Lyapunov Families Research Articles

A method for continuing families of periodic solutions of Lagrangian systems

An autonomous multidimensional Lagrangian system of differential equations depending on parameters is considered. A predictor-corrector method is proposed for constructing a family of periodic solutions (including retrograde solutions) obtained from a given solution by variation of the parameters. In the context of the wide range of problems in classical and celestial mechanics that are described by Lagrangian systems of differential equations, it is particularly interesting to study non-isolated periodic solutions of such equations, parametrized by both extrinsic and intrinsic parameters (the latter are represented by the initial conditions of the solution, such as the energy constant). A family parametrized by the energy constant is known as a natural family /1, 2/. The classical example of a natural family of periodic orbits is provided by the Lyapunov periodic motions /3/ originating from the equilibrium position of a Hamiltonian system. Existing methods of investigating them /4, 5/ are based on the introduction of local coordinates in the neighbourhood of a periodic solution and subsequent normalization of the equations of the perturbed motion, and the construction of the solution as a series with respect to a small parameter, where the latter characterizes the deviation of the motion from the equilibrium position. In a more complicated situation the generating solution is known only in terms of its initial conditions and period, the solution itself being obtained by numerical integration of the original system. Consequently, any method for continuing a (not necessarily Lyapunov) family by expansion with respect to the parameters is necessarily numerical. On the available variety of such methods we mention a series of papers by Sarychev and Sazonov (for references see /6/), who have worked out a highly efficient method for solving the boundary-value problems that arise in this context. This paper draws on ideas similar to those of /7–9/ ∗∗, (∗∗See also Sokol'skii A.G. and Khovanskii S.A., A computation algorithm for continuing periodic solutions of two-dimensional Hamiltonian systems as functions of the parameters, Moscow, MAI, 1986. Deposited at VINITI, 4.05.86, 4042-86.) where a predictor-corrector method for continuation of periodic solutions is worked out for systems with two degrees of freedom. This method will be generalized here to multidimensional systems, for which Birkhoff 's theorem about the reduction of a two-dimensional system to canonical form is no longer valid; it is nevertheless possible to reduce the boundary-value problem to a Cauchy problem by a special choice of local coordinates.

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2. A Lyapunov family of periodic orbits near L 2, the so-called family ( c), is born for C = C 2 and exists for values of C less than C 2. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.

First-order perturbations in the motion of artificial satellites due to the oblateness of the earth: V. F. Proskurin and Iu. V. Batrakov: (pp. 32–38)

The problem of a spacecraft orbiting the Neptune–Triton system is presented. The new ingredients in this restricted three body problem are the Neptune oblateness and the high inclined and retrograde motion of Triton. First we present some interesting simulations showing the role played by the oblateness on a Neptune’s satellite, disturbed by Triton. We also give an extensive numerical exploration in the case when the spacecraft orbits Triton, considering Sun, Neptune and its planetary oblateness as disturbers. In the plane a × I (a = semi-major axis, I = inclination), we give a plot of the stable regions where the massless body can survive for thousand of years. Retrograde and direct orbits were considered and as usual, the region of stability is much more significant for the case of direct orbit of the spacecraft (Triton’s orbit is retrograde). Next we explore the dynamics in a vicinity of the Lagrangian points. The Birkhoff normalization is constructed around L2, followed by its reduction to the center manifold. In this reduced dynamics, a convenient Poincaré section shows the interplay of the Lyapunov and halo periodic orbits, Lissajous and quasi-halo tori as well as the stable and unstable manifolds of the planar Lyapunov orbit. To show the effect of the oblateness, the planar Lyapunov family emanating from the Lagrangian points and three-dimensional halo orbits are obtained by the numerical continuation method.