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2013
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Volume 18, Number 5

Volume 18, Number 5, 2013

Miguel N.,  Simo C.,  Vieiro A.
From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values
Abstract
In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense.
First, we present a study of some dynamical properties of orientation-preserving and orientationreversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects.
Then the phase space of the standard map, for large values of the parameter k, is studied. There are some stable orbits which appear periodically in $k$ and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as $k \to \infty$. Elementary considerations about diffusion properties of the standard map are also presented.
Keywords: Hénon maps, measure of regular and chaotic dynamics domains, islands in the standard map for large parameter, accelerator modes
Citation: Miguel N.,  Simo C.,  Vieiro A., From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 469-489
DOI:10.1134/S1560354713050018
Borisov A. V.,  Mamaev I. S.
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
Abstract
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
Keywords: vortex motion, nonholonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid
Citation: Borisov A. V.,  Mamaev I. S., The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 490-496
DOI:10.1134/S156035471305002X
Combot T.
Non-integrability of a Self-gravitating Riemann Liquid Ellipsoid
Abstract
We consider the motion of a triaxial Riemann ellipsoid of a homogeneous liquid without angular momentum. We prove that it does not admit an additional first integral which is meromorphic in position, impulsions, and elliptic integrals which appear in the potential. This proves that the system is not integrable in the Liouville sense; we actually show that even its restriction to a fixed energy hypersurface is not integrable.
Keywords: Morales–Ramis theory, elliptic functions, monodromy, differential Galois theory, Riemann surfaces
Citation: Combot T., Non-integrability of a Self-gravitating Riemann Liquid Ellipsoid, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 497-507
DOI:10.1134/S1560354713050031
Kazakov A. O.
Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane
Abstract
We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.
Keywords: mixed dynamics, strange attractor, unbalanced ball, rubber rolling, reversibility, twodimensional Poincaré map, bifurcation, focus, saddle, invariant manifolds, homoclinic tangency, Lyapunov’s exponents
Citation: Kazakov A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520
DOI:10.1134/S1560354713050043
Gonchenko A. S.,  Gonchenko S. V.,  Kazakov A. O.
Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone
Abstract
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Keywords: celtic stone, nonholonomic model, strange attractor, discrete Lorenz attractor, Shilnikov-like spiral attractor, mixed dynamics
Citation: Gonchenko A. S.,  Gonchenko S. V.,  Kazakov A. O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
DOI:10.1134/S1560354713050055
Yehia H. M.,  El-Hadidy E. G.
On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor
Abstract
One of the most notable effects in mechanics is the stabilization of the unstable upper equilibrium position of a symmetric body fixed from one point on its axis of symmetry, either by giving the body a suitable angular velocity or by adding a suitably spinned rotor along its axis. This effect is widely used in technology and in space dynamics.
The aim of the present article is to explore the effect of the presence of a rotor on a simple periodic motion of the rigid body and its motion as a physical pendulum.
The equation in the variation for pendulum vibrations takes the form
$$\frac{d^2 \gamma_3}{du^2}+\alpha [\alpha \nu^2+\frac{1}{2}+\rho^2- (\alpha + 1)\nu^2 sn^2u+2\nu \rho \sqrt{\alpha} cnu]\gamma_3=0,$$ in which α depends on the moments of inertia, $\rho$ on the gyrostatic momentum of the rotor and $\nu$ (the modulus of the elliptic function) depends on the total energy of the motion. This equation, which reduces to Lame’s equation when $\rho = 0$, has not been studied to any extent in the literature. The determination of the zones of stability and instability of plane motion reduces to finding conditions for the existence of primitive periodic solutions (with periods $4K(\nu)$, $8K(\nu)$) with those parameters. Complete analysis of primitive periodic solutions of this equation is performed analogously to that of Ince for Lame’s equation. Zones of stability and instability are determined analytically and illustrated in a graphical form by plotting surfaces separating them in the three-dimensional space of parameters. The problem is also solved numerically in certain regions of the parameter space, and results are compared to analytical ones.
Keywords: stability, pendulum-like motions, planar motions, periodic differential equation, Hill’s equation, Lame’s equation
Citation: Yehia H. M.,  El-Hadidy E. G., On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 539-552
DOI:10.1134/S1560354713050067

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