Lyapunov Orbits Research Articles

Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincare recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite ‘minimum recurrence time’. We find the minimum recurrence time as a function of the energy.

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three-Body Problem - A Computer Assisted Proof

The restricted circular three-body problem is considered for the following parameter values C=3.03, μ=0.0009537 – the values for the Oterma comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of a homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle.

Space and security: a US policy quandary

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. This work proposes and describes an intuitive polyhedral interpolative approach for each state component associated with manifold trajectories, both in two and in three dimensions. An adequate grid of data, coming from the numerical propagation of a finite number of manifold trajectories, is employed. Accuracy of this representation is evaluated with reference to the invariant manifolds associated with a two-dimensional Lyapunov orbit and a three-dimensional Halo orbit, and is proven to be satisfactory, with the exclusion of limited regions of the manifolds. As a first, preliminary application, the polyhedral interpolation technique allows identifying the orbits in the proximity of the interior collinear libration point as either asymptotic, transit, or bouncing trajectories. Then, two applications to orbital maneuvering are addressed. First, the globally optimal two-impulse transfer between a specified low Earth orbit and a Lyapunov orbit (through its stable manifold) is determined. Second, the minimum-time low-thrust transfer from the same terminal orbits is found using again the stable manifold. These applications prove the effectiveness of the polyhedral interpolative technique and represent the premise for its application also to different problems involving invariant manifold dynamics.

Dynamic systems arising on the invariant tori of the Kowalewska problem: PMM vol. 39, n≗ 1. 1975, pp. 24–29

The dynamics near the Lagrange equilibria L1 and L2 of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at L1 and L2 of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at L1 or L2 computed with the Cartesian variables is limited in amplitude by |1−μ−xL1| (μ denoting the reduced mass of the problem), we investigate if regularizations allow us to overcome this limit. In particular, we consider the Hamiltonian describing the planar three-body problem in the Levi-Civita regularization and we compute its normalization for the Sun–Jupiter reduced mass for an interval of energy which overcomes the limit of Cartesian normalizations. As a result, for the largest values of the energy that we consider, we notice a transition in the structure of the tube manifolds emanating from the Lyapunov orbit, which can contain orbits that collide with the secondary body before performing one full circulation around it. We discuss the relevance of this transition for temporary captures.

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.