On topological synchronization of Rössler systems

 Abstract

We consider a master/slave system of two interacting almost identical Rössler oscillators], in order to illustrate the concept of topological synchronization. This is a physical concept, which also garnered some press attention, established with the purpose to explain the numerically observed phenomenon where the attractors of the components of a coupled dynamical system becomes very similar as the strength of the coupling is increased, and which is believed to be at the heart of the mechanism of self organization of agent based models composed by many interacting chaotic dynamical systems. We prove that, in the limit of the coupling parameter tending to infinity, the physical measure of the slave system weakly converges to the physical measure of the master one. We also show that for a system diffeomorphically conjugated to a suspension on a Poincaré surface Σ, the spectrum of generalized dimensions (D_q (μ), q ∈ R) of the physical measure μ can be explicitly derived from (D_q (μ R ), q ∈ R), that of the physical measure μ_R associated with the return map R on Σ, provided the roof function (first return time map) is μ_R-integrable. In particular, for the Rössler flow, for a suitable Σ, we show that, from a numerical point of view, R has a skew product structure, namely

Hence D_q (μ_R ) can be recovered from the dimension spectrum of the invariant measure μ_T of the one-dimensional map T. The latter turns out to be a unimodal map, for which the generalized dimensions can be computed explicitly.

Talk by Théophile Gaby