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Volume 21, Number 2

Volume 21, Number 2, 2016

Manevitch L. I.,  Kovaleva A.,  Sigalov G.
Nonstationary Energy Localization vs Conventional Stationary Localization in Weakly Coupled Nonlinear Oscillators
Abstract
In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.
Keywords: nonlinear oscillations, coupled oscillators, nonlinear resonances, systems with slow and fast motions
Citation: Manevitch L. I.,  Kovaleva A.,  Sigalov G., Nonstationary Energy Localization vs Conventional Stationary Localization in Weakly Coupled Nonlinear Oscillators, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 147-159
DOI:10.1134/S1560354716020015
Kuznetsov S. P.,  Kruglov V. P.
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
Abstract
Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale – Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.
Keywords: dynamical system, chaos, attractor, hyperbolic dynamics, Lyapunov exponent, Smale – Williams solenoid, parametric oscillations
Citation: Kuznetsov S. P.,  Kruglov V. P., Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
DOI:10.1134/S1560354716020027
Esen O.,  Choudhury A. G.,  Guha P.,  Gümral H.
Superintegrable Cases of Four-dimensional Dynamical Systems
Abstract
Degenerate tri-Hamiltonian structures of the Shivamoggi and generalized Raychaudhuri equations are exhibited. For certain specific values of the parameters, it is shown that hyperchaotic Lü and Qi systems are superintegrable and admit tri-Hamiltonian structures.
Keywords: first integrals, Darboux polynomials, Jacobi’s last multiplier, 4D Poisson structures, tri-Hamiltonian structures, Shivamoggi equations, generalized Raychaudhuri equations, Lü system and Qi system
Citation: Esen O.,  Choudhury A. G.,  Guha P.,  Gümral H., Superintegrable Cases of Four-dimensional Dynamical Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 175-188
DOI:10.1134/S1560354716020039
Grines V. Z.,  Malyshev D. S.,  Pochinka O. V.,  Zinina S. K.
Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms
Abstract
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.
Keywords: Morse – Smale diffeomorphism, gradient-like diffeomorphism, topological classification, three-color graph, directed graph, graph isomorphism, surface orientability, surface genus, polynomial-time algorithm, magnetic field
Citation: Grines V. Z.,  Malyshev D. S.,  Pochinka O. V.,  Zinina S. K., Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 189-203
DOI:10.1134/S1560354716020040
Przybylska M.,  Rauch-Wojciechowski S.
Dynamics of a Rolling and Sliding Disk in a Plane. Asymptotic Solutions, Stability and Numerical Simulations
Abstract
We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.
Keywords: rigid body, nonholonomic mechanics, rolling disk, sliding disk
Citation: Przybylska M.,  Rauch-Wojciechowski S., Dynamics of a Rolling and Sliding Disk in a Plane. Asymptotic Solutions, Stability and Numerical Simulations, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 204-231
DOI:10.1134/S1560354716020052
Borisov A. V.,  Mamaev I. S.
Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics
Abstract
The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré – Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).
Keywords: adiabatic invariants, Liouville system, transition through resonance, adiabatic chaos
Citation: Borisov A. V.,  Mamaev I. S., Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 232-248
DOI:10.1134/S1560354716020064

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