Nonlinear Dynamics at the Free University Berlin

Winter 2020/2021

Oberseminar Nonlinear Dynamics



Tuesday, November 17th, 15:15 Prof. Dr. Angela Stevens (Universität Münster) Mathematical modeling of regeneration phenomena in biology
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.
Tuesday, November 24th Dr. Giulia Ruzzene (Universitat Pompeu Fabra, Barcelona and IFISC, Palma de Mallorca) Control of chimera states in multilayer networks
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.
Tuesday, December 1st Dr. Felix Kemeth (Johns Hopkins University) 2-Cluster Fixed-Point Analysis of Mean-Coupled Stuart-Landau Oscillators in the Center Manifold
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.
Tuesday, December 8th Elisenda Feliu (University of Copenhagen) Understanding of bistability and Hopf bifurcations in biochemical reaction networks
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, Yürük, Wiuf and de Wolff.
Tuesday, January 12th Serhiy Yanchuk (Technical University Berlin) Delay systems and machine learning applications
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.
Tuesday, January 19th Dr. 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
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.
Tuesday, January 26th Tibor Krisztin (University of Szeged, Hungary) Periodic and connecting orbits for the Mackey-Glass equation
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.
Monday, February 1st, 13.00 Carlos Rocha (CAMGSD, Instituto Superior Técnico, Universidade de Lisboa, Portugal) A minimax property for global attractors of scalar parabolic equations
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.
Bernold Fiedler (Freie Universtität Berlin, Germany) Global attractors of scalar parabolic equations: the Thom-Smale complex
We consider the global attractors of scalar parabolic equations
under Neumann boundary conditions. The Thom-Smale complex decomposes a global attractor into the unstable manifolds of its (hyperbolic) equilibria ut=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 ut=0 are characterized by a meander curve, which arises from a shooting approach to the ODE boundary value problem uxx+f(x,u,ux) =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.
Tuesday, February 2nd David Müller-Bender (Chemnitz University of Technology) Laminar chaos in systems with time-varying delay
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.
Tuesday, February 9th Svetlana Gurevich (Westfälische Wilhelms-Universität Münster) Dynamics of temporal and spatio-temporal localized states in time-delayed systems
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.
Tuesday, February 23rd Prof. Dr. Alexandre Nolasco de Carvalho (University of São Paulo at São Carlos) Gradient Non-autonomous Dynamics
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.

Time and Place

Because of the current Corona virus situation, the events take place online.

Guests are always welcome !


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