Properties of generalized synchronization in smooth and non-smooth identical oscillators
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https://doi.org/10.1140/epjst/e2020-000010-5
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Properties of generalized synchronization in smooth and non-smooth identical oscillators M. Balcerzak, A. Chudzik, and A. Stefanskia Division of Dynamics, Lodz University of Technology, ul. Stefanowskiego 1/15 Lodz, Poland Received 24 January 2020 / Accepted 8 June 2020 Published online 28 September 2020 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.
1 Introduction In analysis and theory of dynamical systems, the research of interactions between them plays an important role. Such interactions often lead to an appearance of some synchronization effects. First time this phenomenon has been observed by C. Huygens in the second half of 17th century [1] and it was [2–4] and still is actively studied by many researchers to this day. Especially interesting are oscillators exhibiting chaotic or stochastic dynamics. In recent years, the chaotic synchronization has become an object of great interest in many areas of science [5–11]. Consequently, a number of new types of synchronization have been also identified, e.g., generalized synchronization [12,13], phase [14–16] and imperfect phase synchronization [17,18], anticipated synchronization [19,20] and ragged synchronizability [21]. One of the most interesting proposals appearing in the context of chaos synchronization, is a concept called the generalized synchronization (GS). This term has been introduced by Rulkov et al. [12] as a generalization of the synchronization idea for unidirectionally coupled systems:
a
x˙ = f (x),
(1a)
y˙ = g(x, y),
(1b)
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The European Physical Journal Special Topics
Fig. 1. Common drive – a star configuration with a unidirectional coupling.
where x ∈ Rm , y ∈ Rk . Such unidirectional link of dynamical systems is also called the master–slave coupling. We can say that the GS of these systems occurs if there exists a static functional relation ψ between their states, i.e., y(t) = ψ[x(t)].
(2)
In general, the GS phenomenon has been considered both in the context of identical (when separated) systems (1a) and (1b), and also 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 [13,22–24]. This phenomenon can be also observed in discrete time systems [25–28]. However, in any of these cases the phenomenon of co
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