Volume 19, Number 6
Volume 19, Number 6, 2014
Kozlov V. V.
On Rational Integrals of Geodesic Flows
Abstract
This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.

Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
Abstract
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the wellknown experimental and theoretical results in this area.

Gelfreich V. G., Lerman L. M.
Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation
Abstract
We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a nonsemisimple double zero one. It is well known that a oneparameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of singleround homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.

Dragović V., Gajić B.
FourDimensional Generalization of the Grioli Precession
Abstract
A particular solution of the fourdimensional Lagrange top on $e(4)$ representing a fourdimensional regular precession is constructed. Using it, a fourdimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.

Delshams A., Gonchenko M. S., Gutierrez P.
Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio
Abstract
We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega = \sqrt{2}−1$. We show that the Poincaré–Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter ε satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.

Turaev D. V.
Hyperbolic Sets near Homoclinic Loops to a Saddle for Systems with a First Integral
Abstract
A complete description of dynamics in a neighborhood of a finite bunch of homoclinic loops to a saddle equilibrium state of a Hamiltonian system is given.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Three Vortex Sources
Abstract
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.

Delshams A., Gonchenko M. S., Gonchenko S. V.
On Bifurcations of Areapreserving and Nonorientable Maps with Quadratic Homoclinic Tangencies
Abstract
We study bifurcations of nonorientable areapreserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable twodimensional surfaces. We consider one and twoparameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.

Borisov A. V., Kazakov A. O., Sataev I. R.
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
Abstract
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the socalled figureeight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

Sevryuk M. B.
Translation of the V. I. Arnold Paper "From Superpositions to KAM Theory" (Vladimir Igorevich Arnold. Selected–60, Moscow: PHASIS, 1997, pp. 727–740)
Abstract
V.I.Arnold (12 June 1937 – 3 June 2010) published several papers where he described, in the form of recollections, his two earliest research problems (superpositions of continuous functions and quasiperiodic motions in dynamical systems), the main results and their interrelations: [A1], then [A2] (reprinted as [A4, A6]), and [A3] (translated into English by the author as [A5]). The first exposition [A1] has never been translated into English; however, it contains many details absent in the subsequent articles. It seems therefore that publishing the English translation of the paper [A1] would not be superfluous. What follows is this translation. In many cases, the translator gives complete bibliographic descriptions of various papers mentioned briefly in the original Russian text. The English translations of papers in Russian are also pointed out where possible. A related material is contained also in Arnold’s recollections “On A.N.Kolmogorov”. Slightly different versions of these reminiscences were published several times in Russian and English [A7–A12]. The early history of KAM theory is also discussed in detail in the recent brilliant semipopular book [A13].

Martinez R., Simó C.
Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion
Abstract
Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the twobody problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant $C$ large enough are compared to direct numerical computations showing improved agreement when $C$ increases. A return map to the pericenter passage is derived, and using an approximation by standardlike maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing $C$. Several anomalous phenomena are described.
