Microscopic description of chaotic systems' synchronization: Gradual process of topological adjustment in phase space, in which the strange attractors of the two coupled systems continuously converge

Synchronization of chaotic systems: A microscopic description. Nir Lahav, Irene Sendina-Nadal, Chittaranjan Hens, Baruch Ksherim, Baruch Barzel, Reuven Cohen, and Stefano Boccaletti. Phys. Rev. E 98, 052204, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.052204

Abstract: The synchronization of coupled chaotic systems represents a fundamental example of self organization and collective behavior. This well-studied phenomenon is classically characterized in terms of macroscopic parameters, such as Lyapunov exponents, that help predict the system's transitions into globally organized states. However, the local, microscopic, description of this emergent process continues to elude us. Here we show that at the microscopic level, synchronization is captured through a gradual process of topological adjustment in phase space, in which the strange attractors of the two coupled systems continuously converge, taking similar form, until complete topological synchronization ensues. We observe the local nucleation of topological synchronization in specific regions of the system's attractor, providing early signals of synchrony, that appear significantly before the onset of complete synchronization. This local synchronization initiates at the regions of the attractor characterized by lower expansion rates, in which the chaotic trajectories are least sensitive to slight changes in initial conditions. Our findings offer an alternative description of synchronization in chaotic systems, exposing its local embryonic stages that are overlooked by the currently established global analysis. Such local topological synchronization enables the identification of configurations where prediction of the state of one system is possible from measurements on that of the other, even in the absence of global synchronization.