I.: The vanishing twist in the restricted three body problem, arXiv:1309.1280 [math-ph (original) (raw)

The vanishing twist in the restricted three-body problem

Physica D: Nonlinear Phenomena, 2014

This paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three-body problem. Near the Lagrangian equilibrium L4 a twistless torus is created near the tripling bifurcation of the short period family. Decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3/10 and 2/7. This scenario is studied by numerically integrating the regularised Hamiltonian flow, and finding rotation numbers of invariant curves in a two-dimensional Poincaré map. To corroborate the numerical results the Birkhoff normal form at L4 is calculated to eighth order. Truncating at this order gives an integrable system, and the rotation numbers obtained from the Birkhoff normal form agree well with the numerical results. A global overview for the mass ratio µ ∈ (µ4, µ3) is presented by showing lines of constant energy and constant rotation number in action space.

Invariant manifolds of L 3 and horseshoe motion in the restricted three-body problem

Nonlinearity, 2006

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L 3 , L 4 and L 5 ), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L 3 . So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L 3 , and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail.

Generic twistless bifurcations

Nonlinearity, 2000

We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an areapreserving map, there is generically a bifurcation that creates a "twistless" torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.

(Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

Physica D, 2003

The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and 1:2 (period-doubling). The twist, which is the derivative of the rotation number with respect to the action, is studied near these bifurcations. When the twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is not satisfied, with interesting consequences for the dynamics. We show that near the saddle-centre bifurcation the twist always vanishes. At this bifurcation a ``twistless'' torus is created, when the resonance is passed. The twistless torus replaces the colliding periodic orbits in phase space. We explicitly derive the position of the twistless torus depending on the resonance parameter, and show that the shape of this curve is universal. For the period doubling bifurcation the situation is different. Here we show that the twist does not vanish in a neighborhood of the bifurcation.

Geometry of Homoclinic Connections in a Planar Circular Restricted Three-Body Problem

International Journal of Bifurcation and Chaos, 2007

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1 between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1 and the other collinear libration points L2, L3 is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

Oscillatory motions for the restricted planar circular three body problem

Inventiones mathematicae, 2015

Mouvements oscillatoires dans le problème plan circulaire restreint des trois corps Résumé : Dans cet article, nousétudions le problème restreint des trois corps, qui modélise le mouvement d'un corps de masse nulle sous l'influence des forces de gravitation newtonienne créées par deux autres corps, appelés les primaires, qui eux se déplacent le long d'orbites képlériennes circulaires. Dans un système de coordonnées convenable, ce système possède deux degrés de liberté et l'énergie conservée est habituellement appelée la constante de Jacobi. En 1980, J. Llibre et C. Simó [LS80b] ont démontré l'existence de mouvements oscillatoires dans le problème plan restreint des trois corps, c'est-àdire d'orbites qui sortent de n'importe quelle région bornée mais qui rentrent une infinité de fois dans une certaine région bornée fixée. Pour démontrer ce résultat, les auteurs avaient besoin de supposer que le rapport des masses des deux primaires est exponentiellement petit par rapportà la constante de Jacobi. Dans le présent travail, nous généralisons ce théorèmeà toute valeur des rapports de masses. Pour obtenir de tels mouvements, nous montrons que, quel que soit le rapport des masses, si la constante de Jacobi est assez grande, il existe des intersections transverses des variétés stable et instable de l'infini, ce qui garantit l'existence d'une dynamique symbolique, puis celle de mouvements oscillatoires. Le principal résultat est de prouver rigoureusement l'existence de ces orbites sans supposer que le rapport de masses est petit, puisqu'alors la transversalité ne peut pasêtre vérifiée par les méthodes de la théorie classique des perturbations relativement au rapport de masses. Comme notre méthode est valable pour toutes les valeurs des rapports de masses, nous parvenonsà détecter une courbe, dans l'espace des paramétres, c'est-à-dire dans l'espace des rapports de masses et de la constante de Jacobi, sur laquelle apparaissent des tangences homoclines cubiques entre les variétés invariantes de l'infini. Contents 1 Introduction 3 2 The invariant manifolds of infinity 6 2.1 The RPC3BP as a nearly integrable Hamiltonian System with two time scales. .. .. . .

Some periodic orbits in the restricted three-body problem, for m>0, from the m = 0 case

In the present work, we deal with horseshoe motion in the frame of the Restricted Three-Body Problem (RTBP) for different values of the mass parameter mm. On one hand, we study numerically families of periodic horseshoe orbits for m small and how they are organised. We figure out the mechanism of the organisation of such families from the two-body problem (m = 0). On the other hand, we study the existence of horseshoe periodic orbits for other values of m. We claim that the behaviour of the invariant manifolds associated to the equilibrium point L3 as well as the existence of homoclinic orbits play an important role.

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

Nonlinearity, 2004

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a 'Petit Grand Tour' of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case.

Computing Invariant Manifolds and Connecting Orbits in the Circular Restricted Three Body Problem

Arxiv preprint arXiv: …, 2011

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the Circular Restricted Three-Body Problem (CR3BP), which models the motion of a satellite in an Earth-Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar Vertical and Halo orbits. We compute the unstable manifolds of selected Vertical and Halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary, leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits could be of potential interest in space-mission design.