Adjoint orbits of the odd real symplectic group

In this paper we classify the adjoint orbits of the odd symplectic group over the field of real numbers. We need a noneigenvalue modulus to classify certain orbits.      

The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic. Project 10071040 supported by NNSF, 200014 supported by Excellent. Ph.D. Funds of ME of China, and PMC Key Lab. of ME of China     

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.     

Density of first Poincaré returns,periodic orbits,and Kolmogorov–Sinai entropy

It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.     

A shadowing lemma for quasi-hyperbolic strings of flows

In this paper we prove a shadowing lemma for pseudo orbits made by quasi-hyperbolic strings. We allow singularities in question and hence, in particular, the quasi-hyperbolic strings are formulated by the rescaled linear Poincaré flow instead of the usual linear Poincaré flow. We also introduce the sectional Poincaré map and rescaled sectional Poincaré map for Lipschitz vector fields on Banach spaces in the article.     

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.     

Global periodic structure of integrable hyperbolic map

Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincaré–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincaré–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity.     

Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system

Let M be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose M consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the λ-lemma) describing the behavior of trajectories near M. Using this result, trajectories shadowing chains of homoclinic orbits to M are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order µ. As µ → 0, double collisions of small bodies correspond to a symplectic critical manifold M of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem.     

Pseudochaotic dynamics near global periodicity

In this paper, we study a piecewise linear version of kicked oscillator model: saw-tooth map. A special case of global periodicity, in which every phase point belongs to a periodic orbit, is presented. With few analytic results known for the corresponding map on torus, we numerically investigate transport properties and statistical behavior of Poincaré recurrence time in two cases of deviation from global periodicity. A non-KAM behavior of the system, as well as subdiffusion and superdiffusion, are observed through numerical simulations. Statistics of Poincaré recurrences shows Kac lemma is valid in the system and there is a relation between the transport exponent and the Poincaré recurrence exponent. We also perform careful numerical computation of capacity, information and correlation dimensions of the so-called exceptional set in both cases. Our results show that the fractal dimension of the exceptional set is strictly less than 2 and that the fractal structures are unifractal rather than multifractal.     

Symplectic homology and periodic orbits near symplectic submanifolds

We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.     

Preservation and destruction of periodic orbits by symplectic integrators

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.     

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.      

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers e ^±iϕ . On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.      

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Geodesic flow, connecting orbits and almost full foliation

We study in this article a special dynamical behavior of geodesic flow on T2. Our example shows that there is an area-preserving monotone twist map for which all minimal periodic orbits can be connected, and at the same time for a certain rational rotation number the minimal set is almost an invariant curve.     

Denominators for the Poincaré series of invariants of matrices with involution

The goal of this paper is to study some Poincaré series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables. The first author gratefully acknowledges Support from DePaul University Research Council. The second author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-Ilan University.     

Calculating the Poincaré Map for Two-Dimensional Periodic Systems and Riccati Equations

Differential Equations - Formulas for calculating the Poincaré period map for a two-dimensional linear periodic system of differential equations and for the Riccati differential equation are...     

Global Symplectic Polynomial Approximation of Area-Preserving Maps

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult.     

Periodic Image Trajectories in Earth–Moon Space

The problem of identifying orbits that enclose both the Earth and the Moon in a predictable way has theoretical relevance as well as practical implications. In the context of the restricted three-body problem with primaries in circular orbits, periodic trajectories exist and have the property that a third body (e.g. a spacecraft) can describe them indefinitely. Several approaches have been employed in the past for the purpose of identifying similar orbits. In this work the theorem of image trajectories, proven five decades ago, is employed for determining periodic image trajectories in Earth–Moon space. These trajectories exhibit two fundamental features: (i) counterclockwise departure from a perigee on the far side of the Earth, and (ii) counterclockwise arrival to a periselenum on the far side of the Moon. An extensive, systematic numerical search is performed, with the use of a modified Poincaré map, in conjunction with a numerical refinement process, and leads to a variety of periodic orbits, with various interesting features for possible future lunar missions.     

Normal forms of families of maps in the Poincaré domain

An analog of Brushlinskaya’s theorem about normal forms of deformations of vector fields in the Poincaré domain is proved; namely, it is proved that for each analytic map whose linear part at a fixed point belongs to the Poincaré domain and has different eigenvalues, the analytic normal form of a deformation of this map is polynomial and contains (in addition to the linear part) only monomials that are resonant for the unperturbed map. A global (with respect to the parameter) version of this theorem is also proved.