Albert 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.

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).

The problem of global behavior of solutions in system of two Duffing–Van der Pole equations close to nonlinear integrable is considered. For regions without unperturbed separatrixes we give partially averaged systems which describe the behavior of solutions of original system in resonant zones. The finiteness of number of non-trivial resonant structures is established. Also we give fully averaged systems which describe the behavior of solutions outside of neighborhoods of nontrivial resonant structures. The results of numerically investigation of these systems are resulted.

We consider a problem about interaction of the two Duffing—van der Pol equations close to nonlinear integrable. The average systems describing behaviour of the solutions of the initial equation in resonant zones are deduced. The conditions of existence of not trivial resonant structures are established. The results of research in cases are resulted, when at the uncoupled equations exist and there are no limiting cycles.

In this paper we consider time-periodic perturbations of self-oscillating pendulum equation which arises from analysis of one system with two degrees of freedom. We derive averaged systems which describe the behavior of solutions of original equation in resonant areas and we find existence condition of Poincare homoclinic structure. In the case when autonomous equation has 5 limit cycles in oscillating region we give results of numerical computation. Under variation of perturbation frequency we investigate bifurcations of phase portraits of Poincare map.