Constructing generalized synchronization manifolds by manifold equation

Sun Jie*, Erik M. Bollt, Takashi Nishikawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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 for computing the generalized synchronization manifolds for nonidentical 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 nonidentical Van der Pol oscillators. By using the manifold equation, we also discuss the design of coupling to achieve desired synchronization.

Original languageEnglish (US)
Pages (from-to)202-221
Number of pages20
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
StatePublished - 2009


  • Computational invariant manifold
  • Coupled dynamical systems
  • Generalized synchronization

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation


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