Topological synchronization of chaotic systems

Nir Lahav,Irene Sendiña-Nadal,Reuven Cohen,Baruch Barzel,Chittaranjan Hens,Baruch Ksherim,Stefano Boccaletti

doi:10.1038/s41598-022-06262-z

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.

Highlights

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure
Local synchronization initiates in the sparse areas of the attractor and as the local synchronizations accumulate, phase synchronization occurs for σps ≥ 0.1 and complete synchronization is obtained for σcs ≥ 2.0
The zipper effect, implies that as in the Rössler case, in the Logistic map case, topological synchronization starts at low coupling strength at areas of the attractor that have a low probability of points and, only when these areas complete their local synchronization, the attractor will topologically synchronize in areas with a high probability of points at a large coupling. In both discrete map and continuous systems, we see the same distinctive pattern of the zipper effect in the multifractal structure, where topological synchronization starts from the sparse to the dense areas of the attractor, suggesting that this trait might be an essential feature in chaotic synchronization

Summary

Introduction

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. The multifractal structures of each strange attractor of the two coupled oscillators continuously converge to a similar form until complete topological synchronization ensues.

Results

Conclusion