A weightless string without preliminary tension with two symmetric discrete masses, which are influenced by elastic supports with cubic characteristics, is investigated both by numerical and analytical methods. The most important limit case corresponding to domination of resonance lowenergy transversal oscillations is considered. Since such oscillations are described by approximate equations only with cubic terms (without linear ones), the transversal dynamics occurs n the conditions of acoustic vakuum. If there is no elastic supports nonlinear normal modes of the system under investigation coincide with (or are close to) those of corresponding linear oscillator system. However within the presence of elastic supports one of NNM can be unstable, that causes formation of two another assymmetric modes and a separatrix which divides them. Such dynamical transition which is observed under certain relation between elastic constants of the string and of the support, relates to stationary resonance dynamics. This transition determines also a possibility of the second dynamical transition which occurs when the supports contribution grows. It relates already to non-stationary resonance dynamics when the modal approach turns out to be inadequate. Effective description of both dynamical transitions can be attained in terms of weakly interacting oscillators and limiting phase trajectories, corresponding to complete energy echange between the oscillators.

We outline a possibility of implementation of Smale–Williams type attractors with different stretching factors for the angular coordinate, namely, $n=3,\,5,\,7,\,9,\,11$, for the maps describing the evolution of parametrically excited standing wave patterns on a nonlinear string over a period of modulation of pump accompanying by alternate excitation of modes with the wavelength ratios of $1:n$.

The detection of coupling presence and direction between various systems using their time series is a common task in many areas of knowledge. One of the approaches used to solve it is nonlinear Granger causality method. It is based on the construction of forecasting models, so its efficiency defends on selection of model parameters. Two parameters are important for modeling signals with a main time scales: lag that is used for state vector reconstruction and prediction length.
In this paper, we propose two criteria for evaluating performance of the method of nonlinear Granger causality. These criteria allow to select lag and prediction length, that provide the best sensitivity and specificity. Sensitivity determines the weakest coupling method can detect, and specificity refers to the ability to avoid false positive results. As a result of the criteria application to several etalon unidirectionally coupled systems, practical recommendations for the selection of the model parameters (lag and prediction length) were formulated.

Theoretical and numerical study of nonlinear wave processes in media with hysteretic nonlinearity are carried out. The phenomena of amplitude-dependent damping as well as change of the propagation velocity of harmonic wave and its second and third harmonic generation are considered. It was shown that the hysteretic media possess nonlinear dispersion that become apparent in the difference between the phase velocities of strong harmonic pump wave and its weak high harmonics. The dispersion leads to both spatial beatings and non-monotonically rate of growth of an amplitude of high harmonic at the increase in amplitude of basic frequency wave.

The solution of the second task of Stokes for the swirled knitting of incompressible liquid is provided. The found solutions represent the elliptic polarized cross waves. The solution of the second Stokes problem for the swirl flow of a viscous incompressible fluid is presented.

In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.

This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.

This work deals with stability relative to three-dimensional disturbances of a compound rotationalaxial shear flow of Newtonian viscous fluid inside a cylindrical clearance. The corresponding linearized problem on stability is stated with the sticking conditions. On the basis of the integral relation method permitting to obtain sufficient estimates of stability as well as lower estimates for critical Reynolds numbers, the general upper estimate of real part of a spectral parameter (responding to stability) is derived. This estimate is defined more exactly for cases of both threedimensional axially symmetric disturbances and two-dimensional non-axially symmetric ones.

This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.

We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.