In this paper, we establish new integrable systems which are similar to WZNW (Wess–Zumino–Novikov–Witten) systems and non-abelian affine Toda models. One of these systems is a new integrable extension of the well known sine-Gordon equation.

Equations of motion of vortex sources (examined earlier by Fridman and Polubarinova) are studied, and the problems of their being Hamiltonian and integrable are discussed. A system of two vortex sources and three sources-sinks was examined. Their behavior was found to be regular. Qualitative analysis of this system was made, and the class of Liouville integrable systems is considered. Particular solutions analogous to the homothetic configurations in celestial mechanics are given.

We give a constructive description of quadratic stochastic operators which act to the set of all probability measures on some measurable space. Our construction depends on a probability measure $\mu$ and cardinality of a set of cells (configurations) which here can be finite or continual. We study behavior of trajectories of such operators for a given probability measure $\mu$ which coincides with a Gibbs measure. For the continual case we compare the quadratic operators which correspond to well-known Gibbs measures of the Potts model on $Z^d$. These investigations allows a natural introduction of thermodynamics in studying some models of heredity. In particular, we show that any trajectory of the quadratic stochastic operator generated by a Gibbs measure $\mu$ of the Potts model converges to this measure

In this paper we consider a discrete dynamical system generated by bounded affine mappings on Banach spaces and on Montel locally convex spaces. We show that if these dynamical systems have bounded trajectories then they have periodic ones. Different applications are considered.

In this paper we study the following problem. Let a particle move in a central Newtonian gravitational field near a massive gravitationally oriented body. The body describes a circular orbit and is equipped with a "leier constraint", i.e. a cable with both ends fixed on the body (from the Dutch maritime term "leier"— a rope with both ends fixed). We study the possibility of capturing the particle (i.e. to force the particle to move periodically) by a gripper coasting on the "leier constraint". Any discontinuities in the velocity and acceleration are not allowed. We show that using a sufficiently long cable the capture is possible for almost all values of the system parameters. We also construct a modified algorithm that allows many-fold reduction in the required cable length.

The generation of topographic eddies and wave wakes over low seamounts in vertically and horizontally sheared zonal currents is studied on the $f$- and $\beta$-plane within the context of the baroclinic quasigeostrophic model. Water stratification and the vertical shear in velocity are shown to result in the joint effect of baroclinicity and a current velocity shift ('JEBACS'), which essentially transforms the manifestation of $\beta$-effect on the $\beta$-plane and can bring about pseudo $\beta$-effect on the $f$-plane. Similar to the $\beta$-effect, 'JEBACS' plays an important part in the generation of above- mountain topographic eddies and wave wakes. The spectrum of the Sturm–Liouville operator for vertical modes can have negative eigenvalues in the beginning of the spectrum not only on the $\beta$-plane but also on the $f$-plane. The negative eigenvalues in the spectrum result in the appearance of wave modes in the respective Helmholtz operator. The wave modes account for wave wakes behind seamounts. Captured Rossby waves, which always appear in any homogeneous eastward flow behind a seamount on the $\beta$-plane, may not appear in flows with vertical shear in velocity, even though these flows are also directed eastward. In this context, it was shown that the use of averaged velocities with the aim to derive the conclusion regarding the generation of waves that form the wave wake can be incorrect in the case of flows with vertical shift. This is an important distinction of the flows with velocity shift from homogenous flow in the case of flows around seamounts.
Another important type of stratified shear currents is represented by two-layer flows with differently directed flows in different layers. The notion of eastern or western flows makes no sense in this case. Nevertheless, wave modes also form in such flows. The topographic eddy in such two-layer flows has the form of an eddy lens concentrated near the interface between layers. In one layer, the wave wake may be located upstream of the mountain, rather than downstream of it.