Analytical-Numerical Methods for Hidden Attractors’ Localization: The 16th Hilbert Problem, Aizerman and Kalman Conjectures, and Chua Circuits

Abstract

This survey is devoted to analytical-numerical methods for hidden attractors’ localization and their application to well-known problems and systems. From the computation point of view, in nonlinear dynamical systems the attractors can be regarded as self-exciting and hidden attractors. Self-exciting attractors can be localized numerically by the following standard computational procedure : after a transient process a trajectory, started from a point of an unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it. In contrast, a hidden attractor is an attractor whose basin of attraction does not contain neighborhoods of equilibria. In well-known Van der Pol, Belousov-Zhabotinsky, Lorenz, Chua, and many other dynamical systems classical attractors are self-exciting attractors and can be obtained numerically by the standard computational procedure. However, for localization of hidden attractors it is necessary to develop special analytical-numerical methods, in which at the first step the initial data are chosen in a basin of attraction and then the numerical localization (visualization) of the attractor is performed. The simplest examples of hidden attractors are internal nested limit cycles (hidden oscillations) in two-dimensional systems (see, e.g., the results concerning the second part of the 16th Hilbert’s problem). Other examples of hidden oscillations are counterexamples to Aizerman’s conjecture and Kalman’s conjecture on absolute stability in the automatic control theory (a unique stable equilibrium coexists with a stable periodic solution in these counterexamples). In 2010, for the first time, a chaotic hidden attractor was computed first by the authors in a generalized Chua’s circuit and then one chaotic hidden attractor was discovered in a classical Chua’s circuit.