We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.

The horizontal plane-parallel flow with an inflection-free velocity profile is considered in ideal, incompressible fluid which is stably stratified in a thin layer. Such a flow is linearly unstable for an arbitrary bulk Richardson number, and it is three-dimensional disturbances that are most unstable within a wide range of parameters. In the paper, the weakly-nonlinear temporal development of an unstable disturbance in the form of a pair of oblique waves is studied. For this purpose, the evolution equation is derived which has the form of a nonlinear integral equation and is valid for both thin and thick critical layers, including the case where the critical layer width exceeds the stratification layer thickness. Solutions of this equation are studied asymptotically and numerically, and it is shown that during the nonlinear stage of development the disturbance grows, as a rule, explosively.

The problem of 2-D and 3-D convection of viscous and incompressible fluid between two horizontal stress-free isothermal planes at heating from below has been considering. It is received that in 3-D turbulent convection the mean vorticity scale decreases at growing supercriticality, but in 2-D convection the mean vorticity scale grows and it does the 2-D convection more large-scale and smooth. The growing of the mean vorticity scale at increasing supercriticality $r$ in 2-D convection is conditioned by the red (inverse) energy cascade formed at $r > 4000$ and transferring the kinetic energy from generation scale to the large scales. The appearance of the red cascade is conditioned by additional conservation law for enstrophy in 2-D flows.

We consider nonlinear dynamics of random nonhomogeneous elastic medium. By random nonhomogeneous media we mean composite materials, granular materials, porous rocks with chaotic components distribution. To describe such medium we need to use Lagrangian coordinates instead of Eulerian coordinates. In this case Piola—Kirchhoff tensor should be used as strain tensor. It is asymmetrical tensor and defined as derivative of an energy with respect to dilatation, i. e. gradient of displacement vector. Here we use the most simple approach to Lagrangian representation which was developed in the Landau—Lifshitz model.

The Landau—Lifshitz approach is generalized here for nonlinear random nonhomogeneous elastic medium. So, equations motion of contain random coordinate dependent coefficients. In this article we considered effect of inherent stresses and finite deformations on medium oscillations near areas with high inherent stresses. Equations of wave propagation near stressed area are derived. As a result of medium random nonhomogeneity these equations describe not only wave propagation but also all multiple reflections from nonhomogeneities.

For averaging in this work we have used the Feynman diagram technique. This technique makes it possible to derive precise equation for average elastic field, which characterizes coherent propagation of waves subject to multiple reflections. This equation is integro-differential. It’s kernel (correlation operator) contains contributions from random nonhomogeneities correlation functions of any order. This operator directly defines velocities of P- and S-waves in random nonhomogeneous elastic medium. These velocities depends on inherent stresses and our approach allows approximate calculation of this dependence. In inverse case one can use experimental velocities of sound in areas with stresses near to critical for material breaking. Using these velocities state of stressed medium can be defined and it’s effective parameters. This article doesn’t cover inverse case. We only derive basic equation here which make it possible to state the inverse problem.

The paper concerns non-one-dimensional structures described by nonlinear SchrЁodinger equation with additional potential term. A method for numerical construction of structures of such kind is suggested. The method is based on dynamical interpretation of the equation under consideration. Some exact statements are formulated; they allow (in some cases) to perform demonstrative computation and to list all the types of structures mentioned above. Physical applications of the problem are associated with the theory of a Bose—Einstein condensate. In this context the considered equation is called Gross—Pitaevskii equation and the structures under consideration correspond to macroscopic wave function of the condensate.

The effect of synchonization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The bifurcation analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in the paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study. We illustrate the bifurcational mechanisms obtained from numerical experiment by the results of physical experiment. The synchronization phenomenon in the vicinity of resonances on a torus with winding numbers 1 : 1 and 1 : 3 is considered in the physical experiment.

The pulse driven Rossler system before the saddle-node bifurcation in the regime of divergence is considered. It is shown that external pulses initiate stable periodic and quasi-periodic regimes in non-autonomous system. The effect of synchronous response due interaction between external signal and own rhythm of autonomous system concerned with the «rotation» of the representation point in the three-dimensional phase space is observed. It is revealed that the torus doublings in the stroboscopic section exist in the certain area on the parameter plane of external force in this system.

In the terms of Lyapunov functions we obtain the conditions that allow to estimate the relative frequency of occurrence of the attainable set of a controllable system in a given set $\mathfrak{M}$. The set $\mathfrak{M}$ is called statistically invariant if the relative frequency of occurrence in $\mathfrak{M}$ is equal to one. We also derive the conditions of the statistically weak invariance of $\mathfrak{M}$ with respect to controllable system, that is, for every initial point from $\mathfrak{M}$, at least one solution of the controllable system is statistically invariant. We obtain the conditions for the attainable set to be non-wandering as well as the conditions of existence of the minimal attraction center.