Naturally invariant measure of chaotic attractors and the conditionally invariant measure of embedded chaotic repellers

We study local and global correlations between the naturally invariant measure of a chaotic one-dimensional map f and the conditionally invariant measure of the transiently chaotic map f_H. The two maps differ only within a narrow interval H, while the two measures significantly differ within the images f^l(H), where l is smaller than some critical number l_c. We point out two different types of correlations. Typically, the critical number l_c is small. The χ^2 value, which characterizes the global discrepancy between the two measures, typically obeys a power-law dependence on the width ε of the interval H, with the exponent identical to the information dimension. If H is centered on an image of the critical point, then l_c increases indefinitely with the decrease of ε, and the χ^2 value obeys a modulated power-law dependence on ε

The Stability and Control of Stochastically Switching Dynamical Systems

Inherent randomness and unpredictability is an underlying property in most realistic phenomena. In this work, we present a new framework for introducing stochasticity into dynamical systems via intermittently switching between deterministic regimes. Extending the work by Belykh, Belykh, and Hasler, we provide analytical insight into how randomly switching network topologies behave with respect to their averaged, static counterparts (obtained by replacing the stochastic variables with their expectation) when switching is fast. Beyond fast switching, we uncover a highly nontrivial phenomenon by which a network can switch between two asynchronous regimes and synchronize against all odds. Then, we establish rigorous theory for this framework in discrete-time systems for arbitrary switching periods (not limited to switching at each time step). Using stability and ergodic theories, we are able to provide analytical criteria for the stability of synchronization for two coupled maps and the ability of a single map to control an arbitrary network of maps. This work not only presents new phenomena in stochastically switching dynamical systems, but also provides the first rigorous analysis of switching dynamical systems with an arbitrary switching period

Escape rates and conditionally invariant measures

We consider dynamical systems on domains that are not invariant under the dynamics—for example, a system with a hole in the phase space—and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue. Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject

Synchronization of fluctuating delay-coupled chaotic networks

We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. Focusing on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone, we compare the synchronization properties of static and fluctuating networks in the regime of large delays. We find that random network switching may enhance the stability of synchronized states. Synchronization appears to be maximally stable when fluctuations are much faster than the time-delay, whereas it disappears for very slow fluctuations. For fluctuation time scales of the order of the time-delay, we report a resynchronizing effect in finite-size networks. Moreover, we observe characteristic oscillations in all regimes, with a periodicity related to the time-delay, as the system approaches or drifts away from the synchronized state