Asymmetric Periodic Orbits Research Articles

Dynamics around non-spherical symmetric bodies – I. The case of a spherical body with mass anomaly

ABSTRACT The space missions designed to visit small bodies of the Solar system boosted the study of the dynamics around non-spherical bodies. In this vein, we study the dynamics around a class of objects classified by us as non-spherical symmetric bodies, including contact binaries, triaxial ellipsoids, and spherical bodies with a mass anomaly, among others. In this work, we address the results for a body with a mass anomaly. We apply the pendulum model to obtain the width of the spin–orbit resonances raised by non-asymmetric gravitational terms of the central object. The Poincaré surface of section technique is adopted to confront our analytical results and to study the system’s dynamics by varying the parameters of the central object. We verify the existence of two distinct regions around an object with a mass anomaly: a chaotic inner region that extends beyond the corotation radius and a stable outer region. In the latter, we identify structures remarkably similar to those of the classical restrict and planar three-body problem in the Poincaré surface of sections, including asymmetric periodic orbits associated with 1:1+p resonances. We apply our results to a Chariklo with a mass anomaly, obtaining that Chariklo rings are probably related to first kind periodic orbits and not with 1:3 spin–orbit resonance, as proposed in the literature. We believe that our work presents the first tools for studying mass anomaly systems.

Puzzling out the coexistence of terrestrial planets and giant exoplanets

Aims. Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Methods. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Using the detrended fast Lyapunov indicator as a chaotic indicator, we constructed maps of dynamical stability by varying both the eccentricity of the outer giant planet and the semi-major axis of the inner terrestrial planet, and identify the boundaries of the stability domains. Guided by the computation of families of periodic orbits, the phase space is unravelled by meticulously chosen stable periodic orbits, which buttress the stability domains. Results. We provide all possible stability domains for coplanar symmetric configurations and show that a terrestrial planet, either in mean-motion resonance or not, can coexist with a giant planet, when the latter moves on either a circular or an (even highly) eccentric orbit. New families of symmetric and asymmetric periodic orbits are presented for the 2/1 resonance. It is shown that an inner terrestrial planet can survive long time spans with a giant eccentric outer planet on resonant symmetric orbits, even when both orbits are highly eccentric. For 22 detected single-planet systems consisting of a giant planet with high eccentricity, we discuss the possible existence of a terrestrial planet. This study is particularly suitable for the research of companions among the detected systems with giant planets, and could assist with refining observational data.

Inclined asymmetric librations in exterior resonances

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.

Families of asymmetric periodic solutions in the restricted four-body problem

Very recently, we presented five of the basic families of the network of periodic orbits of the restricted four-body problem which are simple, i.e. one intersection with the horizontal $x$ -axis at the half period, symmetric with respect to the same axis and asymmetric with respect to the vertical $y$ -axis. In the present work, using these families, we found series of asymmetric critical orbits for various values of the primaries $m_{2}$ and $m_{3}$ . From these critical orbits we calculate and present five new families of simple periodic orbits which are asymmetric with respect to both the $x$ - and $y$ -axis. Additionally, we describe a grid method in the $(x_{0}, \dot{x}_{0})$ plane and we obtain initial conditions for new asymmetric double-periodic orbits. We determine ten families of asymmetric double-periodic orbits from the bifurcations of the previous five asymmetric families using the special generating horizontally critical periodic orbits. The stability of each calculated asymmetric periodic orbit is also studied. Characteristic curves as well as stability diagrams of these families are illustrated. In the last section we present the evolution of the five basic families of simple asymmetric periodic orbits when the primaries are the Sun the Jupiter and the 2797 Teucer Asteroid.

Synchronization transition with coexistence of attractors in coupled discontinuous system

The studies of extended dynamics systems are relevant to the understanding of spatiotemporal patterns observed in diverse fields. One of the well-established models for such complex systems is the coupled map lattices, and several features of pattern formation including synchronization, unsynchronization, traveling waves and clustering synchronization are found. Among the above-mentioned patterns, chaotic synchronization has been intensively investigated in recent years. It has been demonstrated that two or more chaotic systems can be synchronized by linking them with mutual coupling or a common signal or some signals. Over the last decade, a number of theoretical methods have been presented to deal with this problem, such as the methods of master stability functions and eigenvalue analysis. While much effort has been devoted to the networks with different topological structures in continuous systems. The coupled discontinuous maps have been investigated with increasing interest in recent years, they showed that the complete synchronization in coupled discontinuous systems is more complicated than in coupled continuous systems. However, a similar problem of synchronization transition in coupled discontinuous systems is much less known.The synchronization transition in coupled discontinuous map lattices is studied. The average order parameter and maximal Lyapunov exponent are calculated to diagnose the synchronization of coupled piecewise maps. The results indicate that there exist the periodic clusters and the synchronization state, and a new transition style from periodic cluster states to complete synchronization states is found. The periodic cluster states consist of two kinds of periodic orbits: symmetric periodic orbits and asymmetric periodic orbits.Based on the pattern analysis, the common features of the patterns are constructed by the two periodic attractors, and the periodic orbit is an unstable periodic orbit of the isolate map. The discontinuities in a system can divide the phase space into individual zones of different dynamical features. The interactions between the local nonlinearity and the spatial coupling confine orbit into certain spaces and form a dynamic balance between two periodic clusters. The system can reach complete synchronization states when the balance is off. It is shown that synchronization transition of the coupled discontinuous map can exhibit the different processes, which depends on coupling strength. Four transition modes are found in coupled discontinuous map: 1) the transition, from the coexistence of chaotic synchronization and chaotic un-synchronization states to the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits; 2) the transition from the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits; 3) the transition from the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization and symmetric periodic orbits; 4) the transition from the coexistence of chaotic synchronization and symmetric periodic orbits to the chaotic synchronization. Because the local dynamics has discontinuous points, the coupled system shows a riddle basin characteristic in the phase space, and the synchronization transition of coupled piecewise maps looks more complex than continuous system.

Periodic orbits of the Hill problem with radiation and oblateness

The modification of Hill’s problem where the primary is radiating and the secondary is an oblate spheroid is considered. The evolution of the network of the basic families of planar periodic orbits for various values of the parameters of the problem is studied. For specific values of the parameters these families are determined accurately together with their stability properties. The stability of retrograde satellites in an appropriate space of initial conditions is also determined by means of surface of section portraits of the Poincare map and higher order resonances are studied. Simple asymmetric periodic orbits of the problem are also determined.

ON THE BIFURCATION AND CONTINUATION OF PERIODIC ORBITS IN THE THREE BODY PROBLEM

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem, asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These families of periodic orbits continue existing and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario first described by Bozis and Hadjidemetriou.

Planar asymmetric periodic orbits in four dipole problem

In the rings of Saturn the Lorentz field, co-acting with the gravity field of the planet, limits the motion of all charged particles and small size grains with surface charges inside a layer of about 200 meters thickness. In these regions there is enormous three dimensional instability. So, the motion of the charged particles cannot be studied by the theory of Stormer’s problem of the magnetic field of a single dipole, where generally there is not instability in the third dimension. For this reason we follow the assumption that the magnetic field of Saturn is composed by many dipoles in rotation and we have extended Sormer’s problem considering four magnetic dipoles in motion, where in this environment there is three dimensional instability. In this way we study the motions of the charged particles and we reveal families of planar asymmetric periodic orbits, we calculate their horizontal and vertical stability and we give their bifurcations with families of three dimensional asymmetric periodic orbits.

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.

Erratum to: Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

In our effort to extend our present results on the asymmetric periodic orbits in the restricted three-body problem, we found that the results in this paper concerning the case of the Earth-Moon mass distribution (subsection 2.2) are not correct. The equations of motion of the restricted problem have the symmetry property, with respect to the x-axis, that if x = x(t), y = y(t) is a particular solution, then x = x(-t), y = -y(-t) is also a solution (see Szebehely 1967, p. 426). The corresponding symmetry property, with respect to y-axis, is also true in the case where the two primary bodies are equal, i.e. l = 0.5 (subsection 2.1 in our results) but it is not true for l = 0.5 (subsection 2.2 in our results). So the results of subsection 2.2 should be ignored. The numerical determination of the asymmetric periodic orbits in this case (l = 0.5), needs a different procedure which we are applying at this time and will present in a future article.

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.

Asymmetric periodic orbits in the photogravitational Copenhagen problem

In this paper we study the O x -asymmetric solutions of the planar photogravitational restricted three-body problem in the case of primaries with equal masses and equal values of the radiation pressure parameters. In particular, we are concerned with the families of asymmetric orbits which bifurcate from the well known families a, b, and c. Their evolution is examined via the numerical construction of series of the critical bifurcation points of a, b, and c with respect to the variation of the common radiation parameter q . We also present some illustrative cases of these families for several values of this parameter. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, the equations of motion of the problem are regularized by using the Levi-Civita transformations.

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L1, L2 and L3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.

Chaos, Order, and Periodic Orbits in 3 : 1 Resonant Planetary Dynamics

This study addresses the long-term evolution of possible two-planet extrasolar systems that are initially trapped in 3:1 mean motion resonance. A planar general three-body problem model is used, and its resonant dynamics are examined by computing periodic orbits in a rotating frame, Poincare maps, and maps of dynamical stability. We computed the families of symmetric resonant periodic orbits that obey bifurcations, giving rise to families of asymmetric periodic orbits. The linear stability of such orbits has also been computed, and their relation to the long-term stability of their nearby phase-space domain has been studied. The maps of dynamical stability reveal a complicated structure in the phase space, where chaos and order coexist and alternate as the initial eccentricities or the phases of the planets change. The regular orbits are classified into various types according to the librating or rotating evolution of the resonant angles. Apsidal symmetric librations are common in the domain of resonant motion, but asymmetric ones are associated exclusively with the existence of asymmetric periodic orbits. Such a stable asymmetric configuration seems to correspond to the companions b and c of the 55 Cnc extrasolar system, which are trapped in the 3:1 mean motion resonance according to the study of McArthur et al. However, a recent study by Fischer et al. shows the existence of a new planet (the companion f) in the system, and that the planets b and c are not in mean motion resonance.

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n − 1t), (3:1, 5:3, ...) and (3) all other resonan topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.

Motion Around The Triangular Equilibrium Points Of The Restricted Three-Body Problem Under Angular Velocity Variation

We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L3, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2 $$\sqrt2$$ , at which the triangular equilibria disappear by coalescing with the inner collinear equilibrium point L1. We also compute the horizontal and the vertical stability of these families for the angular velocity parameter ω under consideration. Series of horizontal–critical periodic orbits of the short-Trojan families with the angular velocity ω and the mass ratio μ as parameters, are given.

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m 1>m 2, the system HD82943 for the new masses, for the case m 1=m 2 and the same system HD82943 for the values of the masses m 1<m 2 given in previous work. In the case m 1≥ m 2 there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m 1<m 2 all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by $$\varpi_2-\varpi_1$$ and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.

Symmetric and Nonsymmetric Periodic Orbits in the Exterior Mean Motion Resonances with Neptune

We present families of periodic orbits and their stability for the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune in the framework of the planar circular restricted three-body problem. We found that in each resonance there exist two branches of symmetric elliptic periodic orbits with stable and unstable segments. Asymmetric periodic orbits bifurcate from the corresponding symmetric ones. Asymmetric periodic orbits are stable and the motion in their neighbourhood is a libration with respect to the resonant angle variable. In all the families of asymmetric periodic orbits the eccentricity extends to high values. Poincare sections reveal the changes of the topology in phase space.

Computer analysis of periodic orbits of discontinuous vector fields

Computer algebra provides powerful tools to perform a precise and qualitative analysis of the phase portraits of a class of discontinuous vector fields of R 4 . We derive from the analysis of semi-algebraic varieties various results about the existence of families of periodic orbits. The existence of both symmetric and asymmetric periodic orbits for reversible discontinuous vector fields is proved.