Regular and Chaotic Dynamics最新文献

The (2)-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the (2)-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the (2)-body problem on hyperbolic 3-space and discuss their stability for a strictly attractive potential.

A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system inthe vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.

In this paper, we obtain a classification of gradient-likeflows on arbitrary surfaces by generalizing the circularFleitasscheme. In 1975 he proved that such a scheme is a completeinvariant of topological equivalence for polar flows on 2- and 3-manifolds.In this paper, we generalize the concept of a circular schemeto arbitrary gradient-like flows on surfaces. We prove that theisomorphism class of such schemes is a complete invariant oftopological equivalence. We also solve exhaustively therealization problem by describing an abstract circularscheme and the process of realizing a gradient-like flow onthe surface. In addition, we construct an efficient algorithmfor distinguishing the isomorphism of circular schemes.

This paper treats the problem of a spherical robot with an axisymmetric pendulum driverolling without slipping on a vibrating plane. The main purpose of the paper isto investigate the stabilization of the upper vertical rotations of the pendulumusing feedback (additional control action). For the chosen type of feedback,regions of asymptotic stability of the upper vertical rotations of the pendulum are constructedand possible bifurcations are analyzed. Special attention is also given to the question ofthe stability of periodic solutions arising as the vertical rotations lose stability.

The space of (2)-jets of a real function of two real variables, denoted by (J^{2}(mathbb{R}^{2},mathbb{R})), admits the structure of a metabelian Carnot group, so (J^{2}(mathbb{R}^{2},mathbb{R})) has a normal abelian sub-group (mathbb{A}). As any sub-Riemannian manifold, (J^{2}(mathbb{R}^{2},mathbb{R})) has an associated Hamiltonian geodesic flow. The Hamiltonian action of (mathbb{A}) on (T^{*}J^{2}(mathbb{R}^{2},mathbb{R})) yields the reduced Hamiltonian (H_{mu}) on (T^{*}mathcal{H}simeq T^{*}(J^{2}(mathbb{R}^{2},mathbb{R})/mathbb{A})), where (H_{mu}) is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian (H_{mu}) is non-integrable by meromorphic functions for some values of (mu). This result suggests the sub-Riemannian geodesic flow on (J^{2}(mathbb{R}^{2},mathbb{R})) is not meromorphically integrable.

The paper deals with multi-layer canard cycles, extending the results of [1]. As a practical tool we introduce the connection diagram of a canard cycle and we show how to determine it in an easy way. This connection diagram presents in a clear way all available information that is necessary to formulate the main system of equations used in the study of the bifurcating limit cycles. In a forthcoming paper we will show that both the type of the layers and the nature of the connections between the layers play an essential role in determining the number and the bifurcations of the limit cycles that can be created from a canard cycle.

We study relative equilibria ((RE)) for the three-body problemon (mathbb{S}^{2}),under the influence of a general potential which only depends on(cossigma_{ij}) where (sigma_{ij}) are the mutual anglesamong the masses.Explicit conditions formasses (m_{k}) and (cossigma_{ij})to form relative equilibrium are shown.Using the above conditions,we study the equal masses caseunder the cotangent potential.We show the existence ofscalene, isosceles, and equilateral Euler (RE), and isoscelesand equilateral Lagrange (RE).We also show thatthe equilateral Euler (RE) on a rotating meridianexists for general potential (sum_{i<j}m_{i}m_{j}U(cossigma_{ij}))with any mass ratios.

We consider a slow-fast Hamiltonian system with one fast angle variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles in an inhomogeneous magnetic fieldunder the influence of high-frequency electrostatic waves. Trajectories of the system averaged over the fast phase cross the resonant surface.The fast phase makes (simfrac{1}{varepsilon}) turns before arrival at the resonant surface ((varepsilon) is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival at the resonancewas derived earlier in the context of study of charged particle dynamics on the basis of heuristicconsiderations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove that its accuracy is (O(sqrt{varepsilon})) (up to a logarithmic correction). This estimate for the accuracy is optimal.

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones.This gives rise to an (upper semicontinous) torsion function,defined on the base of the system, which is a new(C^{0}) (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum ofthe torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.We examine the relevance of these results in the context of integrable Hamiltoniansystems or diffeomorphisms, with the particular cases of (C^{0})-integrable twist maps on the annulus and geodesic flows.Finally, we bound from below the polynomial entropy of (ell)-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.