Synchronization of chaotic systems: A microscopic description

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 a fresh and novel 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.

An Integrated World Modeling Theory (IWMT) of Consciousness: Combining Integrated Information and Global Neuronal Workspace Theories With the Free Energy Principle and Active Inference Framework; Toward Solving the Hard Problem and Characterizing Agentic Causation

“…This generalized synchrony (Strogatz, 2012 ) has also been characterized in thermodynamic terms (Kachman et al, 2017 ; Friston, 2019 ), where systems spontaneously self-organize into resonant modes with the environments with which they couple—i.e., absorb work and minimize free energy according to Hamilton's principle of least action—where coordinated dynamics have been observed to contain mutually predictive information (Friston, 2013 ). Notably, coupled attractors have recently been found to adjust their dynamics beginning at sparsely frequented areas of phase space (Lahav et al, 2018 ). If these synchronizing manifolds begin to nucleate from improbable (and so surprising) alignments, this flow of (mutual-information maximizing) influence might be functionally understood as updating via “prediction-errors.” While admittedly speculative, these considerations suggest that generalized HPP (and selection for integrated information) could represent a universality class whose potential extensions are nearly as widespread as generalized synchrony itself.…”

On the importance of being flexible: dynamic brain networks and their potential functional significances

“…To avoid this maximally likely outcome of monotonically increasing disorder, systems must exhibit intelligent adaptivity in exchanging matter and energy with their environments. While some systems are simply subject to being entrained by environmental stochasticity and some dynamical rule, such as pendulums (Kirchhoff et al, 2018;Lahav et al, 2018;Oliveira & Melo, 2015), other systems can interact with their environment in order to achieve more desirable states for survival. Living systems such as cells, organs, and organisms, keep surprisal (i.e., cybernetic entropy) at bay by engaging in behaviour that translates into stable minimal points (i.e., uncertainty minima) for a set of potential states bounded by minima and maxima that can mean life and death, respectively.…”

Integrated World Modeling Theory (IWMT) Revisited

“…6. During periods of minimal synchronization, we may expect synchronizing ensembles to be maximally sensitive to signals at any phase (Lahav et al, 2018), but with minimal abilities to drive coupling systems. 7.…”