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Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 447--455 | DOI:10.5890/DNC.2016.12.009

M. Belger$^{1}$, S. De Nigris$^{2}$, X. Leoncini$^{1}$,$^{3}$

$^{1}$ Aix Marseille Univ, Univ Toulon, CNRS, CPT, Marseille, France

$^{2}$ Department of Mathematics and Namur Center for Complex Systems-naXys, University of Namur, 8 rempart de la Vierge, 5000 Namur, Belgium

$^{3}$ Center for Nonlinear Theory and Applications, Shenyang Aerospace University, Shenyang 110136, China

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Abstract

The so-called chaotic states that emerge on the model of XY interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic time-scales to emerge. We find that these time scales τ (N) display a critical slowing down, i.e they diverge as N →ꝏ. The scaling law is analyzed for different energy densities and the behavior τ (N) ∼ √ N is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamicequations of the motion and analyzing the eigenvalues of the adjacency matrix.

Acknowledgments

S.D.N and X.L. would like to thank S. Ogawa for fruitful discussions and remarks during the preparation of this manuscript.

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