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Volume 22, Number 2

Volume 22, Number 2, 2017

Caşu I.,  Lăzureanu C.
Stability and Integrability Aspects for the Maxwell–Bloch Equations with the Rotating Wave Approximation
Abstract
Infinitely many Hamilton–Poisson realizations of the five-dimensional real valued Maxwell–Bloch equations with the rotating wave approximation are constructed and the energy-Casimir mapping is considered. Also, the image of this mapping is presented and connections with the equilibrium states of the considered system are studied. Using some fibers of the image of the energy-Casimir mapping, some special orbits are obtained. Finally, a Lax formulation of the system is given.
Keywords: Maxwell–Bloch equations, Hamiltonian dynamics, energy-Casimir mapping, homoclinic orbits, periodic orbits, elliptic functions
Citation: Caşu I.,  Lăzureanu C., Stability and Integrability Aspects for the Maxwell–Bloch Equations with the Rotating Wave Approximation, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 109-121
DOI:10.1134/S1560354717020010
Grines V. Z.,  Gurevich E. Y.,  Pochinka O. V.
On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics
Abstract
Separators are fundamental plasma physics objects that play an important role in many astrophysical phenomena. Looking for separators and their number is one of the first steps in studying the topology of the magnetic field in the solar corona. In the language of dynamical systems, separators are noncompact heteroclinic curves. In this paper we give an exact lower estimation of the number of noncompact heteroclinic curves for a 3-diffeomorphism with the so-called “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori.
Keywords: separator in a magnetic field, heteroclinic curves, mapping torus, gradient-like diffeomorphisms
Citation: Grines V. Z.,  Gurevich E. Y.,  Pochinka O. V., On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 122-135
DOI:10.1134/S1560354717020022
Polekhin I. Y.
Classical Perturbation Theory and Resonances in Some Rigid Body Systems
Abstract
We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.
Keywords: Poincaré method, Poincaré sets, resonances, periodic solutions, small divisors, rigid body, Kirchhoff’s equations
Citation: Polekhin I. Y., Classical Perturbation Theory and Resonances in Some Rigid Body Systems, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 136-147
DOI:10.1134/S1560354717020034
Semenova N. I.,  Rybalova E. V.,  Strelkova G. I.,  Anishchenko V. S.
“Coherence–incoherence” Transition in Ensembles of Nonlocally Coupled Chaotic Oscillators with Nonhyperbolic and Hyperbolic Attractors
Abstract
We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Hénon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Hénon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Hénon and Lozi maps.
Keywords: ensemble of nonlocally coupled oscillators, chimera states, solitary states, hyperbolic and nonhyperbolic attractors, coupling function
Citation: Semenova N. I.,  Rybalova E. V.,  Strelkova G. I.,  Anishchenko V. S., “Coherence–incoherence” Transition in Ensembles of Nonlocally Coupled Chaotic Oscillators with Nonhyperbolic and Hyperbolic Attractors, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 148-162
DOI:10.1134/S1560354717020046
Tsiganov A. V.
Bäcklund Transformations for the Nonholonomic Veselova System
Abstract
We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.
Keywords: nonholonomic dynamical system, bi-Hamiltonian geometry, Bäcklund transformations
Citation: Tsiganov A. V., Bäcklund Transformations for the Nonholonomic Veselova System, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 163-179
DOI:10.1134/S1560354717020058
Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
The Hess–Appelrot Case and Quantization of the Rotation Number
Abstract
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
DOI:10.1134/S156035471702006X

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