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# Volume 18, Number 3, 2013

 Lăzureanu C.,  Bînzar T. Symplectic Realizations and Symmetries of a Lotka–Volterra Type System Abstract In this paper a Lotka$ndash;Volterra type system is considered. For such a system, bi-Hamiltonian formulation, symplectic realizations and symmetries are presented. Keywords: Lotka–Volterra system, symmetries, Hamiltonian dynamics, Lie groups Citation: Lăzureanu C., Bînzar T., Symplectic Realizations and Symmetries of a Lotka–Volterra Type System, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 203-213 DOI:10.1134/S1560354713030015  Jalnine A. Y. Generalized Synchronization of Identical Chaotic Systems on the Route from an Independent Dynamics to the Complete Synchrony Abstract The transition from asynchronous hyperchaos to complete synchrony in coupled identical chaotic systems may either occur directly or be mediated by a preliminary stage of generalized synchronization. In the present paper we investigate the underlying mechanisms of realization of the both scenarios. It is shown that a generalized synchronization arises when the manifold of identically synchronous states$M$is transversally unstable, while the local transversal contraction of phase volume first appears in the areas of phase space separated from$M$and being visited by the chaotic trajectories. On the other hand, a direct transition from an asynchronous hyperchaos to the complete synchronization occurs, under variation of the controlling parameter, if the transversal stability appears first on the manifold$M$, and only then it extends upon the neighboring phase volume. The realization of one or another scenario depends upon the choice of the coupling function. This result is valid for both unidirectionally and mutually coupled systems, that is confirmed by theoretical analysis of the discrete models and numerical simulations of the physically realistic flow systems. Keywords: synchronization, chaotic dynamics, strange attractors Citation: Jalnine A. Y., Generalized Synchronization of Identical Chaotic Systems on the Route from an Independent Dynamics to the Complete Synchrony, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 214-225 DOI:10.1134/S1560354713030027  Rutstam N. High Frequency Behavior of a Rolling Ball and Simplification of the Separation Equation Abstract The Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential$V (z = \cos\theta, D, \lambda)$that is difficult to analyze. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of$z = \cos\theta$. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency$\omega_3$-dependence of the width of the nutational band, the depth of motion above$V (z_{min}, D, \lambda)$and the$\omega_3$-dependence of nutational frequency$\frac{2\pi}{T}\$. Keywords: rigid body, rolling sphere, integrals of motion, elliptic integrals, tippe top Citation: Rutstam N., High Frequency Behavior of a Rolling Ball and Simplification of the Separation Equation, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 226-236 DOI:10.1134/S1560354713030039
 Bounemoura A. Normal Forms, Stability and Splitting of Invariant Manifolds I. Gevrey Hamiltonians Abstract In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians. Keywords: perturbation of integrable Hamiltonian systems, normal forms, splitting of invariant manifolds Citation: Bounemoura A., Normal Forms, Stability and Splitting of Invariant Manifolds I. Gevrey Hamiltonians, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 237-260 DOI:10.1134/S1560354713030040
 Bounemoura A. Normal Forms, Stability and Splitting of Invariant Manifolds II. Finitely Differentiable Hamiltonians Abstract This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasiperiodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori. Keywords: perturbation of integrable Hamiltonian systems, normal forms, splitting of invariant manifolds Citation: Bounemoura A., Normal Forms, Stability and Splitting of Invariant Manifolds II. Finitely Differentiable Hamiltonians, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 261-276 DOI:10.1134/S1560354713030052
 Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A. The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere Abstract In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior. Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution Citation: Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328 DOI:10.1134/S1560354713030064

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