Three-dimensional bifurcations of periodic solutions around the triangular equilibrium points of the restricted three-body problem

Vertically critical, planar periodic solutions around the triangular equilibrium points of the Restricted Three-Body Problem are found to exist for values of the mass parameter in the interval [0.03, 0.5]. Four series of such solutions are computed. The families of three-dimensional periodic solutions that branch off these critical orbits are computed for µ = 0.3 and are continued till their end. All orbits of these families are unstable.     

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.     

On relative periodic solutions of the planar general three-body problem

We describe two relatively simple reductions to order 6 for the planar general three-body problem. We also show that this reduction leads to the distinction between two types of periodic solutions: absolute or relative periodic solutions. An algorithm for obtaining relative periodic solutions using heliocentric coordinates is then described. It is concluded from the periodicity conditions that relative periodic solutions must form families with a single parameter. Finally, two such families have been obtained numerically and are described in some detail.The present research was carried out partially at the University of California and partially at the Jet Propulsion Laboratory under contract NAS7-100 with NASA.     

Application of Gauss's method in the formation of periodic solutions in the relativistic restricted problem of three bodies

The paper discusses the existence of periodic and quasi-periodic solutions in the space relativistic problem of three bodies with the help of Poincaré's small parameter method starting from non-Keplerian generating solutions, i.e., using Gauss's method. The main peculiarity of these periodic orbits is the fact that they close, in general, after many revolutions. It is worth noticing that these periodic orbits give a new class of periodic solutions of the classical circular problem of three bodies, if relativistic effects are neglected.     

Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.     

Numerical extrapolation of periodic solutions

Numerical procedures are established for the continuation of families of periodic solutions of non-integrable dynamical systems. They are based on the use of the previous known members of a family for non-linear prediction of the next member to be determined. Both symmetric and asymmetric periodic solutions are considered. The procedures are applied and compared in the case of the restricted three-body problem. They are shown to lead to considerable saving of computer time.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

On the stability parameters of periodic solutions

The stability parametersa, b, c, d of plane symmetric periodic solutions of non-integrable dynamical systems of two degrees of freedom are obtained in terms of their initial states of motion and elements of their variational matrics. Explicit formulae are given in the cases of the Störmer problem and the restricted problem of three bodies.     

Radial periodic perturbations of the Kepler problem

We consider radial periodic perturbations of a central force field and prove the existence of rotating periodic solutions, whose orbits are nearly circular. The proof is mainly based on the Implicit Function Theorem, and it permits to handle some small perturbations involving the velocity, as well. Our results apply, in particular, to the classical Kepler problem.     

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with nonextensive ions

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with q-nonextensive velocity distributed ions are studied through non-perturbative approach. Basic equations are reduced to an ordinary differential equation involving electrostatic potential. After that by applying the bifurcation theory of planar dynamical systems to this equation, we have proved the existence of solitary wave solutions and periodic wave solutions. Two exact solutions of the above waves are derived depending on the parameters. From the solitary wave solution and periodic wave solution, the effect of the parameter (q) is studied on characteristics of dust acoustic solitary waves and periodic waves. The parameter (q) significantly influence the characteristics of dust acoustic solitary and periodic structures.     

Numerical investigation of symmetric and asymmetric periodic oscillations

Global information for the periodic solutions — symmetric and asymmetric — of the ‘gravitational’ spring-pendulum problem is given for the first time. For two different sets of the parameters of this problem, the families of symmetric periodic solutions which emanate from the equilibrium point have been determined. Further families of asymmetric and symmetric solutions which bifurcate from them have also been examined and interesting results for their behaviour have been pointed out.     

Critical generating orbits for second species periodic solutions of the restricted problem

The second species periodic solutions of the restricted three body problem are investigated in the limiting case of μ=0. These orbits, called consecutive collision orbits by Hénon and generating orbits by Perko, form an infinite number of continuous one-parameter families and are the true limit, for μ→0, of second species periodic solutions for μ>0. By combining a periodicity condition with an analytic relation, for criticality, isolated members of several families are obtained which possess the unique property that the stability indexk jumps from ±∞ to ?∞ at that particular orbit. These orbits are of great interest since, for small μ>0, ‘neighboring’ orbits will then have a finite (but small) region of stability.     

Deflation techniques for the determination of periodic solutions of a certain period

The computation of periodic orbits of nonlinear mappings or dynamical systems can be achieved by applying a root-finding method. To determine a periodic solution, an initial guess should be located within a proper area of the mapping or a surface of section of the phase space of the dynamical system. In the case of Newton or Newton-like methods these areas are the basins of convergence corresponding to the considered solution. When several solutions of the same period exist in a particular region, then the deflation technique is suitable for the calculation of all these solutions. This technique is applied here to the Hénon's mapping and the driven conservative Duffing's oscillator.     

On periodic flybys of the moon

This paper considers the plane circular restricted three-body problem for small . Symmetric periodic solutions of the second species (passing near the body of mass ) and their distance from the center of the body of mass are studied by constructing perturbations of arc-solutions (solutions with consecutive collisions) existing for =0. Orbits which also pass near the body of mass 1- are studied in detail. The results are applied to finding periodic orbits in the Earth-Moon system and in the Sun-Jupiter system.Russian version: Preprint No. 91 (1978) of Inst. Appl. Math.; present English translation was made by L. M. Perko and W. C. Schulz (February 1979).     

Periodic solutions of the restricted problem that are analytic continuations of periodic solutions of Hill's problem for small μ>0

This paper establishes the existence and first order perturbation approximation of an infinite number of one-parameter families of symmetric periodic solutions of the restricted three body problem that are analytic continuations of symmetric periodic solutions of Hill's problem for small values of the mass ratio μ>0.     

Numerical determination of asymmetric periodic solutions

A simple predictor-corrector procedure is described for the determination of asymmetric periodic solutions of dynamical systems of two degrees of freedom. An application in the case of the Störmer problem is given. The computed periodic motions of the charged particle are of the open-path type.     

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem.     

Three-dimensional periodic solutions around equilibrium points in Hill's problem

The three-dimensional periodic solutions originating at the equilibrium points of Hill's limiting case of the Restricted Three Body Problem, are studied. Fourth-order parametric expansions by the Lindstedt-Poincaré method are constructed for them. The two equilibrium points of the problem give rise to two exactly symmetrical families of three-dimensional periodic solutions. The family_HL _2v ^e originating at L₂ is continued numerically and is found to extend to infinity. The family originating at L₁ behaves in exactly the same way and is not presented. All orbits of the two families are unstable.     

On the stability of periodic solutions in the problem of the motion of a star inside an elliptical galaxy

The problem of the spatial motion of a star inside an inhomogeneous rotating elliptical galaxy with a homothetic density distribution is considered. Periodic solutions are constructed by the method of a small Poincaré parameter. Linear variational equations with periodic coefficients are used to analyze the Lyapunov stability of these solutions.