Abstract: The problem of hidden oscillations in nonlinear control systems forces to develop new approaches of nonlinear oscillation theory. During initial establishment and development of theory of nonlinear oscillations in the first half of 20th century a main attention has been given to analysis and synthesis of oscillating systems for which the solution of existence problems of oscillating regimes was not too difficult. The structure itself of many systems was such that they had oscillating solutions, the existence of which was "almost obvious". The arising in these systems periodic solutions were well seen by numerical analysis when numerical integration procedure of the trajectories allowed one to pass from small neighborhood of equilibrium to periodic trajectory. Therefore main attention of researchers was concentrated on analysis of forms and properties of these oscillations (the "almost" harmonic, relaxation, synchronous, circular, orbitally stable ones, and so on).
Further there came to light so called hidden oscillations – the oscillations, the existence itself of which is not obvious (which are "small" and, therefore, are difficult for numerical analysis or are not "connected" with equilibrium i.e. the creation of numerical procedure of integration of trajectories for the passage from equilibrium to periodic solution is impossible). So in the midpoint of twentieth century M.A.Aizerman and R.E.Kalman formulated two conjectures, which occupy, at once, attention of many famous scholars.
Similar situation is in attractors localization. The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria.
In this presentation the application of special analytical-numerical algorithms for hidden oscillations and hidden attractor localization are discussed. Construction of counterexamples for Aizerman's and Kalman's conjectures, and existence of hidden attractor in Chua's systems are demonstrated.

Brief Biography of the Speaker:
Gennady A. Leonov received his PhD (Candidate Degree) in mathematical cybernetics from Saint-Petersburg State University in 1971 and Dr.Sci. in 1983.
From 1985 – he is full professor at the Mathematics and Mechanics Faculty. He has been vice-rector of Saint-Petersburg State University from 1986 to 1988.
Now Gennady A. Leonov is Dean of Mathematics and Mechanics Faculty (since 1988), Director of Research Institute of Mathematics and Mechanics of St.-Petersburg State University (since 2004), Head of Applied cybernetics Department (since 2007).
Professor G.A. Leonov was awarded Prize of St.-Petersburg State University (1985), State Prize of USSR (1986), Prize of Technische Universitet Dresden (1990).
He is member (corresponding) of Russian Academy of Science, member of the Russian National Committee of Theoretical Mechanics, member of Directorate of St.-Petersburg Mathematical Society.
Professor G.A. Leonov authored and co-authored 300 books and papers. His research interests, now in qualitative theory of dynamical systems, stabilization, nonlinear analysis of phase synchronization systems and electrical machines.