Stability of synchronized dynamics and pattern formation in coupled systems: review of some recent results

Chen, Yonghong ; Rangarajan, Govindan ; Ding, Mingzhou (2006) Stability of synchronized dynamics and pattern formation in coupled systems: review of some recent results Communications in Nonlinear Science and Numerical Simulation, 11 (8). pp. 934-960. ISSN 1007-5704

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.cnsns.2005.01.011

Abstract

In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. These two problems are often treated separately in the literature. Here, we present a unified framework in which we show that the eigenvalues of the coupling matrix determine the stability of the synchronized state, while the eigenvectors correspond to patterns emerging from desynchronization. Based on this simple framework three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable spatial patterns can be selectively realized by varying the coupling strengths. Distinct temporal evolution of the spatial pattern can be obtained depending on the bifurcating synchronized state. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves. Systems with specific coupling schemes are used as examples to illustrate the general methods.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Coupled Dynamical Systems; Chaotic Synchronization; Generalized Turing Patterns
ID Code:73223
Deposited On:02 Dec 2011 09:51
Last Modified:02 Dec 2011 09:51

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