The model of three linear-coupled logistic maps is examined. The structure of parameter plane (coupling value—period-doubling parameter) is discussed. We select configuration of coupling and parameters so, that regimes of three-frequency quasiperiodicity become possible. Also we consider bifurcations associated with such states.

In the present work we analyze the statistics of a set that is obtained by calculating a stroboscopic section of phase trajectories in a harmonically driven van der Pol oscillator. It is shown that this set is similar to a linear shift on a circle with an irrational rotation number, which is defined as the detuning between the external and natural frequencies. The dependence of minimal return times on the size ε of the return interval is studied experimentally for the golden ratio. Furthermore, it is also found that in this case, the value of the Afraimovich–Pesin dimension is $\alpha_c = 1$.

To implicitly singularly perturbed autonomous systems of ordinary differential equations of second order found some sufficient conditions for the existence of periodic solutions of relaxation (self-oscillation), determined by means of an auxiliary dynamical system that implements a sliding mode. It is shown that so defined periodic motions have typical properties of self-oscillations of relaxation defined autonomous systems of ordinary differential equations with a small parameter at the highest derivative.

We have obtained a solution of the problem within the exact solutions of the Navier–Stokes equations which describes the flow of a viscous incompressible fluid caused by spatially inhomogeneous wind stresses.

Proposed simple model movements thin tubes under the influence of fluid flow. Was obtained a nonlinear equation for the string with the flow. Demonstrated the possibility of tube vibrations under the influence of fluid flow and criterion was found linear instability at a constant flow rate. In the condition where the linear instability conditions are violated the possibility of oscillations detected during the presence of small periodic oscillation of flow.

Direct 3D numerical modeling of the displacement process of a light liquid by a heavy one in a thin isothermal horizontal layer of finite length has been carried out. Initial non-equilibrium step distribution of liquids density generates convective process in this inhomogeneous system. The effect of liquids miscibility is taken into account over the calculation. The conditions of convection excitation in the regions with unstable stratification near the interface between counter propagating fluxes of liquids have been analyzed. The calculation results of concentration fronts velocity in dependence on densities difference are received in the presence and in the absence of the secondary spiral rollers in a flow. The evolution of secondary convective structures has been simulated in details at various stages of the process. The results of numerical modeling confirm previous experimental data.

This paper is concerned with a three-body system on a straight line in a potential field proposed by Tsiganov. The Liouville integrability of this system is shown. Reduction and separation of variables are performed.

We study a particle equilibria with respect to axes of precession and of dynamical symmetry of a rigid body in assumption that the body gravitational field is composed of gravitational fields of two conjugate complex masses being on imaginary distance. We establish that there are not more then two of these equilibria in the plane passing the body mass center orthogonally to the precession axis. Using terminology of the Generalized Restricted Circular Problem of Three Bodies, we call these equilibria the Triangular Libration Points (TLP).We find TLPs’ coordinates analytically and we trace their evolution at changing values of the system parameters. We also prove that TLPs are instable.

We discuss an application of the Poisson brackets deformation theory to the construction of the integrable perturbations of the given integrable systems. The main examples are the known integrable perturbations of the Kowalevski top for which we get new bi-Hamiltonian structures in the framework of the deformation theory.