Distribution of periodic orbits and the homoclinic tangle

We study the distribution of regular and irregular periodic orbits on a Poincaré surface of section of a simple Hamiltonian system of 2 degrees of freedom. We explain the appearance of many lines of periodic orbits that form Farey trees. There are also lines that are very close to the asymptotic curves of the unstable periodic orbits. Some regular orbits, sometimes stable, are found inside the homoclinic tangle. We explain this phenomenon, which shows that the homoclinic tangle does not cover the whole area around an unstable orbit, but has gaps. Inside the lobes only irregular orbits appear, and some of them are stable. We conjecture that the opposite is also true, i.e. all irregular orbits are inside lobes.     

Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

Periodic orbits of galactic motion

Poincaré's surface of section method is used to find and classify the main periodic orbits in a two-dimensional galactic potential first introduced by Hénon and Heiles. The stability of these periodic orbits is studied. Numerical integration with Bulirsch-Stoer method is used.     

Periodic orbits of the Sitnikov problem via a Poincaré map

In this paper by means of a Poincaré map, we prove the existence of symmetric periodic orbits of the elliptic Sitnikov problem. Furthermore, using the presence of the Bernoulli shift as a subsystem of that Poincaré map, we prove that not all the periodic orbits of the Sitnikov problem are symmetric periodic orbits.This revised version was published online in October 2005 with corrections to the Cover Date.     

Application of Gauss's method in the formation of periodic solutions in the relativistic restricted problem of three bodies

The paper discusses the existence of periodic and quasi-periodic solutions in the space relativistic problem of three bodies with the help of Poincaré's small parameter method starting from non-Keplerian generating solutions, i.e., using Gauss's method. The main peculiarity of these periodic orbits is the fact that they close, in general, after many revolutions. It is worth noticing that these periodic orbits give a new class of periodic solutions of the classical circular problem of three bodies, if relativistic effects are neglected.     

A note on a conjecture of Poincaré

We prove the following weakened version of Poincaré's conjecture on the density of periodic orbits of the restricted three-body problem: The measure of Lebesgue of the set of bounded orbits which are not contained in the closure of the set of periodic orbits goes to zero when the mass parameter does.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Periodic orbits of an electric charge in a magnetic dipole field

Periodic orbits in the Stormer problem are studied using the symmetry lines of the Poincaré map introduced by De Vogelaere. Many known facts are explained by mean of these lines. The dynamics of four special symmetry lines when the Stormer parameter ₁ changes is presented, and we obtain a clear global view of the structure of the simple periodic orbits and their bifurcations, including the asymmetrical ones. New asymmetrical multiple periodic orbits are obtained.     

The Symmetrical One-dimensional Newtonian Four-body Problem: A Numerical Investigation

Numerical simulations of the one-dimensional Newtonian four-body problem have been conducted for the special case in which the bodies are distributed symmetrically about the centre of mass. Simulations show a great similarity between this problem and the one-dimensional Newtonian three-body problem. As in that problem the orbits can be divided into three different categories which form well-defined regions on a Poincaré section: there is a region of quasiperiodic orbits about a Schubart-like periodic orbit, there is a region of fast-scattering encounters and in between these two regions there is a chaotic scattering region. The Schubart-like periodic orbit's stability to perturbation is studied. It is apparently stable in one-dimension but is unstable in three-dimensions.This revised version was published online in October 2005 with corrections to the Cover Date.     

Numerical exploration of the dynamics of Self-Adjoint S-Type Riemann ellipsoids

A Riemann ellipsoid is a self-gravitating fluid whose velocity field is a linear function of the position coordinates. Though the theory of the equilibrium and stability is thoroughly developed, scarse attention has been paid to the dynamical behaviour.In this paper we present a numerical exploration of the phase-space structure for the Self-Adjoint S-Type Riemann ellipsoids via Poincaré surfaces of section, which reveal a rich and complex dynamical behaviour.Both the occurrence of chaos for certain values of the parameters of the system as well as the existence of periodic orbits are observed.We also considered ellipsoids embedded in rigid, homogeneous, spherical halos, obtaining evidence of the stabilizing effect of halos even in the case of finite-amplitude oscillations.Moreover, we show that the approximated equations of motion derived by Rosensteel and Tran (1991) fail to describe properly the phase-space structure of the problem.     

Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.     

Nonplanar Second Species Periodic and Chaotic Trajectories for the Circular Restricted Three-Body Problem

For the circular restricted three-body problem of celestial mechanics with small secondary mass, we prove the existence of uniformly hyperbolic invariant sets of non-planar periodic and chaotic almost collision orbits. Poincaré conjectured existence of periodic ones and gave them the name “second species solutions”. We obtain large subshifts of finite type containing solutions of this type.     

A mapping for the study of the 1/1 resonance in a galactic type Hamiltonian

We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p _x Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations.     

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.     

Twist and Non-Twist Bifurcations in a System of Coupled Oscillators

We consider a system of a harmonic and an unharmonic oscillator with a weak cubic coupling. We study the non-degenerate bifurcations of periodic orbits for the resonant tori of the unperturbed system for which the twist condition holds. We demonstrate that this system also exhibits for certain values of the small parameter non-twist bifurcations, where the rotation number of the Poincaré map attains a minimum value.     

Kolmogorov entropy as a measure of disorder in some non-integrable Hamiltonian systems

This paper is devoted to study the stochastic behaviour of some Hamiltonian systems with closed velocity curves. We investigate Hamiltonians already studied by Ali and Somorjai (1). These authors, by discussing Poincaré's surfaces of section for several energy values, gave a qualitative evaluation of the stochasticity of the systems.Here we present a quantitative study of this stochastic behaviour. For each energy we compute the Lyapunov characteristic exponents of fifty orbits chosen at random, in order to calculate the Kolmogorov entropy by Pesin's formula. Our results are in agreement with those of Ali and Somorjai: the disorder does not increase monotonically with increasing energy. However, we find that the largest entropy does not necessarily correspond to the maximum of the stochastic volume. The Kolmogorov entropy thus appears to be a good measure of the degree of disorder of dynamical systems.     

Stability of artificial and natural satellites

A quantitative measure of stability based on Hill's definition is evaluated for direct and retrograde satellite orbits. These orbits are known as Poincaré's first kind in the restricted problem of three bodies. Onsets of possible instabilities and captures are established. A critical (maximum) value of the satellite's orbital radius is found for stability as a remarkably simple function of the massparameter. The results are applied to the natural satellites of the solar system.     

Investigation of periodic orbits in elliptical galaxies using the method of implicit functions

Existence of periodic orbits inside elliptical galaxies has been investigated. Necessary conditions for regular, small amplitude periodic motion around the center of galaxy have been derived using implicit functions and solved by approximating through Taylor's series. The solution procedure requires to obtain functions of partial derivatives of dependent variables with respect to initial conditions. Derivation of these functions can be accomplished through solving a set of ordinary differential equations by proper choices of associated initial conditions. The results obtained show complete agreement with those obtained through the application of Poincaré-Lindstedt's method.     

Position and velocity sensitivities at the triangular libration points

The model of the circular restricted problem of three bodies is used to investigate the sensitivity of the third body motion when it is given a positional or velocity deviation away from the L₄ triangular libration point. The x-axis is used as a criteria for defining the stability of the third body motion. Poincaré's surfaces of section are used to compare the regions of periodic, quasi-periodic and stochastic motion to the trajectories found using the definition of stability (not crossing the x-axis) defined in this study. Values of the primary/secondary mass ratios () ranging from 0 to the linear critical value 0.038521... are investigated. Using this new form of stability measure, it is determined that certain values of are more stable than others. The results of this study are compared, and found, to give agreeable results to other studies which investigate commensurabilities of the long and short period terms of periodic orbits.