Volume 19, Number 6

Volume 19, Number 6, 2014

Kozlov V. V.
This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional 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.
Keywords: conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem
Citation: Kozlov V. V., On Rational Integrals of Geodesic Flows, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 601-606
Borisov A. V.,  Erdakova N. N.,  Ivanova T. B.,  Mamaev I. S.
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 well-known experimental and theoretical results in this area.
Keywords: dry friction, linear pressure distribution, planar motion, Coulomb law
Citation: 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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Gelfreich V. G.,  Lerman L. M.
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 non-semisimple double zero one. It is well known that a one-parameter 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 single-round 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.
Keywords: Hamiltonian bifurcation, homoclinic orbit, separatrix splitting, asymptotics beyond all orders
Citation: Gelfreich V. G.,  Lerman L. M., Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 635-655
Dragović V.,  Gajić B.
A particular solution of the four-dimensional Lagrange top on $e(4)$ representing a four-dimensional regular precession is constructed. Using it, a four-dimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.
Keywords: rigid body dynamics, Grioli precession, four-dimensional Lagrange case
Citation: Dragović V.,  Gajić B., Four-Dimensional Generalization of the Grioli Precession, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 656-662
Delshams A.,  Gonchenko M. S.,  Gutierrez P.
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.
Keywords: transverse homoclinic orbits, splitting of separatrices, Melnikov integrals, silver ratio
Citation: 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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 663-680
Turaev D. V.
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.
Keywords: Hamiltonian system, nonintegrability and chaos, resonance crossing, Arnold diffusion
Citation: Turaev D. V., Hyperbolic Sets near Homoclinic Loops to a Saddle for Systems with a First Integral, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 681-693
Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
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.
Keywords: integrability, vortex sources, shape sphere, reduction, homothetic configurations
Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Dynamics of Three Vortex Sources, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
Delshams A.,  Gonchenko M. S.,  Gonchenko S. V.
We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable twodimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
Keywords: area-preserving map, non-orientable surface, elliptic point, homoclinic tangency, bifurcation
Citation: Delshams A.,  Gonchenko M. S.,  Gonchenko S. V., On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 702-717
Borisov A. V.,  Kazakov A. O.,  Sataev I. R.
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 so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
Citation: Borisov A. V.,  Kazakov A. O.,  Sataev I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Sevryuk M. B.
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 quasi-periodic 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 semi-popular book [A13].
Keywords: Hilbert’s 13th problem, superpositions of continuous functions, invariant tori carrying quasi-periodic motions, KAM theory and its applications, Kolmogorov as a supervisor
Citation: 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), Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 734-744
Martinez R.,  Simó C.
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 two-body 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 standard-like 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.
Keywords: invariant rotational curves, separatrix maps, splitting function, restricted three-body problem
Citation: Martinez R.,  Simó C., Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 745-765

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