Volume 29, Number 6

This paper addresses the problem of a homogeneous ball rolling on the inner surface of a circular cylinder in a field of gravity parallel to its axis. It is assumed that the ball rolls without slipping on the surface of the cylinder, and that the cylinder executes plane-parallel motions in a circle perpendicular to its symmetry axis. The integrability of the problem by quadratures is proved. It is shown that in this problem the trajectories of the ball are quasiperiodic in the general case, and that an unbounded elevation of the ball is impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonances at which the ball moves on average downward with constant acceleration.

This paper is concerned with the plane-parallel motion of an elliptic foil with an attached vortex of constant strength in an ideal fluid. Special attention is given to the case in which the vortex lies on the continuation of one of the semiaxes of the ellipse. It is shown that in this case there exist no attracting solutions and the system is integrable by the Euler – Jacobi theorem. A complete qualitative analysis of the equations of motion is carried out for cases where the vortex lies on the continuation of the large or the small semiaxis of the ellipse. Possible types of trajectories of an elliptic foil with an attached vortex are established: quasi-periodic, unbounded (going to infinity) and periodic trajectories.

We study the dynamics of a multidimensional slow-fast Hamiltonian system in a neighborhood of the slow manifold under the assumption that the frozen system has a hyperbolic equilibrium with complex simple leading eigenvalues and there exists a transverse homoclinic orbit. We obtain formulas for the corresponding Shilnikov separatrix map and prove the existence of trajectories in a neighborhood of the homoclinic set with a prescribed evolution of the slow variables. An application to the 3 body problem is given.

A two-layer quasigeostrophic model is considered in the $f$-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case where the structure consists of a central vortex of arbitrary effective intensity $\Gamma$ and $N$ ($N = 4, 5$ and $6$) identical peripheral vortices. The identical vortices, each having a unit effective intensity, are uniformly distributed over a circle of radius $R$ in the lower layer. The central vortex lies either in the same or in another layer. The problem has three parameters $(R,\Gamma,\alpha)$, where $\alpha$ is the difference between layer nondimensional thicknesses. The cases $N=2, 3$ were investigated by us earlier.
The theory of stability of steady-state motions of dynamical systems with a continuous symmetry group $\mathcal{G}$ is applied. The two definitions of stability used in the study are Routh stability and $\mathcal{G}$-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex structure, and the $\mathcal{G}$-stability is the stability of a three-parameter invariant set $O_{\mathcal{G}}$, formed by the orbits of a continuous family of steady-state rotations of a two-layer vortex structure. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.
The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.

We study a class of integrable inhomogeneous Lotka – Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials and prove a contraction theorem. We then use these results to classify the systems according to the number of functionally independent (and, for some, commuting) integrals. We also establish separability/solvability by quadratures, given the solutions to the 2- and 3-dimensional systems, which we provide in terms of the Lambert $W$ function.

We prove that an $n$-sphere $\mathbb{S}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with nonorientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is the integer part of $x$. In addition, any $n$-sphere $\mathbb{S}^n$, $n\geqslant 3$, admits axiom A diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{3}]\}$. We prove that an $n$-torus $\mathbb{T}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,n-1\}$. We also prove that, given any closed $n$-manifold $M^n$, $n\geqslant 2$, and any $d\in\{1,\ldots,[\frac{n}{2}]\}$, there is an axiom A diffeomorphism $f: M^n\to M^n$ with a $d$-dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.

An arbitrary diffeomorphism $f$ of class $C^1$ acting from an open subset $U$ of Riemannian manifold $M$ of dimension $m,$ $m\ge 2,$ into $f(U)\subset M$ is considered. Let $A$ be a compact subset of $U$ invariant for $f,$ i.e. $f(A)=A.$ Various sufficient conditions are proposed under which $A$ is a hyperbolic set of the diffeomorphism $f.$