Hidden oscillations and hidden attractors in dynamical systems: systems with no equilibria, coexistence of attractors and multistability, Aizerman's and Kalman's conjectures on absolute stability, Chua circuits

Keywords: hidden oscillation, hidden attractor, periodic solution, limit cycle, systems without equilibria, multistability, coexistence of attractors, absolute stability, 16^th Hilbert problem, absolute stability, Aizerman’s conjecture, Kalman’s conjecture, describing function method (DFM), harmonic balance

During initial establishment and development of theory of nonlinear oscillations in the first half of twentieth century a main attention has been given to analysis and synthesis of oscillating systems for which the solution of problems of existence of oscillating solutions was not too difficult. The structure itself of many systems was such that they have oscillating solutions, the existence of which was "almost obvious". The arising in these systems periodic solutions were well seen by numerical analysis when numerical procedure of integrating the trajectories allowed one to pass from small neighborhood of equilibrium to periodic trajectory.

Further there came to light so called hidden oscillations and hidden attractors – oscillations, which are "small" (and, therefore, are difficult for numerical analysis) or are not "connected" with equilibrium (i.e. when the creation of numerical procedure of integration of trajectories for the passage from equilibrium to periodic solution is impossible because the neighbourhoods of equilibria do not belong to such attractor).

For the first time the problem of finding hidden oscillations had been stated by D. Hilbert in 1900 (Hilbert's 16th problem) for two-dimensional polynomial systems. For a more than century history, in the framework of the solution of this problem the numerous theoretical and numerical results were obtained. However the problem is still far from being resolved even for the simple classes of systems. Thus, the problem of finding quadratic systems, in which there exists a limit cycle, is nontrivial. In 40-50s of the 20th century A.N. Kolmogorov became the initiator of a few hundreds of computational experiments, in the result of which the limit two-dimensional quadratic systems would been found. The result was absolutely unexpected: in not a single experiment a limit cycle was found [Arnold V.I., Experimental Mathematics, 2005], though it is known that quadratic systems with limit cycles form open domains in the space of coefficients and, therefore, for a random choice of polynomial coefficients, the probability of hitting in these sets is positive.

Further the problem of analysis of hidden oscillations arose in applied problems of automatic control. In the process of investigation, connected with Aizerman's (1949) and Kalman's (1957) conjectures, it was stated that the differential equations of systems of automatic control, which satisfy generalized Routh-Hurwitz stability criterion, can also have hidden periodic regimes.

At present the new analytic-numerical approaches to investigation, of hidden oscillations in dynamical systems, were developed based on the development of numerical methods, computers, and applied bifurcation theory, which suggests revisiting and revising early ideas on the application of the small parameter method and the harmonic linearization.

Publications

1. G.A. Leonov, N.V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, International Journal of Bifurcation and Chaos, 23(1), 2013, art. no. 1330002 (doi: 10.1142/S0218127413300024