Dynamical Systems and Networks Seminars

This series of seminars hosts talks on the dynamics of interacting systems coupled on networks and related topics. Below is a list of past and upcoming seminars.

29th of April 2021:

"Chimera States"

Time: 9:00 AM (ET)

Speaker: Erik A. Martens (Lund University and Copenhagen University)

Title: Chimera states in modular networks

Abstract: Phase oscillator networks model a variety of dynamical phenomena in nature and in technological applications. Many real-world networks have modular structure, that is, they consist of distinct communities. Such networks support chimera states, localized synchrony patterns where one part of the oscillators are synchronized while others drift incoherently. I discuss how chimera states and other synchronization patterns emerge in modular networks and how their mean-field dynamics can be studied via dimensional reduction methods. These insights are a fist step towards understanding how chimera states may be involved in functional and computational properties of the network. For instance, localized synchronization may encode information and thus provide function in neuro-biology.

Time: 10:00 AM (ET)

Speaker: Edmilson Roque (University of Sao Paulo)

Title: Chimera states through invariant manifold theory

Abstract: We consider chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically coupled stars of identical oscillators with shear and Kuramoto--Sakaguchi coupling. From numerical experiments we observe that the chimera states are metastable, i.e., after a transient time the chimera vanishes. In this talk we show a rigorous proof that supports the numerical evidence. In particular, if the intra-star coupling strength is of order , the chimera states persist on time scales at least of order . The analysis relies on using Möbius group reduction for globally coupled phase oscillators and normal hyperbolicity theory. This is joint work with Jaap Eldering, Jeroen Lamb and Tiago Pereira.

1st of April 2021:

"Response of coupled systems to perturbation or parameter change"

Time: 9:00 AM (ET)

Speaker: Bastien Fernandez (Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université)

Title: Conditioning problems for invariant sets of expanding piecewise affine mappings: Application to loss of ergodicity in globally coupled maps

Abstract: In this talk, I will describe a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using information from direct numerical simulations in order to infer, and then to solve, a so-called piecewise linear conditioning problem for the geometric constraints that characterize the polytopes involved in the union. As a proof of concept, I will subsequently apply this approach to some systems of globally coupled maps for which asymmetric IUP are established and analyzed. Joint work with F. Selley.

Time: 10:00 AM (ET)

Speaker: Caroline Wormell (Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université)

Title: Linear response in high-dimensional globally coupled chaotic dynamics

Abstract: The response of long-term averages of observables of chaotic systems to small, time-invariant dynamical perturbations can often be predicted to first order using linear response theory (LRT), but some very basic chaotic systems are known to have a non-differentiable response. However, complex dissipative chaotic systems' macroscopic observables are widely assumed to have a linear response, but the mechanism for this is not well-understood.

We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional weakly coupled networks of chaotic subsystems, where the weak coupling is via a mean field and the microscopic subsystems may or may not obey LRT. Through a stochastic reduction to mean-field dynamics, we provide conditions for linear response theory to hold both in large finite networks, and in the thermodynamic limit. In particular, we demonstrate that in systems of (even quite small) finite size, linear response is induced by self-generated noise.

Conversely, we will present an example in the thermodynamic limit where the macroscopic observable does not satisfy LRT, despite all microscopic subsystems satisfying LRT when uncoupled. This latter example is associated with emergent non-trivial dynamics of the macroscopic observable.

This is joint work with Georg Gottwald.

25th of February 2021:

"Self-consistent transfer operators and coupled maps in the mean-field limit".

Time: 9:00 AM (Eastern Standard Time)

Speaker: Fanni Selley (CNRS and Sorbonne Université)

Title: Linear response for a family of self-consistent transfer operators

Abstract: A dynamical system is called self-consistent if the discrete-time dynamics is different in each time step according to some current statistics on the phase space. In this talk we study the special case of mean-field coupled smooth circle maps. Our main result is that for sufficiently weak coupling, the unique invariant density (the fixed point of the self-consistent transfer operator) is a smoothly differentiable function of the coupling strength. We present a linear response formula for its derivative, reminiscent of the similar formula for families of smooth uniformly expanding circle maps. This is a joint work with Matteo Tanzi (NYU Courant Institute).

Time: 10:00 AM (Eastern Standard Time)

Speaker: Christian Bick (Vrije Universiteit Amsterdam)

Title: Multi-Population Phase Oscillator Networks with Higher-Order Interactions

Abstract: Kuramoto-type phase oscillator networks have been instrumental to explain synchronization phenomena for coupled oscillators. Recently, the effect of nonpairwise higher-order interactions has received significant attention. We consider phase oscillator networks that are constituted by multiple interacting populations populations with such higher-order interactions. We will first discuss the dynamics of such network dynamical system for finitely many oscillators. Then we will consider the dynamics in the mean-field limit described by the evolution of measures governed by a set of characteristic equations.