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 paper the computer research of four-dimensional dynamic system with three parameters adequately describing behavior of model coupled Dynamo in view of viscous friction is carried out. Is shown, that in this system there are five equilibrium states: four stable are focuses–node and one is saddle (3, 1). Are established the bifurcations of the spatial overwound cycles appropriate to doubling of the period of oscillations dynamic variable and resulting to chaotic oscillations at increase of the relation of factors friction.

In the present paper possibilities of parameters estimation are considered in dynamical systems (DS) with additive noise. Simple and effective algorithms, optimal parameter values of numeric simulation and data filtration methods are proposed that enable one to find the controlling parameter value of a noisy DS with a high accuracy. Different DS are studied, and the accuracy of parameter estimation is examined for various dynamical modes and for different noise intensities.

The phenomenon of forced synchronization of periodic oscillations in the multistable system is studied by the example of two linear coupled modified oscillators with inertial nonlinearity. It was found out that external forcing at certain amplitudes can sufficiently change the structure of the phase space of the system. As a result, the synchronous regime breaking for in-phase and non-in-phase oscillations proceeds in accordance with different scenarios.

The appearance of chaotic regimes near elliptic point in a cell of particles’ chain interacting by means of Lennard–Jones potential is studied. The threshold nature of chaotization advent in the case of single-frequency cell excitation is demonstrated. A method of global chaotization based on multifrequency external excitation is proposed. The results of numerical experiments show that in this case the formation of global chaos is achieved at essentially lower values of external excitation amplitude and frequency, than in the case of single frequency excitation.

Qualitative structure of relative motion of three point vortices on the unbounded plain is studied. A classification of phase portraits is proposed.

With the help of mathematical modelling, we study the dynamics of many point vortices system on the plane. For this system, we consider the following cases:
— vortex rings with outer radius $r = 1$ and variable inner radius $r_0$,
— vortex ellipses with semiaxes $a$, $b$.
The emphasis is on the analysis of the asymptotic $(t → ∞)$ behavior of the system and on the verification of the stability criteria for vorticity continuous distributions.

The paper deals with the rolling motion of a balanced, dynamically asymmetric ball on a plane without sliding and spinning. The problem is natural but was not considered by classicists. Generalizations of the problem are analyzed for the case where gyrostat and force Brun field are added. To investigate the dynamic behavior of the system some peculiar periodic solutions are described and their stability is examined. By integral mapping, bifurcation diagrams and bifurcation complexes are constructed.

We report a novel general method for constructing an approximate solution of the planar motion of solids with an axially symmetric mass distribution and normal stresses over the contact area on a rough horizontal surface. For a disk characterized by Galin distribution of contact stresses we obtain explicit dependence of the angular and sliding velocity of the body as a function of time. The relative errors of the method do not exceed 1,5–2 %. The simplicity and high accuracy of the method let us recommend its applications in the practice of engineering calculations.

In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.