Volume 8, Number 3
Volume 8, Number 3, 2003
Chen Y.
Multibump orbits continued from the antiintegrable limit for Lagrangian systems
Abstract
This paper presents an implicit function theorem version of the results of Bolotin and MacKay (1997) for multibump orbits of timedependent Lagrangian systems on the torus. The antiintegrable limit is the singular (or adiabatic) limit in the singularly (adiabatically, resp.) perturbed problems.

Cabral H. E., Meyer K. R., Schmidt D. S.
Stability and bifurcations for the $N + 1$ vortex problem on the sphere
Abstract
The equations of motion for $N$ vortices on a sphere were derived by V.A.Bogomolov in 1977. References to related work can be found in the book by P.K.Newton. We use the equations of motion found there to discuss the stability of a ring of $N$ vortices of unit strength at the latitude $z$ together with a vortex of strength $\kappa$ at the north pole. The regions of stability are bounded by curves $\kappa = \kappa(z)$. These curves are computed explicitly for all values of $N$.
When the stability of a configuration changes, for example by varying the strength of the vortex at the north pole, bifurcations to new configurations are possible. We compute the bifurcation equations explicitly for $N = 2, 3$ and $4$. For larger values of $N$ the complexity of the formal computations becomes too great and we use a numerical value for the latitude instead. We thus derive the bifurcation equations in a seminumerical form. As expected the new configurations look very similar to those which had been found previously for the planar case. 
Deryabin M. V., Pustyl'nikov L. D.
Generalized relativistic billiards
Abstract
We study generalized relativistic billiards, which is the following dynamical system. A particle moves in the interior of a domain under the influence of some force fields. As the particle hits the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall. Both the motion in the domain and the reflection are considered in the framework of the theory of relativity.
We study the periodic and "monotone" action of the boundary for the particle moving in a parallelepiped and in an arbitrary compact domain respectively, and we also consider an "accelerating" model in an unbounded domain. We prove that under some general conditions an invariant manifold in the velocity phase space of the generalized billiard, where the particle velocity equals the velocity of light, either is an exponential attractor or contains one. Thus for an open set of initial conditions the particle energy tends to infinity. 
Markeev A. P.
On stability of regular precessions of a nonsymmetric gyroscope
Abstract
We consider the motion of a rigid body around a fixed point in homogeneous gravity field. The body is not dynamically symmetric and the center of gravity is situated on the straight line passing through the fixed point perpendicular to circular crosssections of inertia ellipsoid. In 1947, G.Grioli proved that the body with such geometry of mass can be in a state of regular precession around of a nonvertical axis. In this paper we study the stability of this precession.

Meleshko V. V., Dunaeva T. A.
Chaotic advection in Stokes flow in semicircle
Abstract
The twodimensional Stokes flows due to a periodic motion of a circular and rectilinear boundary in a semicircle is considered. An exact analytical of the velocity field is given. The peculiarities of the velocity field and their influence on the motion of dye fluids in these domains are studied. Poincaré sections and trajectories of motion of passive fluid particles are calculated and analyzed for various regimes and values of the boundary velocities. The analysis of Poincaré sections reveals both zones of intensive (chaotic) and weak (regular) mixing of marked passive regions of fluid flow in the semicircle.

Ramodanov S. M.
Motion of two circular cylinders in a perfect fluid
Abstract
A planar analog of the Bjeknes problem of interaction of two spheres in a perfect fluid is considered. For the case of equal circulations around the cylinders, the equations of motion in the Poincare–Chetaev form are obtained, the integrals of motion are indicated. The problem is then reduced to a problem with two degrees of freedom. Most probably, this reduced problem is not integrable.

Deryabin M. V., Hjorth P. G.
High dimensional bowling. $n$dimensional ball rolling on $(n – 1)$dimensional surface
Abstract
We consider the nonholonomic system of a $n$dimensional ball rolling on a $(n – 1)$dimensional surface. We discuss the structure of the equations of motion, the existence of an invariant measure and some generalizations of the problem.

Mamaev I. S.
New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface
Abstract
Some new cases when the invariant measure and an additional first integral exist in the problem of a rigid body rolling on a sphere and on an ellipsoid are discussed in the paper. These cases generalize the results obtained previously by V.A.Yaroshchuk and A.V.Borisov, I.S.Mamaev, A.A.Kilin.

Yehia H. M.
Kovalevskaya's integrable case: generalizations and related new results
Abstract
More than 11 decades have elapsed since S. V. Kovalevskaya discovered her famous case of integrability of the equations of motion of a heavy rigid body about a fixed point [1]. Nevertheless, the last 17 years have witnessed the emergence of some amazing and even unexpected results generalizing this case or valid under the same condition $A = B = 2C$. In this paper we give a summary of the known integrable cases of Kovalevskaya's type and point out some of their generalizations. A total of four general and five conditional integrable cases is listed. Of those, two general and three conditional cases are introduced for the first time.
