Regular and Chaotic Dynamics最新文献

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.

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.