We consider a master/slave system of two interacting almost identical R\"ossler oscillators originally introduced in \cite{L1}, see also \cite{L2}, in order to illustrate the concept of \emph{topological synchronization}. This is a physical concept, which also garnered some press attention \cite{PP1,PP2}, 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 \cite{CG2}. We also show that for a system diffeomorphically conjugated to a suspension on a Poincaré surface $\uSigma$, the spectrum of generalized dimensions $(D_{q}(\umu), q\in \mathbb{R})$ of the physical measure $\umu$ can be explicitly derived from $(D_{q}(\umu_{R}), q\in \mathbb{R})$, that of the physical measure $\umu_{R}$ associated with the return map $R$ on $\uSigma$, provided the roof function (first return time map) is $\umu_R$-integrable. In particular, for the R\"ossler flow, for a suitable $\uSigma$, we show that, from a numerical point of view, $R$ has a skew product structure, namely $$\uSigma \ni (x,y) \mapsto R(x,y)=(\mathbf{T}(x),g(x,y)) \in \uSigma.$$ Hence $D_q(\umu_{R})$ can be recovered from the dimension spectrum of the invariant measure $\umu_{T}$ of the one-dimensional map $mathbf{T}$. The latter turns out to be a unimodal map, for which the generalized dimensions can be computed explicitly \cite{CG1,CGSV}.

On topological synchronization of Rössler systems

Michele Gianfelice
2023-01-01

Abstract

We consider a master/slave system of two interacting almost identical R\"ossler oscillators originally introduced in \cite{L1}, see also \cite{L2}, in order to illustrate the concept of \emph{topological synchronization}. This is a physical concept, which also garnered some press attention \cite{PP1,PP2}, 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 \cite{CG2}. We also show that for a system diffeomorphically conjugated to a suspension on a Poincaré surface $\uSigma$, the spectrum of generalized dimensions $(D_{q}(\umu), q\in \mathbb{R})$ of the physical measure $\umu$ can be explicitly derived from $(D_{q}(\umu_{R}), q\in \mathbb{R})$, that of the physical measure $\umu_{R}$ associated with the return map $R$ on $\uSigma$, provided the roof function (first return time map) is $\umu_R$-integrable. In particular, for the R\"ossler flow, for a suitable $\uSigma$, we show that, from a numerical point of view, $R$ has a skew product structure, namely $$\uSigma \ni (x,y) \mapsto R(x,y)=(\mathbf{T}(x),g(x,y)) \in \uSigma.$$ Hence $D_q(\umu_{R})$ can be recovered from the dimension spectrum of the invariant measure $\umu_{T}$ of the one-dimensional map $mathbf{T}$. The latter turns out to be a unimodal map, for which the generalized dimensions can be computed explicitly \cite{CG1,CGSV}.
2023
Topological synchronisation, coupled dynamical systems, R\"ossler flow, generalized dimensions
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/366817
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