Kirill Morozov

We study nonconservative quasi-periodic $m$-frequency $\it parametric$ perturbations of twodimensional nonlinear Hamiltonian systems. Our objective is to specify the conditions for the existence of new regimes in resonance zones, which may arise due to parametric terms in the perturbation. These regimes correspond to $(m + 1)$-frequency quasi-periodic solutions, which are not generated from Kolmogorov tori of the unperturbed system. The conditions for the existence of these solutions are found. The study is based on averaging theory and the analysis of the corresponding averaged systems. We illustrate the results with an example of a Duffing type equation.

We study quasi-periodic nonconservative perturbations of two-dimensional Hamiltonian systems. We suppose that there exists a region $D$ filled with closed phase curves of the unperturbed system and consider the problem of global dynamics in $D$. The investigation includes examining the behavior of solutions both in $D$ (the existence of invariant tori, the finiteness of the set of splittable energy levels) and in a neighborhood of the unperturbed separatrix (splitting of the separatrix manifolds). The conditions for the existence of homoclinic solutions are stated. We illustrate the research with the Duffing – Van der Pole equation as an example.

We study the role of quasi-periodic perturbations in systems close to two-dimensional Hamiltonian ones. Similarly to the problem of the influence of periodic perturbations on a limit cycle, we consider the problem of the passage of an invariant torus through a resonance zone. The conditions for synchronization of quasi-periodic oscillations are established. We illustrate our results using the Duffing –Van der Pol equation as an example.

A nonautonomous analogue of the FitzHugh–Nagumo model is considered. It is supposed that the system is transitory, i.e., it is autonomous except on some compact interval of time. We first study the past and future vector fields that determine the system outside the interval of time dependence. Then we build the transition map numerically and discuss the influence of the transitory shift on the solutions behavior.

The two-dimensional nonautonomous equations of pendular type are considered: the Josephson equation and the equation of oscillations of a body. It is supposed that these equations are transitory, i.e., nonautonomous only on a finite time interval. The problem of dependence of the mode on the transitory shift is solved. For a conservative case the measure of transport from oscillations to rotations is established.

We consider the two-dimensional system, which occurs in the flutter problem. We assume that this system is transitory (one whose time-dependence is confined to a compact interval). In the conservative case of this problem, we identified measure of transport between the cells filled with closed trajectories. In the nonconservative case, we consider the impact of transitory shift to setting of one or another attractor. We give probabilities of changing a mode (stationary to auto-oscillation).