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Chaos and Hyperchaos in a Symmetric Coupling of Three Quadratic Maps

doi: 10.6062/jcis.2010.01.03.0024(Free PDF)


Joao C. Xavier and Paulo C. Rech


In this paper we investigate the dynamical behavior of a symmetric coupling of three quadratic maps, and identify various interesting features as a function of the nonlinearity and the coupling parameters. In particular, we study the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable periodic orbits pertaining to 1×2 n cascade, namely period-1 and period-2 orbits. Lyapunov exponent plots, parameter-space and three-dimensional phase-space portraits, and bifurcation diagrams are used to study the transition from periodic to quasiperiodic states, from quasiperiodic to mode-locked and to chaotic states, and from chaotic to hyperchaotic states. We also investigate the modifications introduced in the parameter-space, by the increase of the number of maps in the coupling.


Naimark-Sacker bifurcation, Lyapunov exponents, hyperchaos..


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