The theory of hidden oscillations and stability of dynamical systems

The development of the theory of global stability, the theory of bifurcations, the theory of chaos, and new computing technologies made it possible to take a fresh look at a number of well-known theoretical and practical problems in the analysis of multidimensional dynamical systems and led to the emergence of the theory of hidden oscillations which represents the genesis of the modern era of Andronov’s theory of oscillations. The theory of hidden oscillations is based on a new classification of attractors as self-excited or hidden. While trivial attractors (equilibrium points) can be easily found analytically or numerically, the search for periodic or chaotic attractors may turn out to be a challenging problem (see, e.g. famous 16th Hilbert problem on the number and disposition of limit periodic oscillations in two-dimensional polynomial systems which is still unsolved). Self-excited attractors can be easily discovered when observing numerically the dynamics with initial data from the vicinity of the equilibria. While hidden attractors have basins of attraction, which are not connected with equilibria, and their search requires the development of special analytical and numerical methods.
For various applications, the transition of the system’s state to a hidden attractor, caused by external disturbances, may result in undesirable behavior and is often the cause of accidents and catastrophes. For various engineering applications the importance of identifying hidden attractors is related to the classical problems of determining the boundaries of global stability in the space of parameters and in the phase space. Outer estimations of the global stability boundary in the space of parameters and the birth of self-excited oscillations in the phase space can be obtained by the linearization around equilibria and the analysis of local bifurcations and are related to various well-known conjectures on global stability by the first approximation (see, e.g. Andronov’s proof of the conjecture on the Watt regulator global stability by the first approximation, Aizerman and Kalman conjectures). Inner estimations of the global stability boundary can be obtained by classical sufficient criteria of global stability. In the gap between outer and inner estimations, there is an exact boundary of global stability which study requires the analysis of nonlocal bifurcations and hidden oscillations.
This lecture is devoted to well-known theoretical and practical problems in which hidden attractors (their absence or presence and disposition) play an important role [1-7].