Regular and Chaotic Dynamics最新文献

For (n) bodies moving in Euclidean (d)-space under the influence of ahomogeneous pair interaction wecompactify every center of mass energy surface, obtaining a(big{(}2d(n-1)-1big{)})-dimensional manifold with corners in the sense of Melrose.After a time change, the flow on this manifold is globally definedand nontrivial on the boundary.

We consider standard-like/Froeschlé dissipative mapswith a dissipation and nonlinear perturbation. That is,

where (pin{mathbb{R}}^{D}), (qin{mathbb{T}}^{D}) are the dynamicalvariables. We fix a frequency (omegain{mathbb{R}}^{D}) and study the existence ofquasi-periodic orbits. When there is dissipation, havinga quasi-periodic orbit of frequency (omega) requiresselecting the parameter (mu), called the drift.

We first study the Lindstedt series (formal power series in (varepsilon)) for quasi-periodic orbits with (D) independent frequencies and the drift when (gammaneq 0).We show that, when (omega) isirrational, the series exist to all orders, and when (omega) is Diophantine,we show that the formal Lindstedt series are Gevrey.The Gevrey nature of the Lindstedt series above was shownin [3] using a more general method, but the present proof israther elementary.

We also study the case when (D=2), but the quasi-periodic orbitshave only one independent frequency (lower-dimensional tori).Both when (gamma=0) and when (gammaneq 0), we showthat, under some mild nondegeneracy conditions on (V), thereare (at least two) formal Lindstedt series defined to all ordersand that they are Gevrey.

We review some recent results on a class of maps, called Bowen – Series-like maps, obtained from a class of group presentations for surface groups. These maps are piecewise homeomorphisms of the circle with finitely many discontinuities. The topological entropy of each map in the class and its relationship with the growth function of the group presentation is discussed, as well as the computation of these invariants.

The Lambert problem consists in connecting two given points in a given lapse of time under the gravitational influence of a fixed center. While this problem is very classical, we are concerned here with situations where friction forces act alongside the Newtonian attraction. Under some boundedness assumptions on the friction, there exists exactly one rectilinear solution if the two points lie on the same ray, and at least two solutions traveling in opposite directions otherwise.

The purpose of this paper is a pedagogical one. We provide a short and self-contained account of Siegel’s theorem, as improved by Bruno, which states that a holomorphic map of the complex plane can be locally linearized near a fixed point under certain conditions on the multiplier. The main proof is adapted from Bruno’s work.

In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian (N)-body problem.Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds foraction values. Explicit formulae will be proved by iterative arguments.We demonstrate some applications to action-minimizing solutions for three- and four-body problems.

Chirikov’s celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigorous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring, respectively, the distance to resonance overlap and nonintegrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov’s criterion.

We propose a new approach to the theory of normal forms for Hamiltonian systems near a nonresonant elliptic singular point. We consider the space of all Hamiltonian functions with such an equilibrium position at the origin and construct a differential equation in this space. Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along the flow of this equation correspond to canonical coordinate changes. So, we have a continuous normalization procedure. The formal aspect of the theory presents no difficulties.As usual, the analytic aspect and the problems of convergence of series are nontrivial.

It is well known that a planar central configuration of the (n)-body problem gives rise to a solution where eachparticle moves in a Keplerian orbit with a common eccentricity (mathfrak{e}in[0,1)). We callthis solution an ellipticrelative equilibrium (ERE for short). Since each particle of the ERE is always in the sameplane, it is natural to regardit as a planar (n)-body problem. But in practical applications, it is more meaningful toconsider the ERE as a spatial (n)-body problem (i. e., each particle belongs to (mathbb{R}^{3})).In this paper, as a spatial (n)-body problem, we first decompose the linear system of ERE intotwo parts, the planar and the spatial part.Following the Meyer – Schmidt coordinate [19], we give an expression for the spatial part andfurther obtain a rigorous analytical method to study the linear stability ofthe spatial part by the Maslov-type index theory. As an application, we obtain stability results for some classical ERE, including theelliptic Lagrangian solution, the Euler solution and the (1+n)-gon solution.

We present the almost global in time existence result in [13]of small amplitude space periodicsolutions of the 1D gravity-capillary water waves equations with constant vorticityand we describe the ideas of proof.This is based on a novel Hamiltonian paradifferentialBirkhoff normal form approach for quasi-linear PDEs.