Multifractal Invariant Measure of Coupled Chaotic Logistic Maps

Coupled logistic map systems have been used in a wide variety of fields including population dynamics, COVID detection, memristors, chemical physics, detection of natural resources such as oil and digital currency forecasting, to name a few. One phenomenon which is of considerable interest in coupled systems is synchronization and then the question of the critical coupling constant beyond which the system synchronizes. The usual method to estimate this coupling constant is via a linear stability analysis of the synchronization state which is local. It is necessary to develop global methods to study the synchronization transition and one possible way can be through the study of invariant measures. With this general aim, we study dissipatively coupled chaotic logistic map systems. The invariant measure of such a system of two coupled logistic maps is generated numerically in a regime where the transition to chaotic asynchronous state has taken place and is shown to have a multifractal nature. We have considered both types of systems in which the parameters of the logistic maps are identical as well as nonidentical. The invariant measure restricted to the synchronization manifold is then analyzed using the wavelet leaders algorithm preceded by fractional integration. The obtained multifractal spectrum is studied while varying the coupling strength. The results obtained are discussed.