Switching thresholds for multistable systems under strong external perturbation

Communications in Nonlinear Science and Numerical Simulation. 2020. Vol. 83. P. 105067.

Klinshov V., Некоркин В. И.

Multistability is a common feature of various dynamical systems which manifests itself as the possibility to demonstrate various behaviors for the same parameter values. These behaviors or states of the system must be stable against weak perturbation in order to be observable in real life. However, a strong enough perturbation may destroy a certain state and lead to the system switching to another one. From the viewpoint of nonlinear dynamics, the response of a multistable system to a strong stimulus depends on the configuration and the mutual arrangement of basins of different attractors. In the present paper, we introduce a novel measure for characterization of a multistable system, the switching threshold. It equals the amplitude of a minimal perturbation capable of switching the system from one attractor to another. We develop a numerical algorithm for calculation of the switching thresholds and apply it to a number of paradigmatic models including dynamical networks. We show that the values of switching thresholds provide important information about multistable systems and their responses to external stimuli. This information allows to develop methods of optimal control of multistable systems by external signals. Surprisingly, it also allows to predict some features of their dynamics under the influence of external noise.

Keywords: Noise, сеть Хопфилда, multistability, мультистабильность, шум, Hoppfield network