Volume 30, Number 3

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 introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body that rolls without slipping on the plane. These examples are constructed by taking as given two vector fields, one on the surface of the body and another on the plane, which specify the velocity of the contact point. We investigate dynamical aspects of the system such as existence of first integrals, smooth invariant measure, integrability and chaotic behavior, giving special attention to special shapes of the convex body and specific choices of the vector fields for which the affine nonholonomic constraints may be physically realized.

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.

Kolmogorov – Arnold – Moser theory started in the 1950s as the perturbation theory for persistence of multi- or quasi-periodic motions in Hamiltonian systems. Since then the theory obtained a branch where the persistent occurrence of quasi-periodicity is studied in various classes of systems, which may depend on parameters. The view changed into the direction of structural stability, concerning the occurrence of quasi-periodic tori on a set of positive Hausdorff measure in a sub-manifold of the product of phase space and parameter space. This paper contains an overview of this development with an emphasis on the world of dissipative systems, where families of quasi-periodic tori occur and bifurcate in a persistent way. The transition from orderly to chaotic dynamics here forms a leading thought.

This paper is devoted to the servoconstraints approach in the problem of periodic motion planning for Euler – Lagrange systems with a single degree of underactuation. We focus on the case where the servoconstraint is not regular and thus leads to the appearance of isolated singularities in reduced dynamics. We demonstrate that, subject to supplementary conditions, the reduced dynamics possess smooth solutions that pass through the singular point and this can be utilized for finding trajectories of the original system. Building upon this outcome, we solve the problem of motion planning of the Pendubot system with an imposed eight-shaped servoconstraint. To verify the feasibility of the discovered trajectory, we present computer simulation results of the closed-loop system with feedback that enables orbital stabilization for the trajectory.