This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.

In the first part of the article we consider a semiclassical asymptotics for a Cauchy problem for the Schrodinger operator on a metric graph. We discuss the statistical properties of the corresponding classical dynamical system: the behavior of "number of particles" at large times and distribution of "particles" on the graph. We describe the distribution of energy on infinite regular trees. In the second part we describe the asymptotics of the spectrum of the Laplace and Schrodinger operators on a thin torus and on the simplest surfaces with delta-potentials.

We obtain properties of three-dimensional phase space and dynamics of Morse-Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.

Within the framework of the so-called satellite approximation, configurations of the relative equilibrium are built and their stability is analyzed. In this case the elliptic Keplerian motion of the satellite/the spacecraft tight group mass center is predefined. The attitude motion of the system does not influence its orbital motion. The principal central axes of inertia are assumed to move as a rigid body. Simultaneously masses of the body can redistribute in a way such that the values of moments of inertia can change. Thus, all configurations can perform pulsing motions changing it own dimensions. One obtains a system of equations of motion for such a compound satellite. It turns out that the resulting system of equations is similar to the well-known equation of V. V. Beletsky for the satellite in elliptic orbit planar oscillations. We use true anomaly as an independent variable as it is in the Beletsky equation. It turned out that there are planar pendulum-like librations of the whole system which may be regarded as perturbations of the mathematical pendulum. One can introduce action-angle variables in this case and can construct the dynamics of mappings over the non-autonomous perturbation period. As a result, one is able to apply the well-known Moser theorem on an invariant curve for twisting maps of annulus. After that one can get a general picture of motion in the case of the system planar oscillations. So, the whole description in the paper splits into two topics: (a) general dynamical analysis of the satellite planar attitude motion using KAM theory; (b) construction of periodic solutions families depending on the perturbation parameter and rising from equilibrium as the perturbation value grows. The latter families depend on the parameter of the perturbation and are absent in the non-perturbed problem.

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a ﬁxed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-ﬁxed axis is equal to zero. Depending on the system parameters, we ﬁnd cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the eﬀect of reversal, which was observed previously in the motion of rattlebacks.

The planar motion of an equilateral triangle with equal masses at vertices and of a point subjected to mutual Newtonian attraction is considered. Necessary conditions for the stability of “straight”, axial steady configurations, when the massive point is located on one of the symmetry axes of the triangle, are studied. The generation of other, “oblique”, steady configurations is discussed in connection with the variation, for certain parameter values, of the degree of instability of some “straight” steady configurations.

It is well known that several small celestial objects are of irregular shape. In particular, there exist asteroids of the so-called “dog-bone” shape. It turns out that approximation of these bodies by dumb-bells, as proposed by V.V. Beletsky, provides an effective tool for analytical investigation of dynamics in vicinities of such bodies. There remains the question of how to divide reasonably a “dogbone” body into two parts using available measurement data. In this paper we introduce an approach based on the so-called KK-mean algorithm proposed by the prominent Polish mathematician H. Steinhaus.

The problem of the motion of a particle in the gravity field of a homogeneous dumbbell-like

body composed of a pair of intersecting balls, whose radii are, in general, different, is studied.

Approximation for the Newtonian potential of attraction is obtained. Relative equilibria and

their properties are studied under the assumption of uniform rotation of the dumbbells.

Consider the class of diffeomorphisms of three-dimensional manifolds and satisfying aksiomA by Smale on the assumption that the non-wandering set of each diffeomorphism consists of surface two-dimensional basic sets. We find interrelations between the dynamics of such a diffeomorphism and the topology of the ambient manifold. Also found that each such diffeomorphism is Ω-conjugate to a modeling diffeomorphism of the manifold, which is a locally trivial bundle over the circle with torus as a leaf. Under some restrictions on the asymptotic behavior of two-dimensional invariant manifolds of points of basis sets obtained the topological classification of structurally stable diffeomorphisms of the class.

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.

In this paper we present the explicit construction of a continuum family of smooth pairwise not isomorphic foliations of codimension one on a standard three-dimensional sphere, each of which has countable many compact leaves-attractors diffeomorphic to the torus. As it was proved by S. P. Novikov, every smooth foliation of codimension one on a standard three-dimensional sphere contains a Reeb component. Changing this foliation only in the Reeb component by the indicated method we get a continuum family of smooth pairwise non-isomorphic foliations containing a countable set of compact leaves-attractors which coincides with the origin foliation outside this Reeb component.

We study the dynamics in the Suslov problem which describes the motion of a heavy rigid body with a fixed point subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) motions and, using a new method for constructing charts of Lyapunov exponents, detect different types of chaotic behavior such as conservative chaos, strange attractors and mixed dynamics, which are typical of reversible systems. In the paper we also examine the phenomenon of reversal, which was observed previously in the motion of Celtic stones.

In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected by two heteroclinic separators. The method of proof is to create a model of the magnetic field produced by the plasma in the solar corona and to study it using the methods of the theory of dynamical systems. Namely, in the space of vector elds on the sphere *S*3 with two sources, two sinks and two saddles, we construct a simple arc with two saddle-node bifurcation points that connects the system without heteroclinic curves to a system with two heteroclinic curves. Discretization of this arc is also a simple arc in the space of diffeomorphisms. The results are new.

In the frameworks of a class of exact solutions of the Navier–Stokes equations with linear dependence of part the speed components on one spatial variablethe axisymmetricalnonselfsimilar ﬂows of viscous ﬂuid in the cylindrical area which radius changes over the time under some law calculated during the solution are considered. The problem is reduced to two-parametrical dynamic system. The qualitative and numerical analysis of the system allowed to allocate three areas on the phase plane corresponding to various limit sizes of a pipe radius: radius of a pipe and stream velocity tend to inﬁnity for ﬁnite time, the area of a cross section of the cylinder tend to zero during a ﬁnite time span, radius of the tube inﬁnitely long time approaches to a constant value, and the ﬂow tend to the state of rest. For a case of ideal ﬂuid ﬂow the solution of the problem is obtained in the closed form and satisfying the slip condition.