Coexisting attractors in periodically modulated logistic maps

Abstract

We consider the logistic map wherein the nonlinearity parameter is periodically modulated. For low periods, there is multistability, namely two or more distinct dynamical attractors coexist. The case of period 2 is treated in detail, and it is shown how an extension of the kneading theory for one-dimensional maps can be applied in order to analyze the origin of bistability, and to demarcate the principal regions of bistability in the phase space. When the period of the modulation is increased—and here we choose periods which are the Fibonacci numbers—the measure of multistable regions decreases. The limit of quasiperiodic driving is approached in two different ways, by increasing the period and keeping the drive dichotomous, or by increasing the period and varying the modulation sinusoidally. In the former case, we find that multistability persists in small regions of the phase space, while in the latter, there is no evidence of multistability but strange nonchaotic attractors are created.