Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors

Abstract

From a computational point of view it is natural to suggest the classification of attractors, based on the simplicity of finding basin of attraction in the phase space: an attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a self-excited attractor. Self-excited attractors can be localized numerically by the standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of unstable equilibrium, is attracted to the state of oscillation and traces it. Thus one can determine unstable equilibria and check the existence of self-excited attractors. In contrast, for the numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures in which an initial point can be chosen from the basin of attraction analytically. For example, hidden attractors are attractors in the systems with no-equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors); hidden attractors arise in the study of well-known fundamental problems such as 16th Hilbert problem, Aizerman and Kalman conjectures, and in applied research of Chua circuits, phase-locked loop based circuits, aircraft control systems, and others.