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# Volume 20, Number 1, 2015

 Schirinzi G.,  Guzzo M. Numerical Verification of the Steepness of Three and Four Degrees of Freedom Hamiltonian Systems Abstract We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degreesof-freedom Hamiltonian system obtained from the Fermi–Pasta–Ulam problem by setting the number of particles equal to four. Keywords: Nekhoroshev’s theorem, steepness, three-body-problem, Fermi–Pasta–Ulam Citation: Schirinzi G.,  Guzzo M., Numerical Verification of the Steepness of Three and Four Degrees of Freedom Hamiltonian Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 1-18 DOI:10.1134/S1560354715010013
 Zhao L. Kustaanheimo–Stiefel Regularization and the Quadrupolar Conjugacy Abstract In this article, we first present the Kustaanheimo–Stiefel regularization of the spatial Kepler problem in a symplectic and quaternionic approach. We then establish a set of action-angle coordinates, the so-called LCF coordinates, of the Kustaanheimo–Stiefel regularized Kepler problem, which is consequently used to obtain a conjugacy relation between the integrable approximating “quadrupolar” system of the lunar spatial three-body problem and its regularized counterpart. This result justifies the study of Lidov and Ziglin [14] of the quadrupolar dynamics of the lunar spatial three-body problem near degenerate inner ellipses. Keywords: Kustaanheimo–Stiefel regularization, quaternions, symplectic reduction, secular systems, quadrupolar system Citation: Zhao L., Kustaanheimo–Stiefel Regularization and the Quadrupolar Conjugacy, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 19-36 DOI:10.1134/S1560354715010025
 Belykh V. N.,  Petrov V. S.,  Osipov G. V. Dynamics of the Finite-dimensional Kuramoto Model: Global and Cluster Synchronization Abstract Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations. Keywords: phase oscillators, Kuramoto model, global synchronization, existence and stability conditions, asynchronous mode Citation: Belykh V. N.,  Petrov V. S.,  Osipov G. V., Dynamics of the Finite-dimensional Kuramoto Model: Global and Cluster Synchronization, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 37-48 DOI:10.1134/S1560354715010037
 Barreira L.,  Dragičević D.,  Valls  C. Admissibility and Nonuniform Exponential Trichotomies Abstract For a nonautonomous dynamics defined by a sequence of linear operators acting on a Banach space, we show that the notion of a nonuniform exponential trichotomy can be completely characterized in terms of admissibility properties. This refers to the existence of bounded solutions under any bounded time-dependent perturbation of certain homotheties of the original dynamics. We also consider the more restrictive notion of a strong nonuniform exponential trichotomy and again we give a characterization in terms of admissibility properties. We emphasize that both notions are ubiquitous in the context of ergodic theory. As a nontrivial application, we show in a simple manner that the two notions of trichotomy persist under sufficiently small linear perturbations. Finally, we obtain a corresponding characterization of nonuniformly partially hyperbolic sets. Keywords: exponential trichotomy, robustness, partially hyperbolic set Citation: Barreira L.,  Dragičević D.,  Valls  C., Admissibility and Nonuniform Exponential Trichotomies, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 49-62 DOI:10.1134/S1560354715010049
 Bardin B. S.,  Chekina E. A.,  Chekin A. M. On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit Abstract We study the Lyapunov stability problem of the resonant rotation of a rigid body satellite about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the satellite completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the above resonance rotation was started in [4, 6]. In the present paper, rigorous stability conclusions in the previously unstudied range of parameter values are obtained. In particular, new intervals of stability are found for eccentricity values close to 1. In addition, some special cases are studied where the stability analysis should take into account terms of degree not less than six in the expansion of the Hamiltonian of the perturbed motion. Using the technique described in [7, 8], explicit formulae are obtained, allowing one to verify the stability criterion of a time-periodic Hamiltonian system with one degree of freedom in the special cases mentioned. Keywords: Hamiltonian system, symplectic map, normal form, resonance, satellite, stability Citation: Bardin B. S.,  Chekina E. A.,  Chekin A. M., On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 63-73 DOI:10.1134/S1560354715010050
 Tsiganov A. V. Simultaneous Separation for the Neumann and Chaplygin Systems Abstract The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Bäcklund transformation. We also prove that after similar Bäcklund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Hénon–Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations. Keywords: bi-Hamiltonian geometry, Bäcklund transformations, separation of variables Citation: Tsiganov A. V., Simultaneous Separation for the Neumann and Chaplygin Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 74-93 DOI:10.1134/S1560354715010062
 Cresson J.,  Wiggins S. A $\lambda$-lemma for Normally Hyperbolic Invariant Manifolds Abstract Let $N$ be a smooth manifold and $f:N \to N$ be a $C^l$, $l \geqslant 2$ diffeomorphism. Let $M$ be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the $\lambda$-lemma in this case. Applications of this result are given in the context of normally hyperbolic invariant annuli or cylinders which are the basic pieces of all geometric mechanisms for diffusion in Hamiltonian systems. Moreover, we construct an explicit class of three-degree-of-freedom near-integrable Hamiltonian systems which satisfy our assumptions. Keywords: $\lambda$-lemma, Arnold diffusion, normally hyperbolic manifolds, Moeckel's mechanism Citation: Cresson J.,  Wiggins S., A $\lambda$-lemma for Normally Hyperbolic Invariant Manifolds, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 94-108 DOI:10.1134/S1560354715010074

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