Types Of Periodic Orbits Research Articles

The Milky Way Bar Potential Constrained by the Kinematics of SiO Maser Stars in BAaDE Survey

We introduce a novel method that utilizes the longitude–velocity (l − v) envelope to constrain the Milky Way (MW) bar potential. Previous work used the l − v diagram to explain the distribution of the observed high-velocity stars. We successfully reproduce their results but find that their method is limited to only one type of periodic orbits. In contrast, we propose that the l − v envelope provides much more comprehensive constraints. We compare the properties of test particles in the M. Portail et al. MW potential model (P17) with the observed SiO maser stars from the Bulge Asymmetries and Dynamical Evolution (BAaDE) survey. We find that the l − v envelope generated by the bar potential demonstrates reasonable agreement with the observational data, albeit with slight discrepancies near the Galactic center. The inconsistencies suggest that the P17 potential yields a lower central rotation curve, a slightly larger quadrupole strength, or a possibly underestimated pattern speed. We also adopt an updated version of the P17 potential with a modified central mass component proposed by G. H. Hunter et al. (H24). The fitting of the l − v envelope suggests that the H24 potential does not completely address the existing challenges and may hint at a possible underestimation of the central bar mass. Our study demonstrates that the l − v envelope can be used as a valuable tool for constraining the Galactic potential and provides insights into the MW bar potential.

Existence and Number of Figure-Eight Loops in Planar Sector-Wise Linear Systems with Saddle–Saddle Dynamics

In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.

Existence and stability of the periodic orbits induced by grazing bifurcation in a cantilever beam system with single rigid impacting constraint

Grazing, which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by adopting the discontinuous mapping method. Then, by means of shooting method, the periodic orbits are determined, followed by the acquisition of Jacobian matrix in the case of non-impact is obtained subsequently. In addition, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing, with the stability criterion of the grazing-induced periodic orbits given. Numerical results verify the effectiveness of theoretical analysis. What is more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.

Bifurcations of Periodic Orbits in the Gravitational Field of Irregular Bodies: Applications to Bennu and Steins

We investigate the topological types and bifurcations of periodic orbits in the gravitational field of irregular bodies by the well-known two parameter analysis method. Results show that the topological types of periodic orbits are determined by the locations of these two parameters and that the bifurcation types correspond to their variation paths in the plane. Several new paths corresponding to doubling period bifurcations, tangent bifurcations and Neimark–Sacker bifurcations are discovered. Then, applications in detecting bifurcations of periodic orbits near asteroids 101955 Bennu and 2867 Steins are presented. It is found that tangent bifurcations may occur three times when continuing the vertical orbits near the equilibrium points of 101955 Bennu. The continuation stops as the Jacobi energy reaches a local maximum. However, while continuing the vertical orbits near the equilibrium points of 2867 Steins, the tangent bifurcation and pseudo period-doubling bifurcation occur. The continuation can always go on, and the orbit ultimately becomes nearly circular.

Periodic Behaviors and Graceful Transitions Between Them for Bimodal Systems: Example of a Jumper on a Trampoline

This letter sets up a simple model for the controlled jumps of a jumper on a trampoline. Such a model is a paradigm for the impulsive control of more general multi-mode multi-dimensional (M3D) systems. First, necessary and sufficient conditions for a motion to be periodic are set up and different types of periodic orbits, characterized by timing diagrams, are identified. Next, we characterize and identify graceful transitions between such orbits, using our raccordation framework. This generalizes notions of persistency of behavior to the class of impulsively controlled multi-mode multi-dimensional systems.

Periodicity and bifurcation of a bouncing ball system with rigidly connected harmonic limiters

The bouncing ball system with two rigidly connected harmonic limiters is revisited in order to further analyze its periodic movement and bifurcation dynamics. By using the impulsive impact maps, we obtain several sufficient conditions for the existence and local stability of three different types of periodic orbits, respectively, and then plot the bifurcation diagrams in the space of the relative velocity and the restitution coefficient for different parameters of the limiter. The numerical simulation results are consistent with those of the theoretical analysis.

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

Study on periodic orbits around the dipole segment model for dumbbell-shaped asteroids

Equilibrium points and periodic orbits in irregular gravitational fields are significant for an understanding of dynamical behaviors around asteroids as well as deep space exploring missions. The dipole segment is a good alternative model to study qualitative dynamical properties near dumbbell-shaped asteroids. In this paper, the dipole segment model and its equilibrium points are simply introduced. The stability of the two triangular equilibrium points of the system is numerically examined. Next, periodic orbits are presented around the dipole segment model in two different cases, in which triangular equilibria are linearly stable and unstable, respectively. New types of periodic orbits are illustrated in detail, including their orbital shapes, periods and the Jacobi integral. The orbital stability, topological classification and bifurcations of these orbits are also analyzed with numerical continuations.

Periodic orbits in the Chermnykh problem

Periodic orbits in irregular gravitational fields are significant for an understanding of dynamical behaviors around asteroids as well as the engineering aspect for deep space explorations. The rotating mass dipole, referred to as the Chermnykh problem, is a good alternative model to study qualitative dynamical environments near elongated asteroids, like the asteroid 1620 Geographos, 216 Kleopatra, or 25143 Itokawa. In this paper a global searching method is adopted to search for periodic orbits around the dipole model based on the concept of Poincare section of surface. Representative families of periodic orbits are illustrated with respect to all three topological cases of the dipole model. Topological transitions of orbits during iso-energetic continuations are also presented as well as identification of new types of periodic orbits.

Searching for Time Optimal Periodic Orbits Near Irregularly Shaped Asteroids by Using an Indirect Method

Periodic orbits over irregularly shaped asteroids and comets are fundamental for understanding vicinal dynamical behaviors and space explorations. In this paper, a new method is proposed to obtain natural periodic orbits, which is based on the optimal control framework with respect to a general form of the irregular gravitational field. Natural elongated asteroids are taken as representatives, whose potentials are approximated by the rotating mass dipole with appropriate parameters. An indirect method is used to transform the optimal control model into a two-point boundary value problem, which can be solved by using a shooting method. Numerical simulations are performed to validate the effectiveness of the proposed method regarding the asteroid 951 Gaspra. Three types of periodic orbits are identified, including the Lyapunov orbit around the collinear equilibrium point, the equatorial retrograde orbit, and the inclined orbit. The connection between the latter two types of orbits is also briefly discussed via numerical continuation.

Delta-V Assisted Periodic Orbits Around Small Bodies

Delta-V assisted periodic orbits are introduced as a new type of periodic orbit around small bodies subject to strong solar radiation pressure. Delta-V assisted periodic orbits are made periodic by introducing a small deterministic delta-V within each period. This type of orbit has a simpler shape and provides higher flexibility than other periodic orbits and therefore enables missions with higher scientific value. The general theory of delta-V assisted periodic orbits is described, including the orbit design methodology, solution-space analysis, and stability analysis. This paper clarifies that delta-V assisted periodic orbits are useful and feasible options for small-body missions and exhibit unique dynamic characteristics.

Dynamics of Orbits Near 3:1 Resonance in the Earth-Moon System

The Interstellar Boundary Explorer (IBEX) spacecraft is currently in a highly elliptical orbit around Earth with a period near 3:1 resonance with the Moon. Its orbit is oriented so that apogee does not approach the Moon. Simulations show this orbit to be remarkably stable over the next 20 years. This article examines the dynamics of such orbits in the Circular Restricted 3-Body Problem (CR3BP). We look at three types of periodic orbits, each exhibiting a type of symmetry of the CR3BP. For each of the orbit types, we assess the local stability using Floquet analysis. Although not all of the periodic solutions are stable in the mathematical sense, any divergence is so slow as to produce practical stability over several decades. We use Poincare maps with twenty-year propagations to assess the nonlinear stability of the orbits, where the perturbation magnitudes are related to the orbit uncertainty for the IBEX mission. Finally we show that these orbits belong to a family of orbits connected in a bifurcation diagram that exhibits exchange of stability. The analysis of these families of period orbits provides a valuable starting point for a mission orbit trade study.

The Hamiltonian Structure-Preserving Control and Some Applications to Nonlinear Astrodynamics

A systematic research on the structure-preserving controller is investigated in this paper, including its applications to the second-order, first-order, time-periodic, or degenerated astrodynamics, respectively. The general form of the controller is deduced for the typical Hamiltonian system in full feedback and position-only feedback modes, which is successful in changing the hyperbolic equilibrium to an elliptic one. With the poles assigned at any different positions on imaginary axis, the controlled Hamiltonian system is Lyapunov stable. The Floquet multiplier is employed to measure the stability of time-dependent Hamiltonian system, because the equilibrium of periodic system may be unstable even though the equilibrium is always elliptic. One type of periodic orbits is achieved by the resonant conditions of control gains, and another type is making judicious choice in the foundational motions with different frequencies. The control gains are selected from the viewpoint of both the local and global optimizations on fuel cost. This controller is applied to some astrodynamics to achieve some interesting conclusions, including stable lissajous orbits in solar sail’s three-body problem and degenerated two-body problem, quasiperiodic formation flying on aJ2-perturbed mean circular orbit, and controlled frozen orbits for a spacecraft with a high area-to-mass ratio.

Complex bifurcations in a nonlinear system of moving belt

A kind of one-degree-of-freedom nonlinear moving belt system is considered. The analytical research of sliding region and existence conditions of equilibrium are first derived by the theory of piecewise-smooth dynamical system. Then, using numerical method, one- or two-parameter continuation of several types of periodic orbits of the system is calculated. We obtain codimension-1 sliding bifurcation curves, codimension-2 sliding bifurcation points, and global bifurcation diagram in parameter space for the system. The investigation of bifurcation behavior shows that the speed of moving belt and amplitude of friction have a great influence on dynamic behavior, and reveals the complex nonlinear dynamic phenomenon of the moving belt system.

ONE-DIMENSIONAL DYNAMICS OF THE VAN DER POL CIRCUIT

The van der Pol circuit with neon glow lamp operates as a relaxation oscillator driven by a sinusoidal voltage source. Its dynamics is described by the function which maps an arc of circle into itself, preserves orientation on this arc, and has at most one jump-discontinuity point on the circle. Properties of this mapping are discussed with emphasis on rotation numbers, types of periodic orbits, saddle-node and border-collision bifurcations. The so-called period-adding phenomenon is explained. Some remarks are given about dynamics, where the mapping does not preserve the orientation because of too long neon lamp lighting.

Non-Birkhoff Periodic Orbits in a Circle Map

Birkhoff and non-Birkhoff types of periodic orbits are defined in circle maps. The dynamical order relation of non-Birkhoff periodic orbits (NBOs) with period longer than or equal to three is proved. Braids are constructed for NBOs, and a lower bound for the topological entropy is derived.

Stable Orbits of Planets of a Binary Star System in the Three-Dimensional Restricted Problem

The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Henon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents. We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10 000 sets of initial values for periodicity in each case, thus 160 000 integrations, all with z o or zo equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160 000 cases with z o or zo equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1 Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper. We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).

Periodic orbit–Quantum mechanical investigation of the inversion mechanism of Ar3

The inversion mechanism of a T-shaped Ar3 is studied both classically and quantum mechanically. Regular states, localized in the region of the transition state for the inversion of the axial argon atom are found and are assigned by the symmetric stretch stable periodic orbits which emanate from the saddle point of the potential. These states inhibit the inversion process. States which promote the inversion are mainly irregular, but a few of them are localized and they have their nodes perpendicularly arranged along periodic orbits which originate from saddle node bifurcations. The two types of periodic orbits, inhibiting and isomerizing, are used to produce distinctly different spectra and to extract the corresponding eigenfunctions by solving the time dependent Schrödinger equation using a variable order finite difference method [J. Chem. Phys. 111, 10827 (1999), preceding paper].

Periodic and Homoclinic Motions in Forced, Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.

The photodissociation of O3: A classical dynamical approach for the interpretation of the recurrences in the autocorrelation function

Families of periodic orbits and the classical survival probability function are calculated for ozone in the excited 1B2 state. It is shown that some of the observed oscillations in the experimentally extracted and quantum mechanically calculated time autocorrelation functions can be explained by certain types of periodic orbits. Disagreements between the calculated and experimental correlation function are attributed to inaccuracies of the potential function used.