Symmetric Periodic Orbits Research Articles

Periodic orbits around 216-Kleopatra asteroid modelled by a dipole-segment

Asteroid 216-Kleopatra presents a ham-bone shape, that is to say, it is quite elongated body with two protuberances on its end-points, which makes of it the perfect candidate to model its shape and mass distribution by a Dipole-Segment, consisting on a massive segment with two spherical masses at the end-points of the rod and that is rotating about its center of mass with uniform velocity. Once obtained an approximation of the actual values of the involved parameters from recent results of Burov and coworkers, we obtain the complete map of families of symmetric periodic orbits, and describe the orbits of the different families, its parametric evolution, stability and also we compute the heteroclinic orbits connecting unstable isosceles equilibria.

Retrograde orbits associated separatrices in perturbed restricted three-body problem

This study develops single and multivariate nonlinear regression models for predicting Jacobi constant related to separatrix associated with simply symmetric retrograde periodic orbits in the framework of CRTBP with and without perturbations effects. These models are constructed using data from real and hypothetical systems considering that the parameter of mass is independent variable, while the Jacobi constant is displayed as a function of mass and oblateness parameters. Third-degree polynomials from single-variable regression and fourth-degree polynomials from multivariate regression accurately predict the Jacobi constant for given mass and oblateness values, facilitating the identification of separatrices in the PSS without intensive construction efforts. Error analysis confirms the model’s high accuracy across all parameters values. Additionally, chaos detection techniques, including the SALI, GALI, and Lyapunov exponent methods, along with bifurcation analysis, are employed to identify the regions of local and global chaos.

Symmetric periodic Reeb orbits on the sphere

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere S 2 n + 1 S^{2n+1} has at least n + 1 n+1 simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least n + 1 n+1 simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex.

Existence and Linear Stability of Symmetric Periodic Orbits in the Generalized Planar Stark–Zeeman Problem

Existence and Linear Stability of Symmetric Periodic Orbits in the Generalized Planar Stark–Zeeman Problem

Analysis of symmetric periodic orbits in a tripolar system with a segment

Analysis of symmetric periodic orbits in a tripolar system with a segment

Generating Symmetric Periodic Orbits and Their Optimal Maintenance in a Binary Asteroid System

In this paper, generating symmetric periodic orbits around a binary asteroid and optimal maintenance of spacecraft on these periodic orbits are investigated. Due to the irregular shape of asteroids, numerical methods are needed to generate periodic orbits. In this regard, the 32 families of symmetric periodic orbits were calculated by grid search and shooting methods utilizing the restricted three-body problem as a basic model. The irregularity of the shape of asteroids and solar radiation pressure are considered perturbations. These perturbations should be compensated by a thruster with a continuous thrust through the continuous control strategy, i.e., the linear quadratic regulator (LQR) method. The effect of different matrix coefficients in the performance of the LQR method is examined. The results show that the LQR method could appropriate control spacecraft on periodic orbits with specular shapes. Also, the practical feasibility of a continuous thrust control strategy is discussed.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

On doubly symmetric periodic orbits

In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e., those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) All covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory (Eliashberg et al. Geom Funct Anal Special Volume Part II:560–673, 2000); (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in Frauenfelder et al. in Symplectic methods in the numerical search of orbits in real-life planetary systems. Preprint arXiv:2206.00627). The above results follow from: (5) A symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in Frauenfelder and Moreno 2021) differ.

Continuation of some nearly circular symmetric periodic orbits in the elliptic restricted three-body problem

Continuation of some nearly circular symmetric periodic orbits in the elliptic restricted three-body problem

Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.

Evolution of the characteristic curves in the restricted three-body problem in terms of the mass parameter

In this work, we study the evolution of the families of simple symmetric periodic orbits in the restricted three-body problem whatever the value of the mass parameter mu . To classify these characteristic curves, we introduce a topological characterization of both orbits and families. Starting from the work of Strömgren for the Copenhagen case, we analyze the evolution of these families, when the mass parameter mu varies in (0, 1/2], focusing on their topological characterization, the existence of asymptotic points and the appearance of certain types of orbits such as horseshoe orbits. Lastly, we consider two samples, the Earth–Moon and Sun–Jupiter systems and classify the different types of orbits for these systems.

On the retrograde planar co-orbital asteroid motion with Jupiter

AbstractWe study the motion of an asteroid being in retrograde 1/1 resonance with Jupiter (co-orbital motion). We consider the planar case (i=180°) and Jupiter is on a circular or elliptic orbit (e′ = 0.048). In the cirular model we compute families of symmetric periodic orbits and their stability type. In the elliptic model we have isolated periodic orbits which affect the orbital modes of motion as it is shown by the FLI dynamical maps.

The phase space structure of retrograde mean motion resonances with Neptune: the 4/5, 7/9, 5/8 and 8/13 cases

We compute planar and three-dimensional retrograde periodic orbits in the vicinity of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and we concentrate on the dynamics of higher-order exterior mean motion resonances with Neptune. By using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 4/5, 7/9, 5/8 and 8/13 resonances. We determine the bifurcation points from the planar circular to the planar elliptic problem and we find all the corresponding families. In order to obtain a global view of the families of periodic orbits, the eccentricity of the primaries takes values in the whole interval 0<e'<1. Then, we find all the possible vertical critical orbits (v.c.o) of the planar circular problem and we proceed to the three-dimensional circular restricted 3-body problem. In this model, retrograde periodic orbits are generated mainly from the retrograde v.c.o. Also, if we continue families of direct orbits for i>90^circ , then we can obtain families of 3D symmetric retrograde periodic orbits. The linear stability is examined too. Stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. In order to study the phase space structure of the above resonances, we construct dynamical stability maps for the whole inclination interval (0<i<180^circ ) by using the well-known “MEGNO Chaos Indicator”. Finally, we discuss about TNOs which are currently located at these resonances.

Nonlinear regression multivariate model for first order resonant periodic orbits and error analysis

In this work, nonlinear multivariate regression models are developed under the structure of the circular restricted three–body problem (CRTBP) with and without perturbations for exterior and interior resonant periodic orbits of order one. The nonlinear multivariate regression can be used for simply symmetric retrograde periodic orbits, which are first–order resonant orbits. Three real systems are taken into consideration for the study namely, Saturn–Hyperion, Saturn–Titan and Earth–Moon with small, moderate and high mass ratio respectively. In the given three systems, the first primary is taken as an oblate spheroid. Also, coefficient of oblateness, Jacobi constant and mass ratio are taken as parameters. The nonlinear multivariate regression model is used to find the initial position of periodic orbit without constructing PSS for time saving and avoid complicated procedure. In this context, fourth-fifth degree polynomial obtained from regression gives the best possible predicted value for the initial condition of periodic orbit by assigning parameter value such as mass ratio, coefficient of the oblateness of first primary and Jacobi constant. Also, error analysis of these models for three real systems Saturn–Hyperion, Saturn–Titan and Earth–Moon is discussed in detail. From error analysis, it can be seen that these models give the best approximation for the initial position of the periodic orbits. • Nonlinear multivariate regression model of the first order resonant periodic orbits in restricted three–body problem. • Analysis of periodic orbit for the Saturn–Hyperion, Saturn–Titan and Earth–Moon systems using Poincaré surface section. • Regression model and error analysis by considering oblateness of first primary body in the restricted three–body problem. • Estimation of maximum and minimum errors of the regression models for the perturbed circular restricted three–body problem.

Short and long period periodic orbits around a stable collinear equilibrium point in the circular restricted three-body problem with a three-body interaction

We deal with a modification of the circular restricted three-body problem (CR3BP), which incorporates an additional effect of a three-body interaction. The parameters determining the configuration of the corresponding dynamical system are the mass ratio μ of the primary bodies and the constant k that scales the contribution of the aforementioned interaction. We are interested in the families of short and long period planar symmetric periodic orbits, which emanate from the collinear equilibrium point L3 in the case that this point is neutrally stable. First, we numerically investigate the correlation between the values of μ and k that result such kinds of families. Subsequently, these families are numerically constructed and presented for an indicative case of these parameters. It is found that both families terminate on some members of the family of short period orbits.

Multiple families of synodic resonant periodic orbits in the bicircular restricted four–body problem

Periodic orbits in gravitational many–body problems have been fundamental special solutions for analyzing phase–space flows not only in autonomous systems but also in time–dependent non–autonomous systems. The growing interest in cislunar space may require deeper understandings of periodic orbits in the Earth–Moon system beyond the simplified dynamics in the circular restricted three–body problem. The present paper deals with two mechanisms of the generation of multiple families of synodic resonant periodic orbits in the bicircular restricted four–body problem through numerical examples adopting planar and three–dimensional retrograde periodic orbits around the Earth. The first part of the paper reveals complex interplays between a synodic resonant periodic orbit and multi–revolutional orbits originated from period–multiplying bifurcations. Extensions of continuation curves beyond the usual limit find another 12:11 synodic resonant planar retrograde periodic orbit around the Earth in addition to the one already found in the previous work. The second part shows that a doubly symmetric periodic orbit in the circular restricted three–body problem can generate four families of synodic resonant periodic orbits in the bicircular restricted four–body problem. This work identifies linearly stable, 1:1 synodic resonant, three–dimensional retrograde periodic orbits around the Earth and indicates their substantially large stability regions under solar gravitational perturbations.

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff’s conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.

Equilibrium Points and Networks of Periodic Orbits in the Pseudo-Newtonian Planar Circular Restricted Three-body Problem

Abstract We explore a pseudo-Newtonian planar circular restricted three-body problem in which the primaries are modeled using an approximate gravitational potential up to the second nonvanishing term of the Fodor–Hoenselaers–Perjés expansion. We aim to understand how the main free parameters of the system affect its dynamical properties. In particular, we determine how the mass of the primaries as well as the transition parameters affect not only the properties of the points of equilibrium (total number, locations, and linear stability) but also the networks of simple symmetric periodic orbits. Our results show that, under this approach, significant variations are observed in the fixed points (number and stability) and periodic orbits of the planar circular restricted three-body problem, even when small contributions of the non-Newtonian terms are considered. We also provide direct applications of the new model potential in real observable binary stellar systems.

The spatial Hill four-body problem I—An exploration of basic invariant sets

In this work we perform a first study of basic invariant sets of the spatial Hill’s four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter μ in such a way that the classical Hill’s problem is recovered when μ=0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter μ by applying a classical predictor–corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill’s three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill’s three-body problem; these families have some desirable properties from a practical point of view.