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Critical parameters of the synchronisation's stability for coupled maps in regular graphs

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  • Gancio, Juan
  • Rubido, Nicolás

Abstract

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters – coupling strength, maximum Lyapunov exponent, and link density – that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: k-cycles (i.e., regular lattices of even degree k, which can be embedded in Tk tori) and k-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree k = 3. Our results highlight differences in the synchronisation manifold's stability of these graphs – even for identical node degrees – in the finite size and infinite size limit.

Suggested Citation

  • Gancio, Juan & Rubido, Nicolás, 2022. "Critical parameters of the synchronisation's stability for coupled maps in regular graphs," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
  • Handle: RePEc:eee:chsofr:v:158:y:2022:i:c:s0960077922002119
    DOI: 10.1016/j.chaos.2022.112001
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    References listed on IDEAS

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    Cited by:

    1. Gancio, Juan & Rubido, Nicolás, 2024. "Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    2. Yao, Xiao-Yue & Li, Xian-Feng & Jiang, Jun & Leung, Andrew Y.T., 2022. "Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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