Symmetric Periodic Orbits Research Articles

Classical chaotic scattering-periodic orbits, symmetries, multifractal invariant sets

The infinite set of periodic orbits of a chaotic system is investigated. Their globally exact symbolic dynamics is related to Birkhoff's method of iterated symmetry lines which produces symmetric periodic orbits of various classes. A complete picture emerges as to what part of the periodic orbits can be obtained by means of the Birkhoff lines. In addition, the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits.

A restricted charged four-body problem

We consider a restricted charged four body problem which reduces to a two degrees of freedom Hamiltonian system, and prove the existence of infinite symmetric periodic orbits with arbitrarily large extremal period. Also, it is shown that an appropriate restriction of a Poincare map of the system is conjugate to the shift homeomorphism on a certain symbolic alphabet.

Periodic orbits for reversible, symplectic mappings

A 2 N-dimensional symplectic mapping which satisfies the twist condition is obtained from a Lagrangian generating function F( q, q′) . The q's are assumed to be angle variables. Reversible maps of this form have 2 N−1 invariant symmetry sets which are Lagrangian manifolds. We show that such maps have at least 2 N symmetric periodic orbits for each frequency ω=( m,n) . Furthermore we show there is at least one orbit which minimizes the periodic action for each ω, and generically at least 2 N−1 other “minimax” orbits. There is a “dominant” symmetry plane on which the minimizing orbit is never observed to occur. As a parameter varies, the symmetric orbits are observed to undergo symmetry breaking bifurcations, creating a pair of non-symmetric orbits. A pair of coupled standard maps provides a four-dimensional example. For this case the two-dimensional symmetry planes give a cross section of the resonances and a visualization of the Arnol'd web.

AN ANALYTICAL STUDY OF EQUAL PERIODIC BIFURCATIONS IN REVERSIBLE AREA PRESERVING MAPS

In this paper, the general structure of linear Jacobian matrices of even periodic orbits for reversible area preserving maps is obtained and two kinds of bifurcation behaviour of symmetric periodic orbits are discussed from the above structure. We present the conditions and the analytical criterions which can distinguish three types for equal periodic bifurcations of reversible area preserving maps. The applications of this analytical method are illustrated with several examples of De Vogelaere map.

Symmetric motions in the Equatorial Magnetic-Binary Problem

Classical numerical techniques are applied to find families of symmetric periodic orbits in the Equatorial Magnetic-Binary Problem. Stability for each orbit is also studied by means of variational methods. Finally an example in a concrete system is given to verify the procedure proposed.

Stability and bifurcations of symmetric periodic orbits

Se un problema Hamiltoniano integrabile non degenere, come si puo ottenere dal problema dei due corpi in coordinate rotanti, e perturbato con una funzione potenziale simmetrica rispetto ad un asse, le proprieta di simmetria delle soluzioni consentono di semplificare la ricerca, di orbite periodiche. In tal modo si ottiene un teorema di continuazione delle orbite periodiche che fornisce piu informazioni di quello classico. La funzione che descrive la simmetria delle orbite e genericamente una funzione di Morse, e le biforcazioni di orbite periodiche simmetriche possono essere descritte in termini di punti singolari e di valori critici di tale funzione. II problema ristretto dei tre corpi ed il problema del satellite in un potenziale non axisimmetrico sono trattati come esempi.

Periodic orbits of the general three-body problem for the Sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 2∶5 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.

Symmetric rectilinear periodic orbits of three bodies

In this paper the existence of families of symmetric periodic orbits in the rectilinear three body problem with the middle mass much larger than the masses on the outside is rigorously established. A number of these families are continued numerically and their stability properties as orbits of the planar general problem of three bodies are studied.

Stability and Bifurcations of Symmetric Periodic Orbits in the Restricted 3-Body Problem

AbstractThe continuation of symmetric periodic orbits can be described in terms of “symmetry functions”; the branching of the zero-level lines in a neighbourhood of a critical point gives rise to the transition from “first kind” to “second kind” periodic orbits. When the families are parametrized with the Jacobi integral, the bifurcations can be described as “catastrophes” of the generating functions. However bifurcations of higher order are more complex than the generic catastrophes with one parameter: both symmetric and asymmetric bifurcations occur.In this way the symmetric periodic orbits that do not have close approaches to the secondary body can be described in an analytic way and their stability can be deduced from simple bifurcation rules. However numerical experiments are required to determine the “natural termination” of the families.

On the Stability of Resonant Asteroid Orbits

AbstractThe stability of the asteroid orbits has been studied by the method of surface of section. Families of simple symmetric periodic orbits of the asteroid and their stability have been computed and this served as a guide for the selection of the energy levels for the surface of section. In this way all possible cases for the structure of phase space have been obtained. It was found that the region in phase space around the resonant orbits at the resonances 1/3, 3/5, 5/7,.... is unstable, but small stability regions of doubly symmetric periodic orbits near the above resonances are also present. At the resonances 1/2, 2/3, 3/4, .... it was found that there exist two separate regions in phase space at about the same resonance 1/2, 2/3, 3/4,...., respectively, one being stable and the other unstable. At certain energy levels only the stable region appears. The above results are consistent with the observed distribution of the asteroids.

The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

Symmetric Periodic Orbits in the Anisotropic Kepler Problem

AbstractThe anisotropic Kepler problem has a group of symmetries with three generators; they are symmetries respect to zero velocity curve and the two axes of motion’s plane. For a fixed negative energy level it has four homothetic orbits. We describe the symmetric periodic orbits near these homothetic orbits. Full details and proofs will appear elsewhere (Casasayas-Llibre).

On the restrited three-body problem when the mass parameter is small

We study some aspects of the restricted three-body problem when the mass parameter μ is sufficiently small. First, we describe the global flow of the two-body rotating problem, μ=0, and we use it for the analysis of the collision and parabolic orbits when μ≳0. Also we show that for any fixed value of the Jacobian constant and for any e>0, there exists a μ0>0 such that if the mass parameter μ∈[0,μ0], then the set of bounded orbits which are not contained in the closure of the set of symmetric periodic orbits has Lebesgue measure less than e.

Examples of predominantly two-body orbits in the Sun-Jupiter asteroid system

Accurate numerical continuation of families of plane symmetric direct periodic orbits around the large primary in the Sun-Jupiter case of the restricted problem of three bodies allows the determination of the ‘vertical branching points’ where families of three-dimensional symmetric periodic orbits bifurcate from the planar ones. Three families of plane periodic orbits, and the initial segments of ten bifurcating families of three-dimensional ones are determined. The stability of these families is examined and examples of their orbits are illustrated.

Application of Hamilton's law of varying action to the restricted three-body problem

The ‘Law of Varying Action’, originally published by Hamilton in 1834, was recently employed by Bailey in devising a technique for generating power series characterizing the motions of dynamical systems. Furthermore, Bailey's method permits one to construct these series in a simple and direct way, without using the associated differential equations of motion. In the present paper, Hamilton's law is developed in its most general form and is used to produce series solutions of the restricted three-body problem. Finally, for illustrative purposes, numerical results are presented for several symmetric periodic orbits.

The family I3υ of the three-dimensional general three-body problem (Parts I and II)

We present the biparametric family I3υ of symmetric periodic orbits of the three-dimensional general three-body problem, found by numerical continuation of the vertical critical orbit I3υ of the circular restricted three-body problem. The periodic orbits refer to a suitably chosen rotating frame of reference.

General three-body problem: Families of three-dimensional periodic orbits (part II)

In this paper we present a two-parametric family of symmetric periodic orbits of the three-dimensional general three-body problem, found numerically by continuation of a vertical critical orbit of the circular restricted three-body problem. The periodic orbits refer to a suitably defined rotating frame of reference.

Numerical investigation of the manifold I qp in the Sun-Jupiter case of the restricted three-body problem

Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.

The structure of periodic solutions in the restricted problem of three bodies

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.

Periodic three body orbits in the case of small third mass

Near a symmetric periodic orbit of the plane circular or elliptic restricted probelm, the conditions for a symmetric periodic orbit of the plane general three body problem are reduced, under a natural nondegeneracy condition, to the vanishing of a single real valued function. The implicit function theorem and Hormander's generalized Morse's lemma are then used to analyze the set of zeros of this function.