Global Analysis of Synchronization in Coupled Maps

Abstract

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.

Fractal Stationary Density in Coupled Maps

“…Using the linear stability analysis it can be shown [8] that this coupled map system synchronizes when 1/4 < ǫ < 3/4. The same result was also obtained by studying the evolution of the support of the invariant measure [7] which is a global result as opposed to the linear stability analysis which is carried out near the synchronized solution. This was done by showing that if ǫ < 1/4 we obtain a quadrilateral of nonzero area as the support of the invariant measure.…”

“…A pointwise Hölder exponent α p (x 0 ) at x 0 is the supremum of the αs for which the inequality (7) holds. We also use the relation between the Hölder continuity and the box dimension of the graph a function, dim B graphf .…”

“…As a next step, one can then consider various coupling topologies. We have demonstrated in [7] that the result of such a system of two coupled maps can be used to derive the result for a globally coupled network of N maps. Here we are concerned with the phenomena of complete synchronization, that is, the dynamics of two systems becomes completely identical after the coupling parameter crosses a certain critical value.…”

Universal Multifractal Stationary Densities in Expanding Piecewise Linear Coupled Maps

“…One can also notice some discontinuities in the measure. In [18], the support of this invariant measure was used to carry out global analysis of synchronization. One can understand the origin of these discontinuties from the analysis carried out there.…”