Properties of generalized synchronization in smooth and non-smooth identical oscillators

M Balcerzak,A Chudzik,A Stefanski

doi:10.1140/epjst/e2020-000010-5

M Balcerzak, A Chudzik + Show 1 more

Open Access

https://doi.org/10.1140/epjst/e2020-000010-5

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Abstract

This paper deals with the phenomenon of the GS only in the context of unidirectional connection between identical exciter and receivers. A special attention is focused on the properties of the GS in coupled non-smooth Chua circuits. The robustness of the synchronous state is analyzed in the presence of slight parameter mismatch. The analysis tools are transversal and response Lyapunov exponents and fractal dimension of the attractor. These studies show differences in the stability of synchronous states between smooth (Lorenz system) and non-smooth (Chua circuit) oscillators.

Highlights

In analysis and theory of dynamical systems, the research of interactions between them plays an important role
Analogous classical examples are well known in the literature of this topic, e.g., references [25,26,27], but we present it in context of comparison with remaining results of our research
After crossing the strong generalized synchronization (GS) threshold at q = 35.7, imperfect synchronous regimes x-y and y-y remain stable even in the presence of small parameter mismatch. These results show that a real condition for the strong GS of identical drive–response systems is λ1T < 0, in spite of the fact that the dimension of the global attractor is still larger than the dimension of the driving system attractor, i.e., the strong GS condition (Eq (16))

Summary

Introduction

In analysis and theory of dynamical systems, the research of interactions between them plays an important role. The GS phenomenon has been considered both in the context of identical (when separated) systems (1a) and (1b), and in cases when the response system is slightly (the same set of ODEs with different values of system parameters) or strictly different (another set of ODEs and attractor dimension) than the driving oscillator [13,22,23,24]. The analysis tools are transversal and response Lyapunov exponents and fractal dimension of the attractor These studies show differences in the stability of synchronous states between smooth (logistic map, Lorenz system) and non-smooth (Chua circuit) oscillators.

Detection of the generalized synchronization

Numerical studies

Logistic maps

Lorenz oscillators

Chua circuits

Summary and conclusions