We consider a model of a point vortex in a two-layer quasi-geostrophic flow. In this model, the chaotization of the phase space strongly depends on the frequency of the external excitation. Numerical experiments show that the degree of chaotization as a function of the excitation frequency has a number of pronounced extrema. Upon examination of rotation frequencies of fluid particles and the corresponding non-linear resonances, we have found a strong connection between these extrema and disappearance of the non-linear resonances. This disappearance phenomenon has been studied using the Poincare surface-of-section technique.

The paper studies the transport, mixing and chaotic advection of passive scalars in a meandering jet flow with a periodic perturbation. The stability of the critical points has been performed. We have found all topologically different regimes of the flow along with their bifurcations. It is shown that the process of mixing of passive scalars exhibits fractal-like patterns. There are some geometric regularities in the relationship between 1) the initial coordinates of scalars and 2) the number of rotations of particles around elliptic points and their escape time from a particular domain in the phase-space. It is shown how these regularities manifest in the evolution of a material line. The results obtained may be used in modelling Lagrangian transport and mixing of water masses with different characteristics in meandering western boundary currents such as the Kuroshio and the Gulf Stream.

The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.

In the paper we present a new integral of motion in the problem of rolling motion of a heavy symmetric sphere on the surface of a paraboloid. We use this integral to study the Lyapunov stability of some trivial steady rotations.

We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.

A short historical overview is given on the development of our knowledge of complex dynamical systems with special emphasis on ergodicity and chaos, and on the semiclassical quantization of integrable and chaotic systems. The general trace formula is discussed as a sound mathematical basis for the semiclassical quantization of chaos. Two conjectures are presented on the basis of which it is argued that there are unique fluctuation properties in quantum mechanics which are universal and, in a well defined sense, maximally random if the corresponding classical system is strongly chaotic. These properties constitute the quantum mechanical analogue of the phenomenon of chaos in classical mechanics. Thus quantum chaos has been found.

It is a well known fact that purely analytical methods are suitable to prove existence results and to get some local or semiglobal information on the dynamics of a system. If a more detailed study is desired or, in many cases, if one has to guess first the properties to be proven, we have to proceed to do a computer assisted study of the problem. In these notes I summarize several domains where a systematic use of a computer can help us to learn about a given family of dynamical systems and provide some examples.