Unstable periodic orbits analysis in the Qi system

We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits,Lyapunov exponents,and Poincarémaps.Based on several periodic orbits with different sizes and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively.In this process,the rule of orbital period changing with parameters is also investigated.The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.

Spherical and Circular Non-Equatorial Photon Orbits around Kerr Black Holes

By analytically solving the equation of azimuthal null geodesics for spherical photon trajectories, a parametric representation of the corresponding segment of the orbit is obtained. The solution parameter is the latitude coordinate. The dependences of the orbital radius on the black hole spinning parameter and the angle of inclination of its plane with respect to the rotation axis are calculated for flat circular non-equatorial orbits. It is proved that all spherical photon trajectories in the Kerr spacetime are unstable, as well as equatorial ones, and the critical photon orbits in the Schwarzschild metric.

Dynamic Electron Orbits in Atomic Hydrogen

Even after extensive research in Quantum Mechanics, we are still unable to visualize instant-to-instant motion of an electron in hydrogen atom. That is because in QM treatment, potential energy term has been mistakenly assumed to be time-independent instead of depending on the instant-to-instant varying position of the orbiting electron [1]. This has led to wrong and weird solutions for the electron motion in hydrogen atom. Before the advent of wave mechanics, Sommerfeld model of elliptical electron orbits was able to explain most features of hydrogen spectra, except for the features associated with electron spin and magnetic moment interactions. However, the Sommerfeld elliptical orbits were of kinematic origin and could not provide visualization of instant-to-instant dynamic motion of the orbiting electron. Contrary to the QM perspective, we find that central core of the electron behaves as a classical particle while its electrostatic field behaves as a wave phenomenon. As such an electron under Coulomb force moves strictly in accordance with Newtonian laws of motion. In this paper, we develop dynamic electron orbits in hydrogen atom by using energy and angular momentum conservation principle in central force field. We have shown that during photon emission, angular momentum of the orbiting electron is changed by ħ due to recoil action. This may be the origin of various quantization rules. During emission of a photon, elliptical orbit transitions are also computed and plotted. Orbit transition time is of the order of 10-16 seconds. We have extended this methodology to compute electron orbits in hydrogen molecular bond and computed the H2 bond energy.

A Positioning Method and Realization on Single Satellites in Different Orbits Using TDOA

The main geolocation technology currently used in COSPAS-SARSAT system is TDOA/FDOA or three-star TDOA,the principle is to determine the location of the signal source by using the difference in arrival time and frequency of the wireless signal between different receivers.Therefore,ground monitoring stations need to be equipped with more than two antenna receiving stations,and multiple satellites should be able to simultaneously relay the distress signal from the target source in order to achieve the geolocation function.However,when the ground receiving system has only one antenna receiving station,or the target source is in a heavily obscured environment,the ground side is unable to receive the forwarded signals from multiple satellites at the same time,which will make it impossible to locate.To address these problems,in this paper,a time-sharing single satellite geolocations method based on different orbits is proposed for the first time.This method uses one or several low-earth orbit satellites(LEO)and mediumearth orbit satellites(MEO)in the visible area,and the receiving station only needs one pair of receiving antennas to complete the positioning.It can effectively compensate for the shortcomings of the traditional TDOA using the same moment and have better positioning accuracy compared with the single satellite in the same orbit.Due to the limited experimental conditions,this paper tests the navigation satellite using different orbit time-sharing single satellite geolocations,and proves that the positioning method has high positioning accuracy and has certain promotion and application value.

Evolution of Periodic Orbits in the Sun-Saturn System

We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitational Sun-Saturn system incorporating actual oblateness of Saturn in the planar circular restricted three body problem. In this paper, we study the effect of solar radiation pressure on the location of Sun centered and Saturn centered orbits, its diameter, semi major axis and eccentricity by taking different values of solar radiation pressure q and different values of Jacobi constant “C”, and by considering actual oblateness of Saturn using Poincare surface of section (PSS) method. It is ob-served that by the introduction of perturbing force due to solar radiation pressure admissible range of Jacobi constant C decreases, it is also observed that as value of C decreases the number of islands decreases and as a result the number of periodic and quasi periodic orbits decreases.Fur-ther, the periodic orbits around Saturn and Sun moves towards Sun by decreasing perturbation due to solar radiation pressure q for a specific choice of Jacobi constant C. It is also observed that due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces, whereas, due to solar radiation pressure uniform change in semi major axis and eccen-tricity of Saturn centered periodic orbits is observed.

Evolution of the “f” Family Orbits in the Photo Gravitational Sun-Saturn System with Oblateness

We analyze the periodic orbits of “f” family (simply symmetric retrograde periodic orbits) and the regions of quasi-periodic motion around Saturn in the photo gravitational Sun-Saturn system in the framework of planar circular restricted three-body problem with oblateness. The location, nature and size of these orbits are studied using the numerical technique of Poincare surface of sections (PSS). In this paper we analyze these orbits for different solar radiation pressure (q) and actual oblateness coefficient of Sun Saturn system. It is observed that as Jacobi constant (C) increases, the number of islands in the PSS and consequently the number of periodic and quasi-periodic orbits increase. The periodic orbits around Saturn move towards the Sun with decrease in solar radiation pressure for given value of “C”. It is observed that as the perturbation due to solar radiation pressure decreases, the two separatrices come closer to each other and also come closer to Saturn. It is found that the eccentricity and semi major axis of periodic orbits at both separatrices are increased by perturbation due to solar radiation pressure.

PERSISTENCE AND PERIODIC ORBITS FOR TWO-SPECIES NON-AUTONOMOUS DIFFUSION LOTKA-VOLTERRA MODELS

A non-autonomous competing system is investigated in this paper,where the species x can diffuse between two patches of a heterogeneous environment with barriers between patches,but for species y,the diffusion does not involve a barrier between patches,further it is assumed that all the parameters are time dependent.It is shown that the system can be made persistent under some appropriate conditions.Moreover,sufficient conditions that guarantee the existence of a unique positive periodic orbit which is globally asymptotic stable are derived.

HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.

Negative Parity States in ^(39)Cl Configured by Crossing Major Shell Orbits

Traditional "magic numbers" were once regarded as immutable throughout the nuclear chart. However, unexpected changes were found for unstable nuclei around N = 20. With both proton and neutron numbers around the magic number of 20, the neutron-rich ^(39)Cl isotope provides a good test case for the study of the quantumstate evolution across the major shell. In the present work, the negative parity states in ^(39)Cl are investigated through the β decay spectroscopy of 39 S. Newly observed γ transitions together with a new state are assigned into the level scheme of ^(39)Cl. The spin parity of 5/2^- for the lowest negative parity state in ^(39)Cl is reconfirmed using the combined γ transition information. These systematic observations of the negative parity states in ^(39)Cl allow a comprehensive comparison with the theoretical descriptions. The lowest 5/2^- state in ^(39)Cl remains exotic in terms of comparisons with existing theoretical calculations and with the neighboring isotopes having similar single-particle configurations. Further experimental and theoretical investigations are suggested.

Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits.Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris.This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories,considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow.Firstly,a set of mean-element orbital dynamic equations of a chaser(spacecraft)and a target(debris)are derived by using the orbital averaging technique,and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem.Subsequently,the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle,and a practical optimization procedure is adopted to find the specific initial costate variables,wherein the necessary conditions of the optimal solutions are all satisfied.Afterwards,the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics.A simple correction strategy about the initialization of the mean elements,specifically the differential mean true longitude,is suggested,which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements.Finally,numerical examples are presented,and specifically,the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems,demonstrating the effectiveness of the optimization method proposed in this article.

EQUIDISTRIBUTION OF CLOSED ORBITS OF BILLIARD FLOWS ON THE PLANE

The authors consider the billiard system with finitely many convex scatters with smooth boundary satisfying the visibility assumption on the plane and prove that the closed orbits for the billiard flow is uniformly distributed.

A Strategy for Magnifying Vibration in High-Energy Orbits of a Bistable Oscillator at Low Excitation Levels

This work focuses on how to maintain a high-energy orbit motion of a bistable oscillator when subjected to a low level excitation. An elastic magnifier （EM） positioned between the base and the bistable oscillator is used to magnify the base vibration displacement to significantly enhance the output characteristics of the bistable oscillator. The dimensionless electromechanical equations of the bistable oscillator with an EM are derived, and the effects of the mass and stiffness ratios between the EM and the bistable oscillator on the output displacement are studied. It is shown that the jump phenomenon occurs at a lower excitation level with increasing the mass and stiffness ratios. With the comparison of the displacement trajectories and the phase portraits obtained from experiments, it is vMidated that the bistable oscillator with an EM can effectively oscillate in a high-energy orbit and can generate a superior output vibration at a low excitation level as compared with the bistable oscillator without an EM.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.

Periodic orbits around areostationary points in the Martian gravity field

This study investigates the problem of areostationary orbits around Mars in three-dimensional space. Areostationary orbits are expected to be used to establish a future telecommunication network for the exploration of Mars. However, no artificial satellites have been placed in these orbits thus far. The characteristics of the Martian gravity field are presented, and areostationary points and their linear stability are cal- culated. By taking linearized solutions in the planar case as the initial guesses and utilizing the Levenberg-Marquardt method, families of periodic orbits around areo- stationary points are shown to exist. Short-period orbits and long-period orbits are found around linearly stable areostationary points, but only short-period orbits are found around unstable areostationary points. Vertical periodic orbits around both lin- early stable and unstable areostationary points are also examined. Satellites in these periodic orbits could depart from areostationary points by a few degrees in longitude, which would facilitate observation of the Martian topography. Based on the eigenval- ues of the monodromy matrix, the evolution of the stability index of periodic orbits is determined. Finally, heteroclinic orbits connecting the two unstable areostationary points are found, providing the possibility for orbital transfer with minimal energy consumption.

Halo Orbits around Sun-Earth L1 in Photogravitational Restricted Three-Body Problem with Oblateness of Smaller Primary

This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.

Periodic Orbits in the Photogravitational Restricted Problem When the Primaries Are Triaxial Rigid Bodies

We have studied periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are triaxial rigid bodies and source of radiation pressure. We have determined periodic orbits for different values of (h is energy constant;μ is mass ratio of the two primaries;are parameters of triaxial rigid bodies and are radiation parameters). These orbits have been determined by giving displacements along the tangent and normal at the mobile co-ordinates as defined in our papers (Mittal et al. [1]-[3]). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of triaxial bodies and source of radiation pressure on the periodic orbits by taking fixed value of μ.

Generating Periodic Orbits for Explorations of Elongated Asteroids

A practical method is proposed to search for periodic orbits of elongated asteroids.The method obtains required initial variables of periodic orbits by using the rotating mass dipole with appropriate parameters,and then implement local iterations to obtain the real orbits over an asteroid in the polyhedral model.In this paper the dipole and polyhedral models,and list detailed procedures of the searching method are introduced.A planar single lobe orbit is presented to evaluate the effectiveness of the method,with the asteroid 216 Kleopatra of the triple asteroid system as a representative elongated body.By applying the above method,ten families of periodic orbits around Kleopatra are identified and discussed with respect to their orbital stabilities and periods.One sample of the sombrero orbit is checked by calculating 1000 hours to examine its orbital behavior.Besides the above orbits,the intriguing head-surrounding orbit is also analyzed.

COADJOINT ORBITS FOR THE CENTRAL EXTENSION OF Diff^+(S^1) AND THEIR REPRESENTATIVES

According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits (?).In the forward direction one applies the methods of geometric quantization to produce a representation, and in the reverse direction one computes a transform of the character of a representation, to obtain a coadjoint orbit. The method of orbits in the representations of Lie groups suggests the detailed study of coadjoint orbits of a Lie group G in the space (?)~* dual to the Lie algebra (?) of G. In this paper, two primary goals are achieved: one is to completely classify the smooth coadjoint orbits of Virasoro group for nonzero central charge c; the other is to find representatives for coadjoint orbits. These questions have been considered previously by Segal, Kirillov, and Witten, but their results are not quite complete. To accomplish this, the authors start by describing the coadjoint action of D-the Lie group of all orientation preserving diffeomorphisms on the circle S^1, and its central extension (?), then the authors will give a complete classification of smooth coadjoint orbits. In fact, they can be parameterized by a subspace Of conjugacy classes of (?)(1,1). Finally, the authors will show how to find representatives f coadjoint orbits by analyzing the vector fields stabilizing the orbits, and describe the amazing connection between the characteristic (trace) of conjugacy classes of (?)(1, 1) and that of vector fields stabilizing orbits.

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a four-dimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincaré map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analYSiS of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface, 2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.

Invariant Relative Orbits Taking into Account Third-Body Perturbation

For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.