Volume 20, Number 1
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 threebody problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the fourdegreesoffreedom Hamiltonian system obtained from the Fermi–Pasta–Ulam problem by setting the number of particles equal to four.

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 actionangle coordinates, the socalled 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 threebody problem and its regularized counterpart. This result justifies the study of Lidov and Ziglin [14] of the quadrupolar dynamics of the lunar spatial threebody problem near degenerate inner ellipses.

Belykh V. N., Petrov V. S., Osipov G. V.
Dynamics of the Finitedimensional Kuramoto Model: Global and Cluster Synchronization
Abstract
Synchronization phenomena in networks of globally coupled nonidentical 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 finitesize 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.

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 timedependent 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.

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 timeperiodic Hamiltonian system with one degree of freedom in the special cases mentioned.

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.

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 threedegreeoffreedom nearintegrable Hamiltonian systems which satisfy our assumptions.
