Stochastic deformations of coupling-induced oscillatory regimes in a system of two logistic maps

Abstract

We consider a system of two coupled identical logistic maps, which in isolation demonstrate stable equilibrium modes. Under increasing coupling strength, this system exhibits transitions from initial equilibrium regime to synchronized oscillatory behavior with complex modes, both regular (periodic or quasiperiodic) and chaotic. Moreover, the coupled subsystem can demonstrate synchronization with anti-phase oscillations in tonic or burst form. A novelty of this paper is related to analysis of noise-induced deformations of these corporative oscillatory regimes. For this model, the following constructive stochastic effects are studied: (i) noise-induced temporal destruction of the anti-phase synchronization with transition from tonic to burst oscillations, (ii) noise-induced destruction of 2-torus with the transition from quasiperiodic oscillations to the bursting, (iii) stochastic transformations of burst oscillations from regular to chaotic. An important role of transients and “riddled” basins in the study of stochastic shifts of crisis bifurcation points is discussed. For the analytical investigation of these phenomena, a method based on the stochastic sensitivity of attractors (discrete cycles and tori) and confidence domains is applied.