Asymmetric Periodic Orbits Research Articles

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities

New kinds of asymmetric periodic orbits in the restricted three-body problem

The procedure of numerical ‘ascent’ from families of planar to three-dimensional periodic orbits and the subsequent ‘descent’ to the plane is proved efficient in determining new families of planar asymmetric periodic orbits in the restricted three-body problem. Two such families are computed and described for values of the mass parameter for which it has been found that they exist. Two new families of three-dimensional asymmetric periodic orbits are also presented in this paper.

Three dimensional periodic orbits in three dipole problem

In the three dipole problem where enormous electromagnetic forces obstruct the three dimensional movement of the charged particle we determined for the first time families of three dimensional asymmetric periodic orbits. We study how these families appear, branching from the planar motion and we develop the procedures we have followed to determine them numerically. Also we give their characteristics and the conical projections and plottings of some orbits.

Planar periodic orbits in three dipole problem

A systematic and detailed discussion of planar periodic orbits, of a charged particle moving under the influence of an electromagnetic field of three celestial bodies, is given for the first time. In this problem the periodic orbits are all asymmetric. Numerical procedures are applied to find the families of these orbits and to study their stability. Moreover, the bifurcations of these families with families of three dimensional asymmetric periodic orbits are given.

The symplectic character of periodic orbits in the three dipole problem

In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations.

Asymmetric Periodic Orbits in the Three-Body Problem and Their Stability

AbstractAn algorithm for the numerical determination of asymmetric periodic solutions of the planar general three body problem is described. The elements of the “variational” matrix which are used in this algorithm are computed by numerical integration of the corresponding “variational” equations. These elements are also used in the study of the linear isoenergetic stability. A number of asymmetric periodic orbits are presented and their stability parameters are given.

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

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 entire range of the mass parameter μ. The asymmetric families emanating from these bifurcation orbits for values of μ=0.05, 0.15, 0.25, 0.35, and 0.45 are completely determined. For μ=0.05 and μ=0.15 for each of the four pairs of critical orbits the family of asymmetric periodic orbits evolves continuously from one bifurcation orbit to the other. The families evolving from each pair of bifurcations for μ=0.25, 0.35, and 0.45 terminate on an asymptotic orbit to one of the two equilateral equilibrium points. The evolution of these four families of asymmetric periodic orbits therefore changes significantly for some value of μ between 0.15 and 0.25. An indication is given on how the evolution with μ of familyl and its branches can be studied.

Numerical investigation of the manifold I qp in the Sun-Jupiter case of the restricted three-body problem

Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.

Asymmetric periodic orbits in three dimensions

A numerical procedure is outlined for the determination of three-dimensional asymmetric periodic orbits in the restricted problem of three bodies. Two families of such orbits are computed as branches of a family of plane asymmetric periodic orbits.

On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

Bifurcations of plane with three-dimensional asymmetric periodic orbits in the restricted three-body problem

Bifurcations of plane with three-dimensional asymmetric periodic orbits in the restricted three-body problem

Some families of periodic oscillations in the restricted problem with small mass-ratios of three bodies

This paper reports on the numerical determination of families of periodic oscillations in the case μ=0.000 95 of the restricted problem. The families emanating out of the collinear Lagrangian pointsL1,L2,L3 are examined as well as some asymmetric periodic oscillations related to them. An effort is made to complete the global picture of simple-periodic symmetric oscillations in the present case of the problem (the S-J case). This is done by examining the orbits with initial conditions such that the infinitesimal body starts from a position on the ξ1-axis (ξ02 = 0) with a negative initial velocity\((\dot \xi _{02} 0. The phase portrait of asymmetric periodic orbits is also examined.

Highly perturbed periodic oscillations around a small primary

A global picture of the families of simple periodic orbits in terms of their characteristics is given in the part of the plane (γ, ξ1) of representation near the singularity corresponding to the small primary, for the case of the restricted problem with a small value of the mass parameter μ. The value used for μ is smaller than the critical value of Routh. The picture is found to be qualitatively different from the one corresponding to μ larger than the critical value. By means of the stability parameters four new families, consisting of asymmetric periodic orbits, are shown to exist as bifurcations of families of symmetric periodic orbits.

Hale's method for the computation of periodic solutions of nonlinear differential equations

Abstract. — Hale's method can be used for the numerical computation of periodic solutions of weakly nonlinear differential equations in standard form. It proves to be a good substitute for Galerkin's method if a limited accuracy is needed. A comparison was made for Van der Pol's equation with a forcing term. It was also successfully applied to the computation of asymmetric periodic orbits of the second kind in the restricted problem. Those orbits are members of a family of periodic solutions with three parameters and are not isolated (their existence will be demonstrated elsewhere). Application of Galerkin's method would be difficult in that case.

Proceedings of the Celestial Mechanics Conference: The search for asymmetric periodic orbits in the restricted problem of three bodies

Proceedings of the Celestial Mechanics Conference: The search for asymmetric periodic orbits in the restricted problem of three bodies