Steady-state Entropy of Dynamical Systems Described by Intermittent Iterated Maps

A stochastic perturbation applied to a set of randomly-phased, non-interacting oscillators is known to induce synchronization provided the noise is very weak [1]. We use iterated phase maps to describe the dynamics of nonlinear oscillators in response to intermittent impulsive forcing. That is, we repeatedly subject all the oscillators to a common phase map imposed at random times. This synchronization can be exploited to create orientational order in a dispersion of forced, rotating, asymmetric colloidal particles [2]. Here we study the case where the forcing creates strong but incomplete synchronization. The strength of the forcing can be characterized by the average Lyapunov exponent Λ of its phase map. We demonstrate a regime of small, positive Λ in which the distribution of oscillator phases and its corresponding entropy H fluctuates wildly. We characterize this novel state numerically for a class of phase maps. The distribution of entropies achieves an apparent steady-state consistent with the form e^HΛ/C, where C is a constant. For small Λ, H shows apparent 1/Λ behavior.

1. Nakao et al. Phys. Rev. E, 2005
2. Eaton et al. Phys. Rev. E, 2016