Topological synchronization of chaotic systems

Abstract

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor. Here we analyze the relation between the two emergent phenomena by shifting the descriptive level of topological synchronization to account for the multifractal nature of the visited attractors. Namely, we measure the generalized dimension of the system and monitor how it changes while increasing the coupling strength. We show that during the gradual process of topological adjustment in phase space, the multifractal structures of each strange attractor of the two coupled oscillators continuously converge, taking a similar form, until complete topological synchronization ensues. According to our results, chaotic synchronization has a specific trait in various systems, from continuous systems and discrete maps to high dimensional systems: synchronization initiates from the sparse areas of the attractor, and it creates what we termed as the ‘zipper effect’, a distinctive pattern in the multifractal structure of the system that reveals the microscopic buildup of the synchronization process. Topological synchronization offers, therefore, a more detailed microscopic description of chaotic synchronization and reveals new information about the process even in cases of high mismatch parameters.

Adaptive coexistence of synchronization and anti-synchronization for a class of switched chaotic systems

“…Theorem 11 Considering the master unified chaotic system (19), if the controller u = (0, k 1 E 2 , 0) T is added into the slave chaotic system (20), where the feedback gain k 1 is updated as (28), then the error system ( 22) is asymptotically stable, i.e. the master unified chaotic system (19) and the slave chaotic system (20) can realize CSAS.…”

“…In the past two decades, increasing attention has been paid to the analysis and synthesis of switched systems due to their significance in both theory and applications, and many significant results have been obtained for the analysis and design of switched systems. In the year of 1990 13 , the discovery of chaos synchronization by Pecora and Carroll greatly enhanced the research enthusiasm for the chaos control problem, such as the complete synchronization [14][15][16] , anti-synchronization [17][18][19] , phase synchronization 20,21 , lag synchronization 22,23 , generalized synchronization 24,25 , projective synchronization 26,27 , topological synchronization 28 , etc. Simultaneously, lots of useful results have been obtained.…”

Topological synchronisation or a simple attractor?

“…The recent paper [1], which also garnered some press attention [2,3], introduced the concept of topological synchronisation which occurs when, in a dynamical system, it is possible to identify two or more attractors which become very similar when the system evolves. This situation is for instance met in coupled lattice map, where each site of the lattice brings its own attractor.…”

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A few recent papers introduced the concept of topological synchronisation. We refer in particular to (Lahav et al 2022 Sci. Rep. 12 2508, where the theory was illustrated by means of a skew product system, coupling two logistic maps.

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Dominant Attractor in Coupled Non-Identical Chaotic Systems

“…Chaos theory is an inviting subject of study in nonlinear dynamics. The concepts in line with chaotic dynamics, such as strange attractors, fractal dimensions, Lyapunov exponents, and synchronization, have been extensively addressed in every research field [ 1 , 2 ]. For example, in neuroscience, the presence of deterministic chaos has been proved both experimentally and theoretically [ 3 , 4 ].…”