Stochastic control of attractor preference in a multistable system

When talking about the size of basins of attraction of coexisting states in a noisy multistable system, one can only refer to its probabilistic properties. In this context, the most probable size of basins of attraction of some coexisting states exhibits an obvious non-monotonous dependence on the noise amplitude, i.e., there exists a certain noise level for which the most probable basin’s size is larger than for other noise values, while the average size always decreases as the noise amplitude increases. Such a behavior is demonstrated through the study of the Hénon map with three coexisting attractors (period 1, period 3, and period 9). Since the position of the probabilistic extrema 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 a system with multiple coexisting states.