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C-oscillators and stability of stationary cluster structures in lattices of diffusively coupled oscillators

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  • Verichev, Nikolai N.
  • Verichev, Stanislav N.
  • Wiercigroch, Marian

Abstract

This paper studies the conditions for existence and stability of stationary cluster structures in lattices of diffusively coupled dynamical systems within the framework of a new interpretation of cluster synchronization as classical synchronization of cluster oscillators (C-oscillators). The study of existence of cluster attractors is based on the linear chains of cluster oscillators, defining possible types of cluster structures in chains. First, we present interval estimates for the range of coupling strengths in which cluster attractors can exist. Then we formulate and prove the basic theorems about the local stability of the various cluster structures. The presented methodology can be extended to study cluster structures on lattices of different geometry and forms such as linear cluster structures in two-dimensional lattices, layered cluster structures in three-dimensional lattices and cluster structures in ring-shaped systems.

Suggested Citation

  • Verichev, Nikolai N. & Verichev, Stanislav N. & Wiercigroch, Marian, 2009. "C-oscillators and stability of stationary cluster structures in lattices of diffusively coupled oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 686-701.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:2:p:686-701
    DOI: 10.1016/j.chaos.2009.01.041
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    1. Verichev, Nikolai N. & Verichev, Stanislav N. & Wiercigroch, Marian, 2007. "Physical interpretation and theory of existence of cluster structures in lattices of dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 34(4), pages 1082-1104.
    2. Verichev, Nikolai N. & Verichev, Stanislav N. & Wiercigroch, Marian, 2009. "Asymptotic theory of chaotic synchronization for dissipative-coupled dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 752-763.
    3. Belykh, V.N. & Belykh, I.V. & Nelvidin, K.V., 2002. "Spatiotemporal synchronization in lattices of locally coupled chaotic oscillators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(4), pages 477-492.
    4. Andrzej Stefanski & Tomasz kapitaniak, 2000. "Using chaos synchronization to estimate the largest lyapunov exponent of nonsmooth systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 4, pages 1-9, January.
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