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# Volume 8, Number 3, 2003

 Chen Y. Multibump orbits continued from the anti-integrable 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 time-dependent Lagrangian systems on the torus. The anti-integrable limit is the singular (or adiabatic) limit in the singularly (adiabatically, resp.) perturbed problems. Citation: Chen Y., Multibump orbits continued from the anti-integrable limit for Lagrangian systems, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 243-257 DOI:10.1070/RD2003v008n03ABEH000242
 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 semi-numerical form. As expected the new configurations look very similar to those which had been found previously for the planar case. Citation: Cabral H. E.,  Meyer K. R.,  Schmidt D. S., Stability and bifurcations for the $N + 1$ vortex problem on the sphere , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 259-282 DOI:10.1070/RD2003v008n03ABEH000243
 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. Citation: Deryabin M. V.,  Pustyl'nikov L. D., Generalized relativistic billiards , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 283-296 DOI:10.1070/RD2003v008n03ABEH000244
 Markeev A. P. On stability of regular precessions of a non-symmetric 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 cross-sections 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. Citation: Markeev A. P., On stability of regular precessions of a non-symmetric gyroscope, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 297-304 DOI:10.1070/RD2003v008n03ABEH000245
 Meleshko V. V.,  Dunaeva T. A. Chaotic advection in Stokes flow in semicircle Abstract The two-dimensional 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. Citation: Meleshko V. V.,  Dunaeva T. A., Chaotic advection in Stokes flow in semicircle , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 305-312 DOI:10.1070/RD2003v008n03ABEH000246
 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. Citation: Ramodanov S. M., Motion of two circular cylinders in a perfect fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 313-318 DOI:10.1070/RD2003v008n03ABEH000247
 Deryabin M. V.,  Hjorth P. G. High dimensional bowling. $n$-dimensional ball rolling on $(n – 1)$-dimensional surface Abstract We consider the non-holonomic 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. Citation: Deryabin M. V.,  Hjorth P. G., High dimensional bowling. $n$-dimensional ball rolling on $(n – 1)$-dimensional surface , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 319-329 DOI:10.1070/RD2003v008n03ABEH000248
 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. Citation: Mamaev I. S., New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 331-335 DOI:10.1070/RD2003v008n03ABEH000249
 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. Citation: Yehia H. M., Kovalevskaya's integrable case: generalizations and related new results , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 337-348 DOI:10.1070/RD2003v008n03ABEH000250

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