Some families of periodic solutions in the planarN-body problem

Recent results and existence proofs concerning periodic motions of circular-elliptic type forN=3 andN=4 are reviewed.     

Nonuniqueness of families of periodic solutions in a four dimensional mapping

If we follow a family of periodic solutions along a closed path in a parameter space of two dimensions we may not return to the original solution when the parameters return to the original values. We study such nonuniqueness phenomena in simple and double period familis. Nonuniqueness appears if a closed path in the parameter space goes around a critical point. In some cases we find Riemann sheets in the same way as in multiply valued functions. In other cases the connections of various families change in a complicated way around the critical point. All these phenomena are explained analytically. At the critical point there is a collision of bifurcations. The changes of the connections of various families at such collisions of bifurcations are studied in some detail.     

Closed families of periodic solutions of a restricted three-body problem

The planar restricted three-body problem has an infinite number of families of symmetric periodic solutions (SPSs). The natural SPS families include certain families which are self-closed with respect to small variations in a parameter. These families remain closed for any admissible variations in the mass parameter μ. However, there are closed SRS families of another type, which exist only in bounded intervals of μ and are formed via self-bifurcations of some SPS families. This type of SPS families is poorly understude. This work describes the initial stage (4 bifurcations) of a bifurcation cascade of the natural family i and points out other closed SPS families known to date.     

On the application of optimization methods to the determination of members of families of periodic solutions

The techniques used for the numerical computation of families of periodic orbits of dynamical systems rely on predictor-corrector algorithms. These algorithms usually depend on the solution of systems of approximate equations constructed from the periodicity conditions of these orbits. In this contribution we transform the root finding procedure to an optimization one which is applied on an objective function based on the exact periodicity conditions. Thus, the determination of periodic solutions and families of such orbits can be accomplished through unconstrained optimization. In this paper we apply and compare some well-known minimization methods for the solution of this problem. The obtained results are promising. This revised version was published online in July 2006 with corrections to the Cover Date.     

A class of periodic solutions of the N-body problem

We prove existence and multiplicity of T-periodic solutions (for any given T) for the N-body problem in ^m (any m 2) where one of the bodies has mass equal to 1 and the others have masses ₂,..., _N , small. We find solutions such that the body of mass 1 moves close to x = 0 while the body of mass _i moves close to one of the circular solutions of the two body problem of period T/k _i, where ki is any odd number. No relation has to be satisfied by k ₂,...,k _N.     

Evolution with the mass parameter of families of asymmetric periodic solutions of the restricted three body problem

Celestial Mechanics and Dynamical Astronomy - The two asymmetric bifurcations associated with the exterior commensurabilities of the formq+1: 1 are found to exist forq=1, 2, 3, 4 throughout the...     

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.     

Direct Taylor presentation of families of periodic orbits

Direct Taylor expansion of the initial states of families of planar symmetric periodic solutions of the restricted problem in terms of the period with numerical examples are given. The computation of the coefficients of the series is based on the integration of the equations of first, second, third, etc. variations. In this work we did not consider the equations of fourth and higher variations.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

Three-dimensional branchings of plane periodic solutions

In the case of the restricted three-body problem with small mass parameter a family of plane symmetric periodic orbits of the direct type around the large primary is found to have branches of three-dimensional periodic orbits. One such branch has been established consisting of stable orbits for small deviations from the plane.     

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.     

A grid search for families of periodic orbits in the restricted problem of three bodies

A method is described for the numerical determination of families of periodic orbits in the planar restricted problem of three bodies. The families are sought in their representation as curves in a two-dimensional space of parameters. A grid search is applied to the study of the evolution of satellite motion when the mass parameter is varied. Only that part of the space of parameters is investigated for which one of them, the relative energy constant, takes values larger than that corresponding to the inner Lagrangian pointL ₂. Critical values of the mass parameter are determined for which new families of simple or double periodic orbits appear inside the closed ovals of zero velocity.     

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.     

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.     

A family of periodic solutions of the planar three-body problem,and their stability

We describe a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with Schubart's (1956) rectilinear orbit and ends in retrograde revolution, i.e. a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the interplay type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.     

New periodic solutions of the three dimensional restricted problem

A global review of the symmetric solutions of the restricted problem made in the Introduction opens a window on new symmetric periodic orbits of the two body problem in rotating axes which could be ‘trivially’ continuable to symmetric periodic orbits of the three dimensional restricted problem for small values of μ (see Figure 3). The proof of this possibility of continuation is given in Sections 1, 2, 3 using regularizing variables.