Volume 12, Number 3
Volume 12, Number 3, 2007
Gonchenko S. V., Gonchenko V. S., Tatjer J. C.
Bifurcations of ThreeDimensional Diffeomorphisms with NonSimple Quadratic Homoclinic Tangencies and Generalized Hénon Maps
Abstract
We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to socalled generalized Hénon maps which have nondegenerate bifurcations of fixed points including those with multipliers $e^{\pm i \phi}$. On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.

Kharlamov M. P.
Separation of Variables in the Generalized 4th Appelrot Class
Abstract
We consider an analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill a fourdimensional surface $\mathfrak{O}$ in the sixdimensional phase space. The constants of the three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in $\mathbb{R}^3$. We point out a pair of partial integrals to obtain explicit parametric equations of this sheet. The induced system on $\mathfrak{O}$ is shown to be Hamiltonian with two degrees of freedom having a thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on $\mathfrak{O}$ and algebraic formulae for the initial phase variables.

Albrecht J.
On the Existence of Invariant Tori in NearlyIntegrable Hamiltonian Systems With Finitely Differentiable Perturbations
Abstract
We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a $2n$times continuously differentiable perturbation ($n$ denotes the number of the degrees of freedom), provided that the moduli of continuity of the $2n$th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the $2n$th partial derivatives of the perturbation are Hölder continuous.

Kuleshov A. S.
Further Development of the Mathematical Model of a Snakeboard
Abstract
This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard. As against to the model, proposed by Lewis et al. [13] and investigated by various methods in [1][13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other. This assumption has been made earlier only by Golubev [13]. Equations of motion of the model are derived in the Gibbs–Appell form. Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles. The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies. All the obtained theoretical results are confirmed by numerical experiments.

Chernov N., Zhang H.
Regularity of Bunimovich's Stadia
Abstract
Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove "regularity" properties.
