Stochastic and Nonlinear Control of Attractor Preference*

Studying the response of multistable systems to stochastic and harmonic perturbations, we observe a surprising and counterintuitive asymptotic behavior, including nonmonotonic dependence in preference of some coexisting attractors with respect to the noise amplitude, i.e., the existence of a certain noise level for which the most probable size of the basin of attraction is larger than for other noise values, while the average size always decreases as the noise amplitude increases. Since the probabilistic size of the basin of attraction depends on the amplitude and frequency of external modulation applied to a system parameter, noise, periodic modulation, and a combination of both provide an efficient control of attractor preference in systems with multiple coexisting states. Such a control is demonstrated in the delayed feedback logistic map with two coexisting attractors and in the Hénon map with three attractors.