Families Of Orbits Research Articles

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.

3D homogeneous potentials generating two-parametric families of orbits on the outside of a material concentration

We study three-dimensional homogeneous potentials V = V(x, y, z) of degree m which are created outside a finite concentration of matter and they produce a preassigned two-parametric family of spatial regular orbits given in the solved form f(x, y, z) = $$c_{1}$$ , g(x, y, z) = $$c_{2}$$ ( $$c_{1}$$ , $$\; c_{2}$$ = {\rm const}). These potentials have to satisfy three linear PDEs; two of them come from the Inverse Problem of Newtonian Dynamics and the last one is the well-known ”Laplace’s equation”. Our aim is to find common solutions for these three PDEs. Besides that we consider that the functions f and g are also homogeneous in the variables $$x,\; y,\;z$$ of any degree and can be represented uniquely by the ”slope functions” $$\alpha (x,y,z)$$ and $$\beta (x,y,z)$$ which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions ( $$\alpha , \; \beta $$ ). If they are satisfied for a specific value of m, then we can find the potential by quadratures. The values obtained for m so far are consistent with familiar gravitational and electrostatic and quadratic potentials. Finally, pertinent examples are given and cover all the cases.

Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

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.

Soliton solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation via the Lie symmetry method and its stability analysis by using bifurcation theory

The objective of the present article is to seek new explicit solutions to the (2+1)-dimensional Nizhnik-Novikov-Veselov(NNV) equation. The NNV system is highly nonlinear in nature and is a known isotropic Lax extension of the Korteweg–de Vries model. The similarity transformation method has been used to systematically reduce the NNV equation into ordinary differential equations(ODEs). The new exact solutions have been obtained by solving the obtained ODEs based on the formed relationships. The resulting soliton solutions contain some arbitrary constants and functions. The use of appropriate functions and constants highlighted that the solutions of the NNV equation might be soliton, multisoliton, parabolic, doubly soliton and trigonometric. Moreover, the stability of the corresponding dynamical system has been investigated using bifurcation theory with different parametric regions. The figures obtained during MATLAB simulation supported the dynamic features of the derived solutions. Finally, we explore incredible aspects of the exact wave solutions via phase portraits. The phase portraits validate the existence of some families of homoclinic and periodic orbits about the equilibrium points, respectively.

Measuring the Orbits of the Arches and Quintuplet Clusters Using HST and Gaia: Exploring Scenarios for Star Formation near the Galactic Center

We present new absolute proper-motion measurements for the Arches and Quintuplet clusters, two young massive star clusters near the Galactic center. Using multiepoch HST observations, we construct proper-motion catalogs for the Arches (∼35,000 stars) and Quintuplet (∼40,000 stars) fields in ICRF coordinates established using stars in common with the Gaia EDR3 catalog. The bulk proper motions of the clusters are measured to be (μ α*, μ δ ) = (−0.80 ± 0.032, −1.89 ± 0.021) mas yr−1 for the Arches and (μ α*, μ δ ) = (−0.96 ± 0.032, −2.29 ± 0.023) mas yr−1 for the Quintuplet, achieving ≳5× higher precision than past measurements. We place the first constraints on the properties of the cluster orbits that incorporate the uncertainty in their current line-of-sight distances. The clusters will not approach closer than ∼25 pc to Sgr A*, making it unlikely that they will inspiral into the nuclear star cluster within their lifetime. Further, the cluster orbits are not consistent with being circular; the average value of r apo/r peri is ∼1.9 (equivalent to an eccentricity of ∼0.31) for both clusters. Lastly, we find that the clusters do not share a common orbit, challenging one proposed formation scenario in which the clusters formed from molecular clouds on the open stream orbit derived by Kruijssen et al. Meanwhile, our constraints on the birth location and velocity of the clusters offer mild support for a scenario in which the clusters formed via collisions between gas clouds on the x 1 and x 2 bar orbit families.

A family of periodic orbits for the extended Hamiltonian system of the Van der Pol oscillator

In this paper we show a new way of using the averaging theory for studying families of periodic orbits of a Hamiltonian system. We do this study computing a new family of periodic orbits of the extension of the Van der Pol oscillator to a Hamiltonian system of two degrees of freedom.

On the existence of limit cycles and invariant surfaces for sewing piecewise linear differential systems on [formula omitted

We consider a class of three-dimensional piecewise linear differential systems with two zones separated by a plane. In this class we show that there exist differential systems having: (i) a unique limit cycle; (ii) a unique one–parameter family of periodic orbits; (iii) scrolls; (iv) invariant cylinders.

Bifurcation of periodic orbits and its application for high-dimensional piecewise smooth near integrable systems with two switching manifolds

In this paper, we study the bifurcation of periodic orbits for high-dimensional piecewise smooth near integrable systems defined in three regions separated by two switching manifolds. We assume that the unperturbed system has a family of periodic orbits which cross two switching manifolds transversely. The expression of Melnikov function is derived based on the first integral. And the conditions of periodic orbits bifurcated from a family of periodic orbits for the high-dimensional piecewise smooth near integrable system are obtained. The theoretical results are applied to the bifurcation analysis of periodic orbits of two-degree-of-freedom piecewise smooth system of nonlinear energy sink. The periodic orbits configurations are presented with numerical method and the number of periodic orbits is three.

Earth–Moon Quasi-Halo Orbit Family: Characteristics and Manifold Applications

Quasi-halo orbits in the circular restricted three-body problem are quasi-periodic orbits that exist about the periodic halo orbits. A quasi-periodic orbit is an orbit diffeomorphic to a quasi-periodic torus that moves on at least a two-dimensional manifold in phase space and does not repeat itself. Given plans for the future Lunar Gateway, the quasi-halo orbits in the Earth–moon system are of interest. About each periodic halo orbit is a one-parameter family of quasi-halos, and thus across the halo orbit family is a two-parameter family of these orbits. This work surveys the two-parameter family of quasi-halo orbits in the circular restricted three-body problem to provide a global picture of the family. Additionally, the hyperbolic manifolds of these objects are explored, and applications of their uses presented. This work contributes results and analysis of the apses, geometry, and stability of the family to evaluate the use of quasi-halo orbits in the early stages of mission design.

Dynamics and integrability of the swinging Atwood machine generalisations

This paper studies the dynamics and integrability of two generalisations of a 3D Swinging Atwood’s Machine with additional Coulomb’s interactions and Hooke’s law of elasticity. The complexity of these systems is presented with the help of Poincaré cross sections, phase-parametric diagrams and Lyapunov exponents spectrums. Amazingly, such systems possess both chaotic and integrable dynamics. For the integrable cases we find additional first integrals and we construct general solutions written in terms of elliptic functions. Moreover, we present bifurcation diagrams for the integrable cases and we find resonance curves, which give families of periodic orbits of the systems. In the absence of the gravity, both models are super-integrable.

Families of periodic orbits about Lagrangian points L1, L2 and L3 with continuation method

This study deals the impact of radiation pressure and albedo from the respective primaries in the context of regularized periodic motion of infinitesimal mass under the frame of restricted three body problem with the help of continuation method. The effect of these perturbing parameters on the orbital properties about the collinear Lagrangian points and their stability property are investigated. An expansion in the structure of Lyapunov orbits are found due to radiation pressure effect and albedo whereas crossing of stability bounds (±1) occurs in the continuation process of Lyapunov orbits indicating the presence of halo bifurcation. This occurrence of halo bifurcation leads to the origin of halo periodic orbits from Lyapunov periodic orbits at different values of radiation parameter ( q ) and albedo ( Q A ). From these bifurcating orbits, families of halo periodic orbits are determined using continuation in radiation parameter and albedo, respectively. Due to the existence of one of the stability index ( ν ), which is either greater than 1 or less than ​− ​1, periodic orbits become unstable. • Regularized restricted three body problem with radiation pressure and albedo are discussed. • Numerical computation of periodic orbits about collinear equilibrium points are discussed. • The parametric evolution of periodic orbit as the function of radiation and albedo parameter are presented. • Families of periodic orbits are found with the help of continuation in radiation and albedo parameter.

Formula omitted] and [formula omitted] axi-symmetric horseshoe periodic orbits about Lagrangian points: A global grid search approach

This paper presents the numerical exploration of planar as well as spatial periodic horseshoe orbits about Lagrangian points in the framework of restricted three-body problem with radiation pressure and albedo as perturbations. The global grid search technique for obtaining both types of periodic horseshoe orbits is described. Further, several families of horseshoe orbits are obtained and then the orbital behaviour of each periodic orbit is investigated. By global grid search method, spatial axi-symmetric horseshoe orbits and their families are obtained via pseudo-arclength continuation. Interestingly, new forms of spatial horseshoe orbits are constructed and their orbital properties are analysed. Moreover, it is found that stable horseshoe orbits exists for different range of x0 in planar as well as in spatial case. Using parameter continuation, the effect of radiation pressure and albedo are discussed for the evolution of horseshoe orbits and found that the radiation pressure affects the shape of horseshoe orbits more then that of albedo. These results are helpful to analyse more generalized problem with other perturbations.

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.

Two and three impulses phasing strategy with a spacecraft orbiting on an Earth–Moon NRHO

The increasing interests in Moon exploration have led in recent years to international collaborations between space agencies aimed to assemble and operate the Gateway, an orbiting spaceport located on a Near Rectilinear Halo Orbit (NRHO) about the second libration point of the Earth–Moon system (EML-2). In this context, transfer strategies between the Earth and the spaceport cover a key role for both assembly and resupply missions. The presented work is thus focused on one leg of the transfer: the phasing maneuver. This work is conducted under the hypotheses of the Circular Restricted Three-Body problem (CR3BP) in which the Earth and the Moon are the only bodies that influence the spacecraft motion. The primaries are assumed to have a circular motion around their common barycenter. Under these hypotheses, is demonstrated the existence of periodic orbits, such as the Halo orbit family, that are exploited to design a proposed phasing maneuver. Two strategies are investigated: Halo parking orbit to NRHO two-impulses transfer and direct phasing with manifold exploitation. The first strategy is an optimal two-impulses transfer departing an EML-2 southern Halo and targeting the baseline NRHO. The second strategy considers the chaser already injected on the Gateway’s NRHO. Poincare maps are employed to identify unstable/stable manifolds intersections in search of low-energy phasing trajectories that leave the reference orbit along the unstable branch before re-entering it via the stable one. Three-impulses transfers with similar costs are found patching together these arcs. The two strategies are thus compared to highlight their advantages and disadvantages with the perspective of a real autonomous cargo mission around the Moon to refurnish the orbiting station.

Bifurcation from Periodic Orbits of n-Dimensional Piecewise Smooth Differential Systems with Two Switching Planes

In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, we obtain the first order Melnikov vector function which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete [Formula: see text]-dimensional piecewise smooth differential system.

Two-parametric families of orbits created by three-dimensional galactic-type potentials

Two-parametric families of orbits created by three-dimensional galactic-type potentials

A numerical study of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in planetary systems

ABSTRACT We present a numerical study on the stability of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in the three-body problem composed of a solar mass star, a Jupiter mass planet, and an additional body with zero mass (elliptic restricted three-body problem) or masses corresponding to either Neptune, Saturn, or Jupiter (planetary three-body problem). For each system, we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they correspond to the libration of either a single resonant argument or all resonant arguments (fixed points). We compare the results obtained in the elliptic restricted three-body problem with previous results in the literature, and we show the differences and similarities between the phase space topology for these retrograde resonances in the circular restricted, elliptic restricted, and planetary three-body problems.

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.