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Non-identical coupled logistic maps

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In a study of coupled systems with complex dynamics the following approach appears to be productive. Let each of the subsystems is characterized by a parameter responsible for transition to chaos. Then, it is convenient to analyze the arrangement of the plane of two parameters of the subsystems. In this context, the simplest model associated with transition to chaos in coupled period-doubling systems is represented by two non-identical coupled logistic maps

On the parameter plane (l₁, l₂) different colors designate periodic regimes (green - период 1, yellow - 2, blue - 4, red - 8, etc.), gray indicates chaos, and white the quasiperiodicity. The chart is plotted for e=0.4.

This system demonstrates two scenarios of transition to chaos: near the main diagonal l₁=l₂ the destruction of quasiperiodicity is observed, and for a case of essential difference of parameters l₁ and l₂ the period doubling transition occurs. The magnified fragment shows a typical for this system synchronization tongue.

Of particular interest is a region on the parameter plane where the approaching each other Feigenbaum critical lines undergo a break and form some bounded domain of quasiperiodic dynamics containing inside a system of Arnold tongues. Some details see in paper of A.P. Kuznetsov, J.V. Sedova, I.R. Sataev: Arrangement of the parameter space of non-identical coupled period doubling systems. (In Russian.)

The picture of transitions to chaos in coupled identical maps is discussed here.

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