Nils Berglund (Orléans), Paolo Dai Pra (Padova), Bastien Fernandez (Marseille), Christof Külske (Bochum), Khashayar Pakdaman (Paris), Arkady Pikovsky (Potsdam).

We will start by recalling classical results on synchronization of two coupled oscillators, and their description by an effective phase dynamics. Noise acting on such a system may induce a temporary loss of synchronicity, called a phase slip. The mathematical description of the distribution of phase slips requires methods going beyond large-deviation theory, and reveals some striking properties. In particular, Martin Day has discovered that exit locations through unstable periodic orbits depend periodically on the logarithm of noise intensity, a phenomenon called cycling. We will explain how the cycling profile is related to the Gumbel distribution, known from extreme-value theory. Based on joint work with Barbara Gentz (Bielefeld).

Paolo Dai Pra (Padova): Collective periodic behavior in dissipative ferromagnetic systems with mean-field interaction.

Various models, mostly inspired by life sciences, consist of many particles driven by random noise and interacting through a mean-field potential; this potential is subject to dissipation and diffusion, so it is random and varying in time. We analyze some particular models, where we show that in the thermodynamic limit the system may exhibit time periodic behavior. We expect this behavior to be somewhat "universal" within ferromagnetic systems.
This research is developing in collaboration with F. Collet (Padova), M. Fischer (Padova), G. Giacomin (Paris), D. Regoli (Pisa).

Bastien Fernandez (Marseille): Properties of synchronization graphs in discontinuous forced systems. pdf

When a contractive map is forced by a chaotic discontinuous system, the asymptotic response function that defines the attracting invariant set can be highly irregular, with a dense set of discontinuities.
In this talk, I'll describe the properties of such function in a basic example of linear real contractions forced by (generalized) Baker's maps.
In this setting, it is also natural to ask whether the invariant distributions of the base and factor systems share the same characteristics and in particular, whether the factor distribution of an absolutely continuous SRB measure in the base can be absolutely continuous. I will show that absolute continuity holds for almost every value of the factor contraction rate.

Giambattista Giacomin (Paris): Synchronization phenomena and statistical mechanics. pdf

The word "synchronization" gathers a variety of phenomena and models coming from very different scientific domains. The first part of the talk will be an overview of synchronization phenomena and models from a mathematical perspective. Notably the emphasis will be on phase reduction and on the arising models (Kuramoto models). The second part of the talk will focus on synchronization for Kuramoto models in the limit of infinitely many interacting components and on the natural connection to nonequilibrium statistical mechanics.

Christof Külske (Bochum): Discrete and Continuous rotators.

We describe connections between systems of discrete and continuous rotators, and show how these connections can be used in the study of some systems with synchronization, both on the lattice and in mean field.
1) B. Jahnel, C. Külske, A class of nonergodic interacting particle systems with unique invariant measure, arXiv:1208.5433v2, to appear in Annals of Applied Probabability. 2) B. Jahnel, C. Külske, Synchronization for discrete mean-field rotators, arXiv:1308.1260

Khashayar Pakdaman (Paris): On some synchronization phenomena in neuronal assemblies.

One of the features of neuronal assemblies in nervous systems is to engage in collective regular or irregular activity, broadly referred to as synchronization. Motivated by experimental observations, this presentation will go over our approaches in modelling and analyzing the conditions for the occurrence of such dynamics in specific systems.

Arkady Pikovsky (Potsdam): Synchronization in Ensembles of Oscillators: Theory of Collective Dynamics. pdf

In the talk theoretical approaches to the problem of synchronization of populations of coupled oscillators are discussed. Theories of Watanabe and Strogatz and of Ott and Antonsen are presented together with their generalizations. Different effects in nonlinearly coupled and strongly heterogeneous populations are discussed.