Fractal Stationary Density in Coupled Maps

Abstract

Summary. We study the invariant measure or the stationary density of a coupled discrete dynamical system as a function of the coupling parameter ǫ (0 < ǫ < 1/4). The dynamical system considered is chaotic and unsynchronized for this range of parameter values. We find that the stationary density, restricted on the synchronization manifold, is a fractal function. We find the lower bound on the fractal dimension of the graph of this function and show that it changes continuously with the coupling parameter.

Global Analysis of Synchronization in Coupled Maps

“…A characterization of complete measures is also needed. We have already taken a step in this direction [39]. It is important to note that the knowledge of the complete measure was not necessary to study the synchronization since we used only the support of the measure.…”