Symmetric Periodic Orbits Research Articles

CONNECTING SYMMETRIC AND ASYMMETRIC FAMILIES OF PERIODIC ORBITS IN SQUARED SYMMETRIC HAMILTONIANS

In this work, we study a generic squared symmetric Hamiltonian of two degrees of freedom. Our aim is to show a global methodology to analyze the evolution of the families of periodic orbits and their bifurcations. To achieve it, we use several numerical techniques such as a systematic grid search algorithm in sequential and parallel, a fast chaos indicator and a tool for the continuation of periodic orbits. Using them, we are able to study the special and generic bifurcations of multiplicity one that allow us to understand the dynamics of the system and we show in detail the evolution of some symmetric breaking periodic orbits.

Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems

We study a one-parameter family of symmetric piecewise linear differential systems in R 3 which is relevant in control theory. The family, which has some intersection points with the adimensional family of Chua’s circuits, exhibits more than one attractor even when the two matrices defining its dynamics in each zone are stable, in an apparent contradiction to the three-dimensional Kalman’s conjecture. For these systems we characterize algebraically their symmetric periodic orbits and obtain a partial view of the one-parameter unfolding of its triple-zero degeneracy. Having at our disposal exact information about periodic orbits of a family of nonlinear systems, which is rather unusual, the analysis allows us to assess the accuracy of the corresponding harmonic balance predictions. Also, it is shown that certain conditions in Kalman’s conjecture can be violated without losing the global asymptotic stability of the origin.

Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

We extend our previous analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized fully symmetric equal mass four-body problem to the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for $m$ between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when $0.54\leq m\leq 1$, and are linearly unstable when $0<m\leq0.53$.

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m1, m2 and m3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.

On motions around the collinear libration points in the elliptic restricted three-body problem

Motions around the collinear libration points in the elliptic restricted three-body problem are studied. Literal expansions of the Lissajous orbits and the halo orbits are obtained. These expansions depend on two amplitude parameters and the orbital eccentricity of the two primaries. Numerical simulations are done to check the validity of these literal series and to compare them with the results in the circular restricted three-body problem. According to the properties of these literal expansions, three kinds of symmetric periodic orbits around the collinear libration points are discussed. Finally, applications of these results in the Earth–Moon system and the Sun–Earth+Moon system are discussed.

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.

The 1/1 resonance in extrasolar systems

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.

Particle Dynamics in a Maxwell’s Ring-Type Configuration with a Radiating Central Primary

Maxwell’s ring-type configuration (i.e. an N-body model where the ν = Ν − 1 bodies have equal masses and are located at the vertices of a regular ν-gon while the N-th body with a different mass is located at the center of mass of the system) has attracted special attention during the last 15 years and many aspects of it have been studied by considering Newtonian and post-Newtonian potentials (Mioc and Stavinschi 1998, 1999), homographic solutions (Arribas et al. 2007) and relative equilibrium solutions (Elmabsout 1996), etc. An equally interesting problem, known as the ring problem of (N + 1) bodies, deals with the dynamics of a small body in the combined force field produced by such a configuration. This is the problem we are dealing with in the present paper and our aim is to investigate the variations in the dynamics of the small body in the case that the central primary is also a radiating source and therefore acts on the particle with both gravitation and radiation. Based on the general outlines of Radzievskii’s model, we study the permitted and the existing trapping regions of the particle, its equilibrium locations and their parametric variations as well as the existence of focal points in the zero-velocity diagrams. The distribution of the characteristic curves of families of planar symmetric periodic orbits and their stability for various values of the radiation coefficient of the central body is additionally investigated.

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem.

Symmetric horseshoe periodic orbits in the general planar three-body problem

We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses. The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal to 35×10−5, corresponding to 105 times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies follow the same path is discussed.

Periodic Motions and their Stability in a N = ν + 1-body Regular Polygonal Configuration

We present a systematic investigation of the parametric evolution of both retrograde and direct families of periodic motions as well as their stability in the inner region of the peripheral primaries of the planar N-body regular polygonal configuration (ring model). In particular, we study the change of the bifurcation points as well as the change of the size and dynamical structure of the rings of stability for different values of the parameters ν = N−1 (number of peripheral primaries) and β (mass ratio). We find some types of bifurcations of families of periodic motions, namely period doubling pitchfork bifurcations, as well as bifurcations of symmetric and non-symmetric periodic orbits of the same period. For a given value of N − 1, the intervals Δx and ΔC of the rings of stability (where the periodic orbits are stable) of both retrograde and direct families increase with β increasing, while for a given value of β, the interval ΔC decreases with increasing N − 1. In general, it seems that the dynamical properties of the system depend on the ratio (N − 1)/β. The size of each ring of stability tends to zero as the ratio (N − 1)/β → ∞, that is, if N − 1→∞ or β → 0, the size of each ring of stability tends to zero (Δx → 0 and ΔC → 0) and, in general, the retrograde and direct families tend to disappear. This study gives us interesting information about the evolution of these two families and the changes of the bifurcation patterns since, for example, in some cases the stability index A oscillates between −1 ≤ Α ≤ + 1. Each time the family becomes critically stable a new dynamical structure appears. The ratios of the Jacobian constant C between the successive critical points, Ci/Ci+1, tend to 1. All the above depend on the parameters N − 1, β and show changes in the topology of the phase space and in the dynamical properties of the system.

Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall

The motion of an organism swimming at low Reynolds number near an infinite straight wall with a finite-length gap is studied theoretically within the framework of a two-dimensional model. The swimmer is modelled as a point singularity of the Stokes equations dependent on a single real parameter. A dynamical system governing the position and orientation of the model swimmer is derived in analytical form. The dynamical system is studied in detail and a bifurcation analysis performed. The analysis reveals,inter alia, the presence of stable equilibrium points in the gap region as well as Hopf bifurcations to periodic bound states. The reduced-model system also exhibits a global gluing bifurcation in which two symmetric periodic orbits merge at a saddle point into symmetric ‘figure-of-eight’ bound states having more complex spatiotemporal structure. The additional effect of a background shear is also studied and is found to introduce new types of bound state. The analysis allows us to make theoretical predictions as to the possible behaviour of a low-Reynolds-number swimmer near a gap in a wall. It offers insights into the use of gaps or orifices as possible control devices for such swimmers in confined environments.

Dynamics of Symmetric Dynamical Systems with Delayed Switching

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value, so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity-induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincaré map near the colliding periodic orbit. The Poincaré map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.

Analysis of Periodic and Quasi-Periodic Orbits in the Earth-Moon System

O NE of the earliest investigations of the Poincare surfaces of section method [1] to discover periodic orbits in the restricted three-body problemwas byHenon [2–5] in a number of publications. Jefferys [6] also used this method to evolve a large number of periodic orbits in the restricted three-body problem. After that, this technique has been used by many researchers to find periodic orbits in the restricted three-body problem. Some of the important contributions are by Smith [7], Smith and Szebehely [8], Tuckness [9,10], Scott and Spencer [11], Howell and Kakoi [12], Demeyer and Gurfil [13], Guilera [14], and Kolemen et al. [15]. The motivation for the present study comes from the work of Winter [16], which deals with a family of simply symmetric retrograde periodic orbits around the moon in the rotating Earth–moon–particle system in the framework of the planar circular restricted three-body problem for the Earth–moon mass ratio (0.01215). This family of periodic orbits is one particular case of the family of periodic orbits classified by Broucke [17].

BIFURCATIONS AND CHAOS IN HAMILTONIAN SYSTEMS

This paper deals with the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom of Hamiltonian systems. These numerical methods are based on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and the skeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems are studied: the Copenhagen and the (n + 1)-body ring problems and the Hénón–Heiles Hamiltonian. All the numerical integrations have been done by using the state-of-the-art numerical library TIDES based on the extended Taylor series method.

The Planar Three-Body Problem, Symmetries and Periodic Orbits

We study the planar three-body problem using the principal axes of inertia frame, determined by one Euler angle in the planar case. Three variables are needed in this frame: two distances R1 and R2, related with the principal moments of inertia on the plane of motion of the particles, and an auxiliary angle σ. We also connect these coordinates with the shape sphere of the similarity class of triangles and we give a geometric interpretation of the angle σ. We then write the Hamiltonian and equations of motion in these coordinates to find the symmetries of involution and symmetric periodic orbits. To exemplify this method, we calculate periodic orbits of a three-body problem with two or three equal masses.

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincare surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j–k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ– C characteristic curves of these families extend to the left of the μ– C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* ( μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun–Jupiter and Earth–Moon systems. We found new μ– C and x– C characteristic curves. Along each μ– C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x– C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ– C or the x– C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L 4 and L 5 are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ– C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x– C curve, that spiral into the points L 4 and L 5, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.

Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent discoveries in the n-body problem. The key ingredient is a generalized Bott-type iteration formula for periodic solution in the presence of finite group action on the orbit. For second order system, we prove, under general boundary conditions, the close formula for the relationship between the Morse index of an orbit in a Lagrangian system and the Maslov index of the fundamental solution for the corresponding orbit in its Hamiltonian system counterpart, and the boundary conditions cover the cases which appeared in the n-body problem. As an application we consider the stability problem of the celebrated figure-eight orbit due to Chenciner and Montgomery in the planar three-body problem with equal masses, and we clarify the relationship between linear stability and its variational nature on various loop spaces. The basic idea is as follows: the variational characterization of the figure-eight orbit provides information about its Morse index; based on its relation to the Maslov index, our stability criteria come into play.

ON PERIODIC ORBITS OF 3D SYMMETRIC PIECEWISE LINEAR SYSTEMS WITH REAL TRIPLE EIGENVALUES

The counter-intuitive appearance of stable periodic orbits in three-dimensional piecewise linear systems has been recently reported for stable saturated control systems with high gain and real triple eigenvalues [Moreno &amp; Suárez 2004]. In this letter using several mathematical tools, the reported sudden phenomenon is explained, and the analysis is completed by providing a global bifurcation diagram of the symmetric periodic orbits. Approximate methods are used only to illustrate the nonlinear behavior with respect to the bifurcation parameter. Analytical methods are employed to rigorously prove the main result. The employed techniques are useful not only for the family studied but also for generic three-dimensional symmetric piecewise linear systems.