the model of temperature control through the interior. In this talk we discuss the L∞-estimates for the second-order space derivatives D^2u near the parabolic boundary ∂pQ. Observe that the case of general Dirichlet data cannot be reduced to zero ones due to non-linearity and discontinuity at u=0 of the right-hand side of the first equation. The talk is based on works in collaboration with Nina Uraltseva.
January 2012
Eugen
Zhang
Oregon State University
Efficient Morse Decomposition of Vector Fields
Abstract.
Traditional vector field topology relies on the ability to accurately compute trajectories, which is difficult to achieve due to noise and error. Morse decomposition addresses this issue. However, computing Morse decomposition given a simulation data set can be challenging due to the complexity in both the flows and the underlying domains. In this talk I will discuss how to effectively compute Morse decomposition in a hierarchical fashion. The results have been applied to a number of simulation data sets.
December 2011
Thomas
Wagenknecht
University of Leeds
Homoclinic snaking: different ways to kill the snakes
Svetlana
Gurevich
Westfälische Wilhelms-Universität Münster
Destabilization of localized structures induced by delayed feedback
November 2011
Carlo
Laing
Massey University New Zealand
Fronts and bumps in spatially extended Kuramoto networks
Abstract.
We consider moving fronts and stationary “bumps” in networks of non-locally coupled phase oscillators. Fronts connect regions of high local synchrony with regions of complete asynchrony, while bumps consist of spatially-localised regions of partially-synchronous oscillators surrounded by complete asynchrony. Using the Ott-Antonsen ansatz we derive non-local differential equations which describe the network dynamics in the continuum limit. Front and bump solutions of these equations are studied by either “freezing” them in a travelling coordinate frame or analysing them as homoclinic or heteroclinic orbits. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable.
Andrey
Muravnik
Peoples' Friendship University Moscow
The Cauchy problem for parabolic differential-difference equations: integral representations of solutions and their long-time behavior
Abstract.
Parabolic equations (including singular ones) containing translation (generalized translation) operators acting with respect to spatial variables are considered. Integral representations of their classical solutions are found and asymptotic closeness (stabilization) theorems are proved for their solutions. It turns out that there are principally new effects of the long-time behavior of the above solutions caused by the non-local nature of the equation. Moreover, those effects hold even in the case where only low-order terms of the equation are non-local.
June 2011
Sergio
Oliva
University Sao Paulo, Brasil
Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary
Carlos
Rocha
Instituto Superior Tecnico, Lisbon, Portugal
Sturm permutations for S^1-equivariant Sturm attractors
Abstract.
We consider semilinear parabolic equations of the form u_t = u_xx + f(u,u_x) defined on the circle x ∈ S^1 = R/2πZ and for dissipative nonlinearity f. Using the Sturm permutation introduced for the characterization of Neumann flows, we obtain a characterization for the Sturm attractors A_f in this class of problems. This is based on a joint work with Bernold Fiedler and Mathias Wolfrum.
Michael
Zaks
Humboldt University Berlin
A mechanism for birhythmicity in ensembles of coupled oscillators
May 2011
Alexey
Osipov
St. Petersburg State University
Random walks on groups
Abstract.
We will talk about 'limit behaviour' of trajectories of random walks on various groups. We will start from simple examples such as random walks on Z^2 or Z^d for d>2. We will see that almost every trajectory of a random walk (with nondegenerate measure) on a free non-abelian group converges to the boundary of the free group (homeomorphic to the Cantor set). Finally, we will consider the case of a random walk on an arbitrary group that has a normal free non-abelian subgroup (or a normal hyperbolic subgroup).
Karsten
Matthies
University of Bath
Time-evolution of Probability Measures on Collision Trees - a Tool for Micro-macro Transitions
Abstract.
A method is presented to show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. Considering a simplified case of hard-spehere dynamics, where particles are removed after the first collision. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are crucial in the convergence proof. Joint work with Florian Theil.
Sergey
Tikhomirov
Free University Berlin
Shadowing. State of art and open questions.
Abstract.
Shadowing theory study properties of approximate trajectories of homeomorphisms and diffeomorphisms. This notion is important for stability theory and for theoretical motivation of numerical simulations. I give an overview of main results in this theory, both classical and recent. As well I will give perspectives for future development in this theory. The Talk is based on joint works with S. Pilyugin, A.Osipov, K. Palmer.
January 2011
Alexander
Skubachevskii
Peoples' Friendship University of Russia
Damping Problem for Control System with Delay and Nonlocal Boundary Value Problems
Abstract.
We consider damping problem for control system with delay. For the first time this problem was studied by N.N. Krasovskii in 1968 for delay differential equation. We consider the damping problem in general case, i.e. for neutral differential difference equation. Such problem can be formulated as a variational problem for nonlocal functional containing derivatives and shifts for unknown function. We reduce a variational problem to a boundary value problem for a second order neutral differential-difference equation and prove a uniqueness and existence of generalized solution of this boundary value problem. Using a connection between boundary value problem for differential-difference equations and nonlocal boundary value problems we obtain the necessary and sufficient conditions for smoothness of generalized solutions.
Stefan
Liebscher
Free University Berlin
Bifurcation without parameters
December 2010
Martin
Väth
Free University Berlin
Bifurcation for a Reaction-Diffusion System with Obstacles and Pure Neumann Boundary Conditions
Abstract.
Consider a reaction-diffusion system which is subject to Turing's effect of diffusion-driven instability (leading to patterns). It is known that the presence of obstacles can lead to bifurcation of stationary solutions in a parameter domain where the system is stable (thus amplifying Turing's effect in a sense). However, usually additional Dirichlet conditions were supposed. For almost 30 years it has been an open problem whether the same result holds without Dirichlet conditions. In the talk the somewhat surprising answer is given, and the difficulties of the proof are sketched.
November 2010
Roman
Shamin
Peoples' Friendship University of Russia, Shirshov Institute of Oceanology
Probability of the occurrence of freak waves
Abstract.
Hydrodynamics of ideal heavy liquid with a free surface in conformal variables is studied. In numeric simulations we show occurrence of freak waves. The statistics of the occurrence of freak waves is investigated. The characteristics of freak waves are considered.
Flavio
Abdenur
PUC, Rio de Janeiro
Geometric mechanisms for robust transitivity
Abstract.
A diffeomorphism is said to be robustly transitive if it is transitive, and moreover it is, err, robustly so. (Meaning of course all diffeomorphisms sufficiently close to it are also transitive.) Robustly transitive (but non-Anosov) systems are in a sense a model for non-hyperbolic but hyperbolic-like global behavior. Though many examples have been constructed, and many consequences deduced, the general mechanism(s) that underlie this phenomenon are still poorly understood. I will report on some progress - meaning actual theorems, not just lemmas or tentative ideas - that has been recently achieved in this direction, in the context of partially hyperbolic systems. The exciting keywords here are blenders, minimality of foliations, and the crossing condition. This is a joint (and ongoing) work with Sylvain Crovisier.
Ilya
Kashchenko
University of Yaroslavl, Russia
Dynamics properties of second-order equations with large delay
Abstract
Carlos
Rocha
Instituto Superior Tecnico, Lisbon, Portugal
Transversality in scalar reaction-diffusion equations on a circle
Abstract.
Stable and unstable manifolds of hyperbolic periodic orbits for scalar reaction-diffusion equations defined on a circle always intersect transversally. Moreover, hyperbolic periodic orbits do not possess homoclinic orbit connections. We review these results that as main tool use Matano's zero number theory dealing with the Sturm nodal properties of the solutions.
October 2010
Dmitry
Glazkov
University of Yaroslavl, Russia
Local dynamics of delay differential equations with long delay feedback
September 2010
Dmitry
Glazkov
University of Yaroslavl, Russia
Qualitative analysis of one class of optoelectronic systems with singularly perturbed models
June 2010
Sergey
Tikhomirov
PUC, Rio de Janeiro
Dynamical systems with hysteresis
Marek
Fila
Comenius University, Bratislava
Homoclinic and Heteroclinic Orbits for a Semilinear Parabolic Equation
Abstract.
We study the existence of connecting orbits for the Fujita equation, u_t=Δu+u^p, with a critical or supercritical exponent p. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.
Michael
Winkler
University of Duisburg-Essen
Irreversibly heading for balance? Some examples of colorful dynamics in simple diffusion processes
Abstract.
Diffusion equations have been thoroughly studied in the past decades, and a large variety of results has given rise to some common belief about effects of diffusion. Typical conclusions of this flavor read as follows. Bounded solutions stabilize towards some equilibrium. Diffusive evolution never returns to a state in which it has already been sometime in the past. Singularities are melted down either instantaneously or never. The intention of the talk is to present some exemplary results which indicate that such dogma-like statements need not be true in general. To underline this, the main focus will be on cases where the mathematical setting, made up by the PDE ingredients, space dimensions and domains, is as simple as possible.
May 2010
Bernold
Fiedler
Free University Berlin
Global attractors of Sturm type: examples and counterexamples
Alexey
Osipov
St. Petersburg State University
(Non)genericity of shadowing properties
Abstract.
We will consider the following shadowing properties of discrete dynamical systems: pseudo-orbit tracing property (or standard shadowing property), orbital shadowing property and weak shadowing property. We will discuss the problems of genericity and density of the above-mentioned shadowing properties in different spaces of discrete dynamical systems on closed smooth manifolds. Particularly we will talk about nondensity of orbital shadowing property with respect to C^1-topology.
February 2010
Przemyslaw
Perlikowski
Technical University of Lodz
Dynamics in a ring of delayed coupled Stuart-Landau oscillators
Abstract.
Results presented in the talk consider the ring of unidirectionally delayed coupled Stuart-Landau oscillators. Firstly, we show investigations of steady state stability, we present its properties with respect to delay in coupling and number of oscillators in the ring. To cope with small and large delay in the coupling we applied analytical techniques, while for intermediate delay we use Lambert function. Then, we present the influence of time delay and size of the ring on appearance of periodic solutions. We find coexisting stable periodic solutions (here, rotating waves), which correspond to Eckhaus instability. Presented results are ongoing joint work with S. Yanchuk.
Flavio
Abdenur
PUC, Rio de Janeiro
Geometric mechanisms for robust transitivity
Abstract.
The existence of robustly transitive but non-Anosov diffeomorphisms has been known ever since Shub's construction in the early seventies. In the past 15 years or so, new examples have been given, many of which rely on the blender studied by Diaz and Bonatti. In addition, for the first time general necessary properties of such systems have been obtained: "robust transitivity implies such-and-such a condition". In this talk I will briefly survey the literature and then discuss some ongoing work that goes in the opposite direction: we ask which conditions are sufficient for - and hopefully in some sense characterize - the phenomenon of robust transitivity. This involves ongoing joint work with S. Crovisier.
January 2010
Sergei
Shapiro
Freie Universität Berlin, FR Geophysik
and
Marc
Timme
MPI Dynamics&Selforganization, Göttingen
Fluid-induced seismicity and pore-pressure diffusion in rocks; Spatio-Temporal Patterns in the Brain: Neural Networks with Nonlinear Dendrites
December 2009
Georgy
Kitavtsev
WIAS Berlin
Reduced ODE models describing coarsening dynamics of slipping droplets
Abstract.
In this talk the topic of reduced ODE models corresponding to a family of one-dimensional lubrication equations derived by M"unch et al. '06 is addressed. This family describes the dewetting process of nanoscopic thin liquid ?lms on hydrophobized polymer substrates due to the presence of several intermolecular forces and takes account of different ranges of slip-lengths at the polymer substrate interface. Reduced ODE models derived from underlying lubrication equations allow for an efficient analytical and numerical investigation of the latest stage of the dewetting process: coarsening dynamics of the remaining droplets. We present a new geometric approach which can be used for an alternative derivation and justi?cation of above reduced ODE models and is based on a center-manifold reduction recently applied by Mielke and Zelik '08 to a certain class of semilinear parabolic equations. One of the main problems for a rigorous justi?cation of this approach is investigation of the spectrum of a lubrication equation linearized at the stationary solution, which describes physically a single droplet. The corresponding eigenvalue problem turns out to be a singularly perturbed one with respect to a small parameter ε tending to zero. For this problem we show existence of an ε-dependent spectral gap between a unique exponentially small eigenvalue and the rest of the spectrum.
November 2009
Nikolay
Zhuravlev
Peoples' Friendship University of Russia
Methoden zur qualitativen Untersuchung periodischer Lösungen von Funktionaldifferentialgleichungen
Abstract.
Die Eigenwerte des Monodromie-Operators beschreiben das Verhalten der Trajektorien in der Naehe einer periodischen Loesung. Im Gegensatz zu gewoehnlichen Differentialgleichungen fuehren Funktionaldifferentialgleichungen jedoch auf Operatoren zwischen unendlich-dimensionalen Raeumen. Es ist daher viel schwieriger, die Eigenwerte des Monodromie-Operators zu bestimmen. Im Vortrag werde ich ueber Methoden zur Bestimmung dieser Eigenwerte berichten.
Julia
Ehrt
WIAS / FU Berlin
Promotionsvortrag / Dissertation Defense: Mathmatics of Cell Motility
Abstract.
The talk will investigate mathematical and modelling aspects of cell motility. After giving a brief overview on the bio-chemical processes of cell motility in general I will focus on crawling cells. The different processes involved in crawling cells have posed challenging problems in terms of their physical and chemical background. In addition crawling cells pose a rich field in terms of mathematical modelling. Several aspects of the physical and chemical background and the difficulties in modeling these will be adressed in the talk. In the last part I will present a very recent model investigated by Gholami, Falcke and Frey (2008) in which interesting mathematical questions arise. I will explain the model and investigate its mathematical background.
Laurette
Tuckermann
PMMH, ESPCI Paris
Turbulent-laminar patterns in plane Couette flow
Abstract.
The greatest mystery in fluid dynamics is probably transition to turbulence. The simplest shear flow, plane Couette flow -- the flow between parallel plates moving at different velocities -- is linearly stable for all Reynolds numbers, but nevertheless undergoes sudden transition to 3D turbulence at Re near 325. At just these Reynolds numbers, it was recently discovered experimentally at CEA-Saclay that the flow takes the form of a steady and regular pattern of wide oblique turbulent and laminar bands. We have been able to reproduce these remarkable flows in numerical simulations of the Navier-Stokes equations. Simulations display a rich variety of variants of these patterns, including spatio-temporal intermittency, branching and travelling states, and localized states analogous to spots. Quantitative analysis of the Reynolds-averaged equations reveals that both the mean flow and the turbulent force are centrosymmetric and can be described by only three trigonometric functions, leading to a model of 6 ODEs. We find that the transition is best described as a bifurcation in the probability distribution function of the power spectrum.
Sergei
Pilyugin
St.Petersburg University
Structural stability of diffeomorphisms and shadowing properties
Abstract.
We will discuss several new results concerning relations between structural stability, Ω-stability, and some shadowing properties of diffeomorphisms of smooth closed manifolds. Variational shadowing property is equivalent to structural stability. Lipschitz shadowing property is equivalent to structural stability. Lipschitz periodic shadowing property is equivalent to Ω-stability. Some of these results are joint with A. Osipov and S. Tikhomirov.
July 2009
Andrey
Shilnikov
Multistability in models of center pattern generators
Abstract.
We study polyrhythmic dynamics in inhibitory-excitatory models of central pattern generators controlling various locomotive behaviors. We analyze various configurations of CPG motifs and describe the universal mechanisms for synergetic bursting patterns.
Thomas
Lorenz
University of Frankfurt
Mutational Analysis - the Cauchy problem in and beyond vector spaces
Abstract.
Thomas Lorenz: Mutational Analysis, 2009 (Habilitationsschrift) Thomas Lorenz: Mutational Analysis, 2009 (Presentation, Berlin)
June 2009
Julia
Ehrt
Weierstrass Institute Berlin
Cascades of heteroclinics in hyperbolic balance laws
Abstract.
The talk will focus on the question of global attractors for viscous balance laws and their hyperbolic limit. I will present a result concerning uniqueness of heteroclinic connections in the hyperbolic case and present a corrected result of Fan and Hale's theorem on persistence of heteroclinic connections for vanishing viscosity. Some consequences of this will be discussed.
Reiner
Lauterbach
University of Hamburg
Bifurcation in Homogeneous Networks: Linear Theory
Abstract.
In this talk we investigate the tensor product structure of the flow on homogeneous networks. We show that this structure has strong implications on the notion of genericity in networks and we find that the dimension of the internal dynamics is crucial for this. Generically center subspaces are linearly isomorphic to generalized eigenspaces of the adjacency matrix if this dimension is at least 2. This relates to genericity results in equivariant bifurcation theory, where the kernels are generically absolutely irreducible group representations. In equivariant bifurcation theory fixed point spaces play an important role. In the case of networks a similar role is played by colorings of an associated graph and corresponding flow invariant spaces. Finally we present some bifurcation results which generalize the Equivariant Branching Lemma to our situation.
Jan
Sieber
University of Portsmouth
Using feedback control to find unstable phenomena in experiments
Markus
Dahlem
Technical University Berlin
Modeling dynamics in large scale neural populations
May 2009
Leonid
Rossovskii
Peoples' Friendship University of Russia
Boundary-value problems connected with the pantograph equation and their elliptic analogs
Abstract.
Elliptic functional differential equations with contracted and expanded arguments in the senior terms will be considered. The equations share some principal features with their prototype, the pantograph equation, which possesses a large number of applications in applied mathematics and engineering. Compared to the relatively well studied differential-difference equations, these equations represent diverse mathematical structures and pose different mathematical challenges. We will discuss necessary and sufficient conditions for the Garding inequality, solvability and regularity of solutions to boundary value problems in Sobolev and weighted spaces.
April 2009
Nicolas
Roy
Humboldt University Berlin
A microlocal approach to Ruelle resonances of Anosov diffeomorphisms
Abstract.
If f is an Anosov diffeomorphism on a compact manifold, the decay of the dynamical correlation functions is governed by the so-called Ruelle resonances. It follows from the works of Baladi & al and Liverani & al, that these resonances can be obtained by a suitable spectral analysis of the composition operator (or another one related to it) called the "Transfer operator". In this talk, we will show how these results can be obtained by a systematic microlocal analysis, and method coming originally from Quantum mechanics. The talk is based on the recent paper: [Faure, Roy, Sjoestrand] "Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances". Open Math. Journal (2008), vol. 1, 35--81.
February 2009
A.L.
Skubachevskii
Moscow State Aviation Institute
Elliptic functional differential equations with degeneration
January 2009
Jens
Rademacher
Centrum Wiskunde & Informatica Amsterdam
Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies
Abstract.
When all nodes in a heteroclinic network are equilibria much is known about the bifurcations. Recently, heteroclinic networks whose nodes can also be periodic orbits have found increasing attention. In the present article we consider finite heteroclinic networks in arbitrary phase space dimensions whose nodes can be an arbitrary mixture of equilibria and periodic orbits. In addition, we allow for tangencies in the intersection of un/stable manifolds. The problem we address is to find solutions that are close to the heteroclinic network for all time, and their parameter values. The main result is a reduction of this problem to a system of algebraic equations for the parameters with leading order expansion in terms of certain geometric characteristics. The only difference for a periodic orbit instead of an equilibrium is that one of these characteristics becomes discrete. The essential assumptions are hyperbolicity of the nodes and transversality of parameter variation.
Juleitte
Hell
Freie Universität Berlin
Conley Index at Infinity
Abstract.
We interpret blow up phenomena as heteroclinic connections to infinity and propose to analyse them with Conley index methods. To apply those at infinity we have to face two main obstacles: the lack of boundedness of neighbourhoods of infinity and the frequent degenerate behaviour at infinity. The first obstacle may be overcome by 'compactification' of the phase space while the second forces us to generalise the definition of the Conley index to a class of degenerate invariant sets at infinity. We show how this new definition fits into the machinery allowing to detect heteroclinic orbits.
December 2008
Frank
Schilder
University of Surrey
Computational bifurcation analysis of Hamiltonian relative periodic orbits
George R.
Sell
University of Minnesota
On the theory and applications of the longtime dynamics of 3-dimensional fluid flows on thin domains
Abstract.
The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are 'good' in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions.
November 2008
Nitsan
Ben-Gal
FUB / Brown University
Asymptotics of Grow-Up Solutions and Global Attractors of Non-Dissipative PDEs
Abstract.
There has been a great deal of work in recent years on the asymptotics of solutions to scalar parabolic pdes which remain bounded or blow up in finite time. In this talk we will discuss recent results addressing the boundary case of grow-up solutions. These results allow for a thorough understanding of the asymptotics of grow-up solutions and a complete decomposition of the global attractor for the ensuing non-dissipative reaction-diffusion systems.
October 2008
Grigory
Bordyugov
Universität Potsdam
Response functions of spiral waves
Alexey
Teplinsky
Academy of Sciences, Kiev
Smooth conjugacy of circle diffeomorphisms with singularities
Victor
Tkachenko
Academy of Sciences, Kiev
Slowly oscillating wave solutions of a single-species reaction-diffusion equation with delay
August 2008
Dmitrii
Turaev
Imperial College, London
The Lorenz attractor does exist
July 2008
Oleh
Omelchenko
Humboldt-Universität Berlin
Chimera States: The Natural Link Between Coherence and Incoherence
June 2008
Jürgen
Knobloch
Ilmenau
Snaking in reversible systems
Oleksandr
Burylko
National Academy of Sciences of Ukraine
Bifurcations to heteroclinic cycles in phase coupled oscillators
Pavel
Gurevich
Moscow
On periodic solutions of thermocontrol problems with hysteresis
May 2008
Arnd
Scheel
University of Minnesota
How robust are Liesegang patterns
April 2008
Alessandro
Torcini
Institute for Complex Systems, Florence
Chaotic synchronization in spatially extended systems
Abstract.
Synchronization transitions in chaotic spatially extended systems with short range interactions will be briefly reviewed [1]. In particular, we will focus on the existing analogies between out-of-equilibrium phase transitions and chaotic synchronizations. Moreover, recent developments, concerning synchronization in systems with power-law coupling, will be presented [2]. In this framework, the transitions are found to be always continuous, while the critical indexes vary with continuity with the power law exponent characterizing the interaction. For discontinuous local dynamics, numerical evidences indicate that the observed transitions belong to the 'anomalous directed percolation' family of universality classes found for Levy-flight spreading of epidemic processes [3]. Instead for continuous local dynamics it is not possible to associate the transitions to previously studied critical phenomena [4].
February 2008
Karsten
Matthies
University of Bath
Longtime Validity of Boltzmann Equations for Hard-sphere Dynamics
Mohamed
Belhaq
Univ. Hassan II, Casablanca
Control of Limit Cycle and Hysteresis Suppression in Driven Self-Sustained Oscillators
Leonid P.
Shilnikov
Nizhny Novgorod State University
Verzweigungen und Seltsame Attraktoren
December 2007
Jan
Sieber
University of Aberdeen
Continuation with feedback control: First results from mechanical experiments
Sergey
Zelik
University of Surrey
Chaotic soliton interaction and space-time chaos in 1D Ginzburg-Landau equations
October 2007
Andrey
Shilnikov
Georgia State University
Complex homoclinic bifurcations in slow-fast neuronal models
January 2007
Stefan
Liebscher
Brown University, Providence
Bifurcation without parameters in PDE
Martin
Vaeth
FU Berlin
Bifurcation of reaction-diffusion systems with inclusions in the stable domain
December 2006
Alexander
Dressel
University Stuttgart
Center manifold reduction for a class of singular ODEs
Jan
Sieber
University of Aberdeen
Bifurcation tracking with feedback control
Sofia
Maczynska
RWTH Aachen
Rigorous numerics for heteroclinic connections
November 2006
Jan
Andres
Olomouc University
Sharkovskii's theorem, differential equations and beyond
Fatihcan
Atay
MPI Leipzig
Asymptotic regimes
Robert
Szalai
University Bristol
Dynamics of milling processes
Michel
Nizette
Free University, Bruxelles
The Q-switching instability in passively mode-locked lasers
October 2006
Marc
Georgi
Free University Berlin
Interaction of homoclinic solutions and Hopf bifurcations in Forward-backward delay equations
July 2006
Dmitry
Turaev
University of Ber Sheva, Israel
On the complex Lorenz attractor
Shiojenn
Tseng
Tamkang University (Taiwan)
A Stochastic Analysis on Dynamic Biological Systems
A.
Doelman
Amsterdam
Semi-strong pulse interaction
June 2006
J.L.
Hudson
University of Virginia
Populations of coupled chemical oscillators
May 2006
A.
Champneys
Bristol
and
E.
Knobloch
Berkeley
Localized Patterns
J.
Rademacher
WIAS Berlin
The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems
S.
Yanchuk
WIAS Berlin
Systems with large delay: Bifurcations, amplitude equations and Eckhaus instability
February 2006
Domnic
Merkt
BTU Cottbus
Spatiotemporal evolution of bounded two-layer films in lubrication approximation
January 2006
Mark
Lichtner
HU Berlin
A spectral gap mapping theorem and smooth invariant manifolds for semilinear hyperbolic systems
Sergei
Zelik
WIAS Berlin
Multi-pulses and space-time chaos in dissipative systems
December 2005
Frank
Schilder
University of Bristol
Computing Arnol'd tongue scenarios
Michel
Nizette
Freie Universität Brüssel
Pulse interaction via gain and loss dynamics in passive mode-locking
Y.
Maistrenko
Forschungszentrum Jülich
and
Borys
Lysyansky
Forschungszentrum Jülich
Complex dynamics in a system of two coupled phase oscillators with delay
November 2005
N.N.
Nefedov
Moscow State University
On moving fronts in periodically forced reaction-diffusion systems
B.
Wagner
WIAS Berlin
Asymptotic regimes for dewetting thin films
October 2005
Kening
Lu
Brigham Young University, Provo
Invariant manifolds for infinite dimensional dynamical systems
February 2005
Michael
Zaks
HU Berlin
Dynamics in noise-driven ensembles of globally coupled excitable systems
January 2005
Victor
Kozyakin
Russian Academy of Sciences, Moscow
Asynchronous systems: A short survey and problems
Jens D.M.
Rademacher
University of British Columbia, Vancouver, Canada
The absolute spectrum of wave trains
Dmitry
Turaev
University of Ber Sheva, Israel, HU Berlin
On Hausdorff dimension of nonuniformly hyperbolic sets
Christof
Melcher
HU Berlin
Motion of magnetic domain walls
December 2004
Arnd
Scheel
University Minneapolis
Coarsening Fronts
Milan
Tvrdy
Akademie der Wissenschaften, Prag
The method of lower and upper functions in singular periodic problems for second order nonlinear differential equations
Jan
Sieber
Bristol University
Delay compensation in the coupling in hybrid experimental/numerical systems
November 2004
Bernd
Krauskopf
Bristol University
Saddle-Node-Hopf Bifurcation with Global Reinjection
Florian
Theil
University Warwick
Crystallization in two dimensions
O.
Omel'chenko
Kiew
Periodic Contrast Structures in Singularly Perturbed Parabolic Equations
October 2004
Michael
Herrmann
WIAS Berlin
Modulated traveling waves in atomic chains
July 2004
Dmitry
Turaev
University of Ber Sheva, Israel
On maps close to identity
June 2004
Mohamed
Belhaq
Univ. Hassan II, Casablanca
Fast Vibrations in Mechanical Systems
Sarah
Day
Cornell University
Conley index techniques for global dynamics: a study of the Swift-Hohenberg equation
Jens
Rademacher
University of Minnesota
Homoclinic orbits near heteroclinic cycles with periodic orbits
Abstract.
New results on homoclinic orbits near certain generic codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit are presented. In the codimension-2 case the global topology of heteroclinic sets determines the number of curves of homoclinic orbits that bifurcate and influences the leading order expansion of parameter curves. The codimension-1 case partially explains the phenomenon of 'tracefiring' in reaction-diffusion equations, which is the bifurcation of a stable excitation pulse to a self-replicating pulse-chain. For the Oregonator model the codimension-2 case can be used to understand the loss of stability of the primary pulse.
May 2004
Dan
Luss
University of Houston
Temperature patterns in catalytic reactors
Leonid
Cherkas
Minsk, State University
Limit cycles of polynomial vector fields
Dmitrii
Rachinskii
WIAS-Berlin
Sector estimates in Hopf bifurcation problems
April 2004
Arnd
Scheel
University of Minnesota, Minneapolis
Corner defects in interface propagation
Abstract.
We study existence and stability of curved interfaces in planar reaction-diffusion systems. We start with a planar, x-independent front and investigate the existence of traveling waves which locally resemble the primary front. The problem of existence of corners is reduced to an ordinary differential equation that can be viewed as the travelling-wave equation to a viscous conservation law or variants of the Kuramoto-Sivashinsky equation. The corner typically but not always points in the direction opposite to the direction of propagation. For the existence and stability problem, we rely on a spatial dynamics formulation with an appropriate equivariant parameterization for relative equilibria. We also comment on oscillatory front propagation, invasion of patterns, and viscous shock waves in anisotropic systems.
Martin
Hasler
Swiss Fed. Inst. of Technology, Lausanne
Complete synchronization of regular and small-world networks of dynamical systems
Abstract.
Diffusively coupled dynamical systems with arbitrary coupling graphs are considered. Explicit upper bounds for the minimal coupling strength (diffusion constant) needed to achieve complete synchronization are derived using Lyapunov functions in the difference variables. These bounds are a product of a term depending only on the individual system dynamics and a term depending only on the coupling graph. The latter is formulated purely in graph theoretical terms. Furthermore, systems where all or part of the couplings are switched on and off in a random fashion are considered. It is proved that for sufficiently fast switching, complete synchronization is almost always achieved, if the averaged system completely synchronizes. This is applied to small-world networks where in addition to fixed couplings "short-cut" couplings are switched on and off. It is shown that even with very low probability of switching short-cuts on, the coupling strength needed to achieve complete synchronization can be considerably lowered.
February 2004
Christian
Rohde
Albert-Ludwig University of Freiburg
Hyperbolic-elliptic conservation laws as models for the dynamics of phase transitions
Sjoerd
Verduyn Lunel
University of Leiden
Forward-backward functional differential equations, holomorphic factorization and applications
January 2004
Markus
Kunze
University of Essen
On the existence of a maximizer for the Strichartz inequality
Hannes
Uecker
University of Karlsruhe
Stability and instability in inclined film flows
Alex
Hutt
WIAS Berlin
Analysis of nonlocal fields including propagation delay effects
December 2003
Björn
Sandstede
Ohio State University
Contact defects: existence and interaction
November 2003
Takao
Ohta
University of Hiroshima
Statistical pulse dynamics in an excitable reaction-diffusion system
Hans-Günter
Bothe
Free University Berlin
Attractors with low Entropy
Barbara
Niethammer
Humboldt University Berlin
On the dynamics of the Beker-Döring Equations
Andrey
Goritsky
State University Moskow
Inertial and integral manifolds for PDE
October 2003
N.N.
Nefedov
State University Moskow
Multidimensional Internal Layers Of Spike Type
July 2003
Marc
Georgi
and
Nihar
Jangle
Spiral waves in reaction-diffusion-systems
Karsten
Matthies
Atomic-scale localization of lattice waves
Arnd
Scheel
Beyond the Gap Lemma - blowup in critical eigenvalue problems
June 2003
M.
Kamenskij
Averaging method via topological degree theory
D.
Luss
Temperature pattern formation in packed bed-reactors
Lutz
Recke
Generic properties of initial boundary value problems with non-smooth data
May 2003
Dmitry
Turaev
On the problem of the genericity of Newhouse phenomenon
Matthias
Wolfrum
Heteroclinic orbits between rotating waves for scalar parabolic PDE
April 2003
Dieter
Armbruster
Dynamics and Control of Supply Chains
February 2003
Igor
Chuechov
Kharkiv National University, Ukraine
Long-time behaviour of solutions to contact parabolic problems in two-layer thin domains
January 2003
Stephan
van Gils
University of Twente
Travelling waves in a singularly perturbed sine-Gordon equation
Holger
Stephan
WIAS Berlin
General Fokker-Planck Equations and their asymptotic behavior
December 2002
Arnd
Scheel
University of Minnesota, Minneapolis
Defects in oscillatory media - towards a classification
Leonid
Shilnikov
Nizhny Novgorod, Russland
Bifurcations of attractors
Genevieve
Raugel
Universite de Paris-Sud
The one-dimensionnal damped wave equation: Analyticity and finite number of determining nodes.
November 2002
Frederic
Laurent-Polz
Institut Non Linéaire de Nice
Dynamics of N point vortices on a sphere
Alexander
Skubachevskiy
Moscow State Aviation Institute
Functional differential equations with partial derivatives and applications
Volker
Reitmann
Max-Planck-Institut Dresden
Attractor crises and dynamic buckling of elastic plates
October 2002
Jürgen
Knobloch
Technische Universität Ilmenau
Lin's method and applications
Carlos
Rocha
Instituto Superior Técnico, Lisbon
Morse-Smale Flows and Order Preserving Semigroups
Katrin
Gelfert
Max-Planck-Institut Dresden
Diffusion approximation for the slow motion in a deterministic multi-scale system
Boris
Hasselblatt
Tufts University, Medford
Hausdorffdimension hyperbolischer Mengen als Summe derer von stabilen und instabilen Schnitten
Astrid
Huber
Technische Universität Wien
Laser: Dynamik auf verschiedenen Zeitskalen im Yamada-Modell
July 2002
Nickolaj
Kuznetsov
Sankt-Peterburg
Stability by first approximation for discrete systems
Jörg
Härterich
FU Berlin
Semidiscretization and Exponential Dichotomies
June 2002
Sergey
Gonchenko
Nizhny Novgorod
On dynamics of multidimensional diffeomorphisms in the Newhouse regions
Laurent
Gosse
Bari, Italien
Well-balanced schemes for discrete kinetic models in the diffusive limit
Nils
Ackermann
Uni Giessen
Multiple Equilibria in Superlinear Parabolic Problems
M.
Zaks
HU Berlin
Delayed bifurcations and slow-fast dynamics in weakly desynchronized chaotic oscillators
May 2002
Bas
Lemmens
Eurandom, Eindhoven
Periods of periodic points of nonexpansive maps
Dmitry
Turaev
WIAS Berlin
Richness of chaos in area-preserving two-dimensional maps
Andrey
Shilnikov
Atlanta
Curious effects in slow-fast flows and maps
Oleg
Sten'kin
London
Conservative and nonconservative dynamics in reversible maps with homoclinic tangencies
April 2002
Andreas
Amann
TU Berlin
Efficient time-delayed feedback control using spatio-temporal Floquet eigenmodes
Nikolaj
Nefedov
Moscow
Reaction-diffusion systems with balanced nonlinearity
Björn
Schmalfuß
FH Merseburg
Invariant Manifolds for Random Dynamical Systems
February 2002
Klaus
Schneider
WIAS Berlin
Canards and other strange birds
Dmitry
Anosov
Steklov Institute, Moscow
On Hilbert 21-st problem
January 2002
Stefan
Liebscher
Freie Universität Berlin
Plane Kolmogorov flows and bifurcation without parameters
December 2001
Norbert
Koksch
Technische Universität Dresden
Inertiale Mannigfaltigkeiten für nichtautonome Systeme
November 2001
Sergey
Gonchenko
Institut für Angewandte Mathematik und Kybernetik, Nizhny Novgorod
Homoclinic bifurcations and generalized Henon-maps
Vladimir
Belych
Nizhny Novgorod
Synchronization
Marta
Lewicka
Max-Planck-Institute for Mathematics in the Sciences, Leipzig
Stability of wave patterns arising from systems of conservation laws
Juri
Suris
TU Berlin
Integrable discretizations for integrable systems
Vered
Rom-Kedar
The Weizmann Institute of Science, Rehovot
On chaotic fluid mixing
October 2001
Martin
Väth
Julius-Maximilians-Universität Würzburg
Einige Anwendungen nichtlinearer Spektraltheorie
Dalibor
Prazak
Charles University, Prague
Large time behavior and the method of trajectories
Andrei
Afendikov
Keldysh Institute of Applied Mathematics, Moscow
Bifurcation from the line of equilibria and Kolmogorov flows on plane torus