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 fivedimensional real valued Maxwell–Bloch equations with the rotating wave approximation are constructed and the energyCasimir 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 energyCasimir mapping, some special orbits are obtained. Finally, a Lax formulation of the system is given.

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 3diffeomorphism with the socalled “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori.

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.

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 discretetime 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 spatiotemporal 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.

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 byproduct one gets two natural integrable systems on the cotangent bundle to the unit twodimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.

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.
