The shadowing problem is related to the following question: under which condition, for any pseudotrajectory approximate trajectory) of a vector field there exists a close trajectory? It is known that in a neighbourhood of a hyperbolic set diffeomorphisms and vector fields have shadowing property. In fact more general statement is correct: structurally stable dynamical systems satisfy shadowing property.

We are interested if converse implication is correct. We consider notion of Lipschitz shadowing property and proved that it is equivalent to structural stability for the cases of diffeomorphisms and vector fields.

Talk is based on joint works with S. Pilyugin and K. Palmer

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model.

By reducing the problem to a discontinuous infinite dynamical system, we construct all periodic solutions with exactly two switchings on the period and study their stability. In the problem under consideration, the hysteresis gap (the difference between the switching temperatures) is of especial importance.

If the hysteresis gap is large enough, then the constructed periodic solution is in fact unique and globally stable. For small values of hysteresis gap coexistence of several periodic solutions with different stability properties is proved to be possible.

This is a joint work with Pavel Gurevich.

In this talk we present a survey of our results on quasi-periodic Jacobi operators whose diagonal and off-diagonal elements are generated from two analytic functions on the circle. Such operators arise as effective Hamiltonians describing the effects of external magnetic fields on a tight binding, infinite crystal layer. The main motivation of our investigations was extended Harper's model (EHM), whose description on both the level of spectral analysis, as well the Lyapunov exponent (LE) had posed an open problem even from the point of view of physics literature. Among the topics that will be addressed are: Singular components of spectral measures for ergodic Jacobi operators, Singular analytic cocycles and joint continuity of the Lyapunov exponent, Recovery of spectral data from rational frequency approximants, Almost constant cocycles and the complexified LE of EHM, Spectral theory of EHM.