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.

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.

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.

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.

Construction of Periodic Orbits and Chaos

In contrast to the main stream of studying the chaotic behavior of a given map, constructing chaotic map has attracted much less attention. In this paper, we propose simple analytical method for constructing one dimensional continuous maps with certain specific periodic points.Further, we obtain results for the existence of chaotic phenomenon in the sense of Li and Yorke. Some examples are also analyzed using the proposed methods.

Detecting Unstable Periodic Orbits in Hyperchaotic Systems Using Subspace Fixed-Point Iteration

We present a numerical method for efficiently detecting unstable periodic orbits(UPO’s)embedded in chaotic attractors of high-dimensional systems.This method,which we refer to as subspace fixed-point iteration, locates fixed points of Poincare maps using a form of fixed-point iteration that splits the phase space into appropriate subspaces.In this paper,among a number of possible implementations,we primarily focus on a subspace method based on the Schmelcher-Diakonos(SD)method that selectively locates UPO’s by varying a stabilizing matrix,and present some applications of the resulting subspace SD method to hyperchaotic attractors where the UPO’s have more than one unstable direction.

THE CRITERION ALGORITHM OF RELATION OF IMPLICATION BETWEEN PERIODIC ORBITS(Ⅰ)

In recent years, there is a wide interest in Sarkovskii's theorem ami the related study. According to Sarkovskii's theoren if the continuous self-mapf of the closed interval has a 3-pcriodic orbit, then fmust has an n-pcriodic orbit for any positive integer n. But f can not has all n-periodic orbits for some n.For example, letEvidently, f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits. This explains that it isn't far enough to uncover the relation between periodic orbits by information which Sarkovskii's theorem has offered. In this paper, we raise the concept of type of periodic orbits, and give a feasible algorithm which decides the relation of implication between two periodic orbits.

Analysis of Effect of Oblateness of Smaller Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of oblateness of smaller primary on these orbits are considered. It is observed that oblateness of smaller primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to planetary oblateness has to be understood and should be taken care of during trajectory design. In this paper, detailed stability analysis of periodic orbit having three loops is given for A2 = 0.0001.

Analysis of Effect of Solar Radiation Pressure of Bigger Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of solar radiation pressure of bigger primary and actual oblateness of smaller primary on these orbits areconsidered. It is observed that solar radiation pressure of bigger primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to solar radiation pressure has to be understood and should be taken care of during trajectory design. It is also verified that stability of such orbits are negligible so they can be used as transfer orbit. For each pair of solar radiation pressure q and Jacobi constant C we get two separatrices where stability of island becomes zero. In this paper, detailed stability analysis of periodic orbit having two loops is given when q = 0.9845.

A review of periodic orbits in the circular restricted three-body problem

This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem(CRTBP),which is a standard ideal model for the Earth-Moon system and is closest to the practical mechanical model.It focuses the attention on periodic orbits in the Earth-Moon system.This work is primarily motivated by a series of missions and plans that take advantages of the three-body periodic orbits near the libration points or around two gravitational celestial bodies.Firstly,simple periodic orbits and their classification that is usually considered to be early work before 1970 are summarized,and periodic orbits around Lagrange points,either planar or three-dimensional,are intensively studied during past decades.Subsequently,stability index of a periodic orbit and bifurcation analysis are presented,which demonstrate a guideline to find more periodic orbits inspired by bifurcation signals.Then,the practical techniques for computing a wide range of periodic orbits and associated quasi-periodic orbits,as well as constructing database of periodic orbits by numerical searching techniques are also presented.For those unstable periodic orbits,the station keeping maneuvers are reviewed.Finally,the applications of periodic orbits are presented,including those in practical missions,under consideration,and still in conceptual design stage.This review article has the function of bridging between engineers and researchers,so as to make it more convenient and faster for engineers to understand the complex restricted three-body problem(RTBP).At the same time,it can also provide some technical thinking for general researchers.

Noise destroys the coexistence of periodic orbits of a piecewise linear map

The effects of Gaussian white noise and Gaussian colored noise on the periodic orbits of period-5（P-5） and period-6（P-6） in their coexisting domain of a piecewise linear map are investigated numerically.The probability densities of some orbits are calculated.When the noise intensity is D = 0.0001,only the orbits of P-5 exist,and the coexisting phenomenon is destroyed.On the other hand,the self-correlation time τ of the colored noise also affects the coexisting phenomenon.When τc〈τ〈τc,only the orbits of P-5 appear,and the stability of the orbits of P-5 is enhanced.However,when τ〉τc（τc and τc are critical values）,only the orbits of P-6 exist,and the stability of the P-6 orbits is enhanced greatly.When τ〈τc,the orbits of P-5 and P-6 coexist,but the stability of the P-5 orbits is enhanced and that of P-6 is weakened with τ increasing.

Bifurcations of Periodic Orbits for a Four-Dimensional System

Consider a four-dimensional system having a two-dimensional invariant surface. By analyzing the solutions of bifurcation equations, this paper studied the bifurcation phenomena of a k multiple closed orbit in the invariant surface. Sufficient conditions for the existence of periodic orbits generated by the k multiple closed orbit were given.

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.

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.

Topological classification of periodic orbits in Lorenz system

We systematically investigate the periodic orbits of the Lorenz flow up to certain topological length. As an alternative to Poincar6 section map analysis, we propose a new approach for establishing one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is stable numerically for cycle searching, and two orbital fragments can be used as basic building blocks for initialization. The topological classification based on the entire orbital structure is revealed to be effective. The deformation of periodic orbits with the change of parameters provides a chart to the periods of cycles. The current research may provide a methodology for finding and systematically classifying periodic orbits in other similar chaotic flows.

COEXISTING PERIODIC ORBITS IN VIBRO-IMPACTING DYNAMICAL SYSTEMS

A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, and in particular, it is investigated in details how to assure that no other impacts will happen in an evolution period of a single impact periodic motion. Furthermore, some criteria for nonexistence of single impact periodic orbits with specific periods are also established. Finally, the stability of coexisting periodic orbits is discussed, and the corresponding computation formula is given. Examples of numerical simulation are in good agreement with the theoretic analysis.

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.

PERIODIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

In this paper, we develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds. We give existence, stability and bifurcation theorems and illustrate our results with a truncated spectral model of the forced, dissipative quasi-geostrophic flow on a cyclic beta-plane.

The Centres of Gravity of Periodic Orbits

Let f ： I → I be a continuous map. If P （n, f） = {x ∈I; fn （x） = x} is a finite set for each n ∈ N, then there exits an anticentered map topologically conjugate to f, which partially answers a question of Kolyada and Snoha. Specially, there exits an anticentered map topologically conjugate to the standard tent map.

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

The current methods for designing periodic orbits in the elliptic restricted three-body problem(ERTBP)have the disadvantages of targeting limited orbits and ergodic searches and considering only symmetric orbits.A universal method for designing periodic orbits is proposed in this paper.First,the homotopy classes of orbits are structured based on their topological structures.Second,a dynamic model based on homotopy classes,ranging from the circular restricted three-body problem(CRTBP)to the ERTBP,can be built using the homotopy method.Third,a multi-and a single-period orbit were selected based on the resonance ratios.Finally,the corresponding orbit in the ERTBP was computed by modifying the initial condition of the orbit in the CRTBP.This method,without an ergodic search,can extend to any orbit,including an asymmetric orbit in the CRTBP,to the ERTBP model,and the two orbits are of the same homotopy class.Examples of the Earth–Moon ERTBP are presented to verify the efficiency of this method.