Nonlinear Dynamics at the Free University Berlin

Sommer 2022

Oberseminar Nonlinear Dynamics



Tuesday, May 3rd Dr. Marius Yamakou (Friedrich-Alexander-Universität Erlangen-Nürnberg) Transitions between weak-noise-induced resonance phenomena in a multiple timescales neural system
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.
Tuesday, May 10th,

fully online

Prof. Dr. Sergey Tikhomirov (St.Petersburg State University and Instituto de Matemá tica Pura e Aplicada, Rio de Janeiro) Mixing zone in miscible displacement: application in polymer flooding and theoretical attempts of improving
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.
Dr. Yulia Petrova (Instituto de Matemá tica Pura e Aplicada, Rio de Janeiro) On the impact of dissipation ratio on vanishing viscosity solutions of Riemann problems for chemical flooding models
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).
Tuesday, May 17th Leonhard Schülen (Technische Universität Berlin) The solitary route to chimera states
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
Tuesday, May 24th Dr. 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
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. s
Tuesday, June 14th Babette de Wolff (VU Amsterdam) Title: Delayed feedback stabilization & unconventional symmetries
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).

Time and Place

The event takes place at WIAS, Mohrenstr. 39, Erhard-Schmidt lecture room, 10117 Berlin. The event is also accessible via Zoom.

Guests are always welcome !


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