Constructing Generalized Synchronization Manifolds by Manifold Equation

TL;DR

This paper develops a PDE-based method to explicitly construct generalized synchronization manifolds in coupled non-identical dynamical systems, advancing understanding beyond the identical case and enabling targeted synchronization design.

Contribution

It introduces a novel PDE approach for computing generalized synchronization manifolds and addresses boundary condition issues, with applications to complex systems like Van der Pol oscillators.

Findings

Derived PDEs whose stationary solutions represent synchronization manifolds

Developed a numerical scheme with boundary condition handling

Applied method to coupled Van der Pol oscillators

Abstract

Full understanding of synchronous behavior in coupled dynamical systems beyond the identical case requires an explicit construction of the generalized synchronization manifold, whether we wish to compare the systems, or to understand their stability. Nonetheless, while synchronization has become an extremely popular topic, the bulk of the research in this area has been focused on the identical case, specifically because its invariant manifold is simply the identity function, and there have yet to be any generally workable methods to compute the generalized synchronization manifolds for non-identical systems. Here, we derive time dependent PDEs whose stationary solution mirrors exactly the generalized synchronization manifold, respecting its stability. We introduce a novel method for dealing with subtle issues with boundary conditions in the numerical scheme to solve the PDE, and we develop first order expansions close to the identical case. We give several examples of increasing sophistication, including coupled non-identical Van der Pol oscillators. By using the manifold equation, we also discuss the design of coupling to achieve desired synchronization.