Christophe LETELLIER
14/10/2023

Théophile Gaby

**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 (

Hence *D _{q} (μ_{R} )* can be recovered from the dimension spectrum of the invariant measure μ