Numerical continuation of families of heteroclinic connections between periodic orbits in a Hamiltonian system

Abstract

Abstract. This paper is devoted to the numerical computation and continuation of families of heteroclinic connections between hyperbolic periodic orbits of a Hamiltonian system. We describe a method that requires the numerical continuation of a nonlinear system that involves the initial conditions of the two periodic orbits, the linear approximations of the corresponding manifolds and a point in a given Poincaré section where the unstable and stable manifolds match. The method is applied to compute families of heteroclinic orbits between planar Lyapunov periodic orbits around the collinear equilibrium points of the Restricted Three-Body Problem in different scenarios. In one of them, for the Sun-Jupiter mass parameter, we provide ranges of energy for which the transition between different resonances is possible.

The arches of chaos in the Solar System

“…The manifolds reported herein act over orbital time scales of several decades, not the tens of thousands to millions of orbital revolutions traditionally considered (13)(14)(15)(16)(17)(18)33) in treating Solar System dynamics. More detailed quantitative studies of the discovered phase-space structures, as revealed by homoclinic-heteroclinic connections and their association with mean-motion resonances (1)(2)(3)11), could provide deeper insight into the transport between the two belts of minor bodies and the terrestrial planet region. Combining observations, theory, and simulation will improve our current understanding of this short-term mechanism acting on the TNO, Centaur, comet, and asteroid populations and merge this knowledge with the traditional picture of the long-term chaotic diffusion through orbital resonances; a formidable task for the large range of energies considered.…”

“…Stable manifolds, despite the epithet, can nevertheless lead to chaotic motion as a result of their complicated interaction with the corresponding unstable manifolds (see the Supplementary Materials). The analytical construction of these manifolds is highly complex, even in a much simpler dynamical models, and is usually described only locally for the very narrow band of three-body energies close to the collinear Lagrange points (2,3,(9)(10)(11)27). Our maps cover a much wider range of energies than previously considered, which encompass the observational datasets of JFCs and Centaurs.…”

“…Chaos in the Solar System is inextricably linked with the existence of stable and unstable manifolds, intricate structures whose mutual intersection plays a crucial role in chaotic transportation (1)(2)(3). The general properties are best described by considering the planar, circular, and restricted three-body problem (PCR3BP) (4), the simplest dynamical model that approximates the motion of both natural and artificial celestial bodies.…”

Transit regions and ejection/collision orbits in the RTBP

“…We describe how to compute a heteroclinic connection P 1 − LP O 1 , (we proceed similarly for the other case, P 2 − LP O 1 ). Since a heteroclinic connection P 1 − LP O 1 is an orbit that belongs to W e (P 1 ) ∩ W s (LP O 1 ), the numerical strategy to compute such connections, given µ > 0 and for a fixed value of C, consists, roughly speaking, in the following steps: to compute the 2D manifold W s (LP O 1 ) (one of the two possible branches) up to a given Poincaré section Σ, giving rise to a curve denoted by γ s (see for example [3] for more details and references therein); to compute the 2D manifold n and H 2 n , ejecting from P 1 and tending to the LP O 1 . We remark that for n = 1 figure right does not appear since there are no heteroclinic orbits in this case as shown on the middle plot.…”

Ejection-collision orbits in the restricted three-body problem

“…The traditional method to study the n-EC orbits would be method I. The study and computation of heteroclinic connections by means of the intersection of the stable and unstable manifolds is a technique that has been widely used (see for example [BMO13] and references therein).…”