Families Of 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.

Solar sail orbital motion at the non-autonomous oblate earth-moon system: family of periodic orbits

Solar sail orbital motion at the non-autonomous oblate earth-moon system: family of periodic orbits

Chaos and Sub-Harmonic Bifurcations in a Simple SIR Model with Periodically and Parametrically Nonlinear Incidence Rate

Chaos and sub-harmonic bifurcations of a simple SIR model with periodically and parametrically nonlinear incidence rate are investigated analytically with the Melnikov method and sub-harmonic Melnikov method, respectively. The critical curves of chaos for the model with the periodically and parametrically nonlinear incidence rate are studied analytically for the first time. Some new interesting dynamic phenomena are presented and proved rigorously, including the nonchaotic band, controllable frequencies and even controllable frequency interval, which may exist for some system parameters in this model. The sub-harmonic periodic orbit family is derived. It is presented that sub-harmonic bifurcations with even order and integer order may exist in the system. The sub-harmonic bifurcation conditions are obtained rigorously for two different starting points. Numerical simulations including phase portraits, Poincaré sections, basins of attraction and Lyapunov exponents are given to verify the chaos threshold obtained by the analytical results. It is also shown that the system can undergo chaos through even-order sub-harmonic bifurcations.

Identification and analysis of interior and exterior resonant orbits in the Sun–Venus system

With Venus being Earth’s nearest neighbor and it being similar in size to Earth, it is the ideal candidate for certain mission types. These missions range from scientific, such as helio-physics and space weather monitoring, to defense focused, such as monitoring Earth’s orbital path for meteoroid threats. Limited research has been done on the periodic orbits in the Sun–Venus system. The first step to understanding which orbits could be useful for such a mission is to identify families of resonant periodic orbits in the Sun–Venus system and conduct a stability analysis on these orbits. This research work identifies 90 periodic, resonant orbits in the Sun–Venus system and their associated families of orbits. The orbits are found within the Circular Restricted 3 Body Problem (CR3BP) dynamical model with solar radiation pressure included as a perturbation. The periodic orbits are found using Poincaré maps, and the families are generated using a continuation method that steps through different Jacobi constants. The stability for each orbit in the family is calculated and the structure of the eigenvalues for each is assessed to determine when the family has crossed a bifurcation point. This research work seeks to generate a catalog of resonant orbits within the Sun–Venus system while providing stability and bifurcation information for each resonant orbit family.

The dynamics about asteroid (162173) Ryugu

The dynamical environment around the asteroid (162173) Ryugu is analyzed in detail using a constant-density polyhedron model based on the measurements from the Hayabusa2 mission. Six exterior equilibrium points (EPs) are identified along the ridge line of Ryugu, and their topological classifications fall into two distinctive categories. The initial periodic orbit (PO) families are computed and analyzed, including distant retrograde/prograde orbit (DRO/DPO) families and fifteen PO families emanating from the exterior EPs. The fifteen PO families are further divided into three categories: seven converge to an EP, seven reach Ryugu’s surface, and one exhibits cyclic behavior during its progression. The existence of initial PO families converging to an EP is analyzed using the bifurcation of a degenerate EP. Connection between these families and similar ones in the circular restricted three-body problem (CRTBP) is also examined. Bifurcated PO families are identified and computed from the initial PO families, including ten families from the DROs, fifteen from the DPOs, and twenty-five associated with the EPs. The bifurcated families are separately analyzed and categorized in terms of their corresponding initial families. Connections established by the same bifurcation points between different bifurcated families are identified. A comparison is made for the dynamical environments of Ryugu and Bennu to evaluate the similarities and differences in the evolution of EPs and the progression of PO families in top-shaped asteroids.

Orbital analysis in the gravitational potential of elongated asteroids

This work studies the motion around irregular elongated asteroids through two approaches. Firstly, it revisits the dipole-segment model, identifying families of periodic orbits for asymmetric mass distribution. Additionally, a new model incorporating variable density for elongated asteroids is introduced and compared to the dipole-segment model. Several families of periodic orbits have been found through continuation of planar orbits and out-of-plane bifurcation processes, obtaining results in agreement with previous studies about the dynamics around irregular asteroids. This highlights the relevance of simple mathematical models in studying asteroid dynamics and the importance of accounting for density and geometric properties. Although the families of periodic orbits studied in this work are not comprehensively sampled, they constitute an example of the variety of orbits that can be followed by a particle orbiting the asteroid, helping us to better understand the dynamics around these elongated bodies.

Periodic orbits in a Hamiltonian system of stellar type

We investigate the existence of periodic orbits in a perturbed Hamiltonian system of stellar type in 1:1 resonance. The perturbation consists of a potential of degree four with two real parameters. We determine six families of periodic orbits using reduction and averaging theories. Also, we characterize the stability of these orbits and their bifurcation curves in terms of the parameters. Finally, we show a complete picture of the choreographies of critical points originating the periodic orbits.

Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem

A novel station-keeping strategy leveraging periodic revolutions of homeomorphic orbits in the Elliptic Restricted Three-Body Problem within the pulsating frame is presented. A systemic approach founded on arc-length continuation is presented for the discovery, computation, and classification of periodic revolutions that morph from their traditional circular restricted three-body counterparts to build an a priori dataset. The dataset is comprehensive in covering all possible geometric architectures of the restricted problem. Shape similarity is quantified using Hausdorff distance and works as a filter for the station-keeping algorithm in relation to appropriate target conditions. Finally, an efficient scheme to quantify impulsive orbit maintenance maneuvers that minimize the total fuel cost is presented. The proposed approach is salient in its generic applicability across any elliptic three-body system and any periodic orbit family. Finally, average annual station-keeping costs using the described methodology are quantified for selected “orbits of interest” in the cis-lunar and the Sun–Earth systems. The robustness and efficacy of the approach instill confidence in its applicability for realistic mission design scenarios.

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Dynamical Structures under Nonrestricted Hierarchical Planetary Systems with Different Mass Ratios

Secular dynamics have been extensively studied in both the inner and outer restricted hierarchical three-body systems. In the inner restricted problem, the quadrupole-order resonance (i.e., the well-known Kozai resonance) causes large coupled oscillations of eccentricity and inclination when the maximum inclination is higher than 39.2°, and the octupole-order resonance leads to the behavior of orbital flips. In the outer restricted problem, the behavior of orbital flips is due to the quadrupole-order resonance. Secular dynamics under the inner and outer restricted systems are distinctly different. The mass ratio of inner and outer bodies could change the ratio of circular orbital angular momenta β, which significantly influences dynamical structures of the system. But this influence is still unclear. In this paper, we focus on nonrestricted hierarchical planetary systems where β > 1 and investigate the secular dynamics by changing mass ratios. Dynamical structures are systematically explored from four aspects: periodic orbits, secular resonances, orbital flips, and chaos detection. We find that (a) it tends to lead to more bifurcations in the host family of prograde periodic orbits associated with Kozai resonance with smaller β; (b) with the decrease of β, fewer orbits inside the octupole-order resonance can realize flip; (c) for given initial conditions, the forbidden region appears in the retrograde region and becomes larger as β decreases, meaning that the mutual inclination cannot reach a very high value if β is small; and (d) chaotic orbits are distributed in the low-eccentricity, high-inclination region when β > 1.

Period-Multiplying Bifurcations in the Gravitational Field of Asteroids

Periodic orbit families around asteroids serve as potential trajectories for space probes, mining facilities, and deep space stations. Bifurcations of these families provide additional candidate orbits for efficient trajectory design around asteroids. While various bifurcations of periodic orbit families around asteroids have been extensively studied, period-multiplying bifurcations have received less attention. This paper focuses on studying period-multiplying bifurcations of periodic orbit families around asteroids. In particular, orbits with periods of approximately 7 and 17 times that of the rotational period of asteroid 216 Kleopatra were computed. The computation of high-period orbits provides insights into the numerical aspects of simulating long-duration trajectories around asteroids. The previous literature uses single-shooting and multiple-shooting methods to compute bifurcations of periodic orbit families around asteroids. Computational difficulties were encountered while using the shooting methods to obtain period-multiplying bifurcations of periodic orbit families around asteroids. This work used the Legendre–Gauss collocation method to compute period-multiplying bifurcations around asteroids. This study recommends the use of collocation methods to obtain long-duration orbits around asteroids when computational difficulties are encountered while using shooting methods.

Periodicity and lifetime of orbits around elongated asteroids

This paper presents dynamics and control of orbits around an asteroid that is modelled as a uniformly rotating homogeneous rectangular parallelepiped (HRP). Aside from the non-uniform gravitational field of the asteroid, we include the solar radiation pressure (SRP) as a perturbing force. A simple model based on the polyhedral method is employed for calculating the gravity around the HRP. First, circular periodic orbits are identified both in the presence and absence of solar radiation pressure. Then, families of general non-circular periodic orbits in the presence of SRP are determined and subsequently analysed for stability. Next, for a general orbit in the equatorial plane, its stability against a catastrophic impact of an orbiting spacecraft onto the asteroid’s surface is checked in the presence of SRP. Orbits, where the solar radiation pressure is less/more dominant over the central body’s gravity, are identified with the aim of finding the best orbits for parking a spacecraft. Then, for long-period orbits, SRP is averaged and the time averaged evolution of the orbital elements is obtained. For a special set of initial condition, a closed form solution for the evolution of the orbital eccentricity is obtained. Using this solution, a lower limit for the life-time of a spacecraft that is left uncontrolled is derived in a closed form. Finally, active control strategies are developed for orbit-keeping ofspacecraft for any choice of orbital parameters. A linear-quadratic regulator (LQR) is designed to track a given Keplerian orbit with minimum fuel expenditure. Fuel expenditure on various orbits are calculated and compared to determine the orbits of cheapest operation. The work presented is applicable to small body (asteroids and comets) exploration missions which require online cancellation of the perturbing forces with minimal knowledge of the accurate shape, size, and composition of the small body prior to the mission.

Sun–Venus CR3BP, part 2: resonance investigation and planar periodic orbit family generation

Sun–Venus CR3BP, part 2: resonance investigation and planar periodic orbit family generation

Expanded Families of Periodic Orbits for the Jupiter–Callisto System

Periodic orbits are widely used in space missions and are fundamental for us to understand the complex dynamic system of circular restricted three-body problem. However, there are still many potential families of periodic orbits that have not been located and computed. This study aims to generate some new and less-known families of periodic orbits for the Jupiter–Callisto system to obtain an extended database. The numerical methods for the computation and continuation of periodic orbit families are provided. Five bifurcation diagrams are explored in depth, and 108 periodic orbit families are located. Of these, 59 families are continued to the entire families. Twenty asymmetrical families also come into view. New discoveries and the characteristics of several particular families are presented and discussed. Synodic resonant periodic orbits appearing under the perturbation of Ganymede are also identified. The expanded families of periodic orbits are expected to provide a reference for future Callisto exploration mission design.

Investigating the albedo effects on the dynamics of infinitesimal mass in the elliptic Sitnikov five-body problem

In this paper, we aim to showcase the significant impacts of albedo and eccentricity of the primaries on the infinitesimal mass within the elliptic case of the Sitnikov five-body problem. The Van der Pol transformation and averaging technique were employed to formulate the averaged equations of motion for the infinitesimal mass. Subsequently, the Hamiltonian equations of motion were derived using the action angle variables I and θ. Additionally, we obtained θ in terms of the Jacobi elliptic function sn through a canonical transformation. The investigation utilized the first return map to explore various orbits such as regular, periodic, quasi-periodic, chaotic, or stochastic in the presence of albedo and eccentricity. It was observed that chaotic and stochastic regions emerged as the albedo effect increased, disrupting the periodic tubes and islands. Moreover, by escalating the albedo effect and eccentricity of the primaries, several families of periodic orbits were revealed.

Grazing-Induced Dynamics of the Piecewise-Linear Chua’s Circuit

In this paper, we consider the piecewise-linear Chua’s circuit, which is well known for its rich variety of bifurcation, chaotic and other nonlinear phenomena. Suitable switching boundaries are introduced based on the piecewise-linear representation of Chua’s diode. In this way, we derive analytical conditions for a grazing bifurcation to occur, when one or two families of periodic orbits have a zero-velocity contact with the switching boundaries. In connection to this phenomenon, we also study the focus-center-limit cycle bifurcation and its implications regarding the system dynamics, from both analytical and numerical points of view. Furthermore, a detailed parametric study of Chua’s circuit is carried out via path-following techniques for nonsmooth dynamical systems, implemented via the continuation software COCO. This study reveals the presence of codimension-one bifurcations of limit cycles, such as those mentioned above, as well as classical (fold and period-doubling) bifurcations. The analysis confirms the presence of coexisting attractors, which are produced by a hysteresis loop induced by the interaction of a fold and a focus-center-limit cycle bifurcation.

Planar spacecraft trajectories in the Didymos–Dimorphos binary asteroid system

The dynamics of a spacecraft in the vicinity of the Didymos and Dimorphos binary system is studied, with specific emphasis on the bounded orbits that exist in such a perturbed gravitational environment. The binary environment is approximated with gradually increasing fidelity, starting with a model that simulates Didymos with an oblate spheroid and Dimorphos with a triaxial ellipsoid. Within these simplifying assumptions, we compute all the families of planar periodic orbits and we study the deviations that arise to them under more realistic perturbations that include the actual shape of Didymos, its rotation, the solar gravity and radiation pressure. To establish the robustness of the different orbital configurations, we perform a classification based on planar dynamical maps and connect the stability regions with the periodic motion. A subset of retrograde orbits around Didymos and a particular type of outer orbits with double multiplicity seem to show long-term stability and robustness, so they can be assumed as safe orbits. On the contrary, orbits around Dimorphos, although stable in the basic model, become strongly unstable in our more realistic model. Our approach is based only on natural dynamics and no active spacecraft control is applied, thus providing a useful methodology in finding ‘safe’ orbits for a spacecraft in the vicinity of a binary asteroid system.

The stability around Chariklo and the confinement of its rings

Context. Chariklo has two narrow and dense rings, C1R and C2R, located at 391 km and 405 km, respectively. Aims. In the light of new stellar occultation data, we study the stability around Chariklo. We also analyse three confinement mechanisms that prevent the spreading of the rings, based on shepherd satellites in resonance with the edges of the rings. Methods. This study was performed through a set of numerical simulations and the Poincaré surface of section technique. Results. From the numerical simulation results, and from the current parameters referring to the shape of Chariklo, we verify that the inner edge of the stable region is much closer to Chariklo than the rings. The Poincaré surface of sections allows us to identify periodic and quasi-periodic orbits of the first kind, and also the resonant islands corresponding to the 1:2, 2:5, and 1:3 resonances. We construct a map of aeq versus eeq space that gives the location and width of the stable region and the 1:2, 2:5, and 1:3 resonances. Conclusions. We find that the first kind periodic orbit family can be responsible for a stable region whose location and size meet that of C1R, for specific values of the ring particle eccentricities. However, C2R is located in an unstable region if the width of the ring is assumed to be about 120 m. After analysing different systems, we propose that the best confinement mechanism is composed of three satellites: two satellites shepherding the inner edge of C1R and the outer edge of C2R, and the third satellite trapped in the 1:3 resonance.

Analyzing the structure of periodic orbit families that exist around asteroid (101955) Bennu

Analyzing the structure of periodic orbit families that exist around asteroid (101955) Bennu

The photo-gravitational concentric Sitnikov problem

The current framework involves a configuration of two pairs of primary celestial bodies engaged in synchronized circular orbits around a central point of mass. Additionally, an infinitesimal mass is positioned along the z-axis, traversing through the system’s center of mass. This distinctive celestial scenario is designated as the Concentric Sitnikov Problem. Notably, it is postulated that the initial pair of primary bodies emit radiation, while the latter pair remains radiation-free. Within the scope of this study, we delve into an exploration of equilibrium points, periodic orbits, and the intriguing Newton–Raphson basins of convergence (N-R BoC) within the concentric Sitnikov model, all subject to the influence of radiation pressure. Remarkably, our investigation uncovers the presence of three equilibrium points, each exhibiting linear instability across the entire spectrum of mass parameter values μ∗∈(0,0.25). To visually comprehend the character of these celestial paths, we employ a graphical analysis technique known as the first return map. Varied values of the mass parameter μ∗ lead to the generation of diverse families of periodic orbits encircling both the primary celestial bodies and their equilibrium positions. Lastly, we embark on an exploration of the intricacies of the N-R BoC, intimately connected with the equilibrium points within this proposed celestial model.