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# Volume 12, Number 3, 2007

 Gonchenko S. V.,  Gonchenko V. S.,  Tatjer J. C. Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple 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 so-called generalized Hénon maps which have non-degenerate 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. Keywords: homoclinic tangency, rescaling, generalized Henon map, bifurcation Citation: Gonchenko S. V.,  Gonchenko V. S.,  Tatjer J. C., Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 233-266 DOI:10.1134/S156035470703001X
 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 four-dimensional surface $\mathfrak{O}$ in the six-dimensional 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. Keywords: Kowalevski top, double field, Appelrot classes, separation of variables Citation: Kharlamov M. P., Separation of Variables in the Generalized 4th Appelrot Class, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 267-280 DOI:10.1134/S1560354707030021
 Albrecht J. On the Existence of Invariant Tori in Nearly-Integrable 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. Keywords: nearly integrable Hamiltonian systems, KAM theory, perturbations, small divisors, Celestial Mechanics, quasi-periodic motions, invariant tori, trigonometric approximation in several variables, Holder condition Citation: Albrecht J., On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems With Finitely Differentiable Perturbations, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 281-320 DOI:10.1134/S1560354707030033
 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. Keywords: Snakeboard, Gibbs–Appell equations, dynamics, analysis of motion Citation: Kuleshov A. S., Further Development of the Mathematical Model of a Snakeboard, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 321-334 DOI:10.1134/S1560354707030045
 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. Keywords: billiards, stadium, hyperbolicity, chaos, absolute continuity, distortion bounds Citation: Chernov N.,  Zhang H., Regularity of Bunimovich's Stadia, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 335-356 DOI:10.1134/S1560354707030057

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