Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

Abstract

We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamiltonian systems.In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from L 1,2,3 -Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.

Structure of the center manifold of the L1 and L2 collinear libration points in the restricted three-body problem

“…More delicate is the behaviour of the normal form in the case of L 3 , where, at low mass ratios, the optimal order seems to be so low to render very uncertain the estimate of the domain. However, we remark that the first-order prediction for the bifurcation of the halo made in Ceccaroni et al (2016) is in good agreement with recent accurate numerical estimates (Walawska & Wilczak 2019) also for µ → 0.…”

“…Historically the study started with the investigation of orbit families with particular emphasis on the 'halo' orbits, using at first pioneering numerical methods (Farquhar & Kamel 1973;Hénon 1973) and subsequently perturbation theory methods like the Poincaré-Lindstedt approach (Richardson 1980;Jorba & Masdemont 1999). Whereas the description of these orbits became satisfactory enough to be usefully exploited as a basis for space missions (Howell 1984;, a deep and general understanding of the general orbit structure has been obtained only quite recently with works devoted to the systematic construction of the invariant manifolds around the equilibria (Gómez & Mondelo 2001;Ceccaroni et al 2016;Delshams et al 2016;Guzzo & Lega 2018;Walawska & Wilczak 2019).…”

Recent advances in rigorous computation of Poincaré maps

“…for parameter c = 250. As shown in [29], this periodic orbit belongs to a family parameterized by c ∈ [ 9 8 , 100000]. The choice of c = 250 is motivated by the fact, that this orbit appears to be ill-conditioned for the method of minimization of the crossing time [t 1 , t 2 ] -see Section 3.4.…”

“…In each case u is a fixed point of P 2 (the orbit crosses section Π twice in opposite directions). Let matrix B be defined as in (29).…”

Torus knot choreographies in the $n$-body problem

“…The relevant literature is rich and we direct the interested reader to the works of [44,45,16,46,47] for a much more complete view of the literature. Other authors have studied center manifolds [48], transverse intersections of stable/unstable manifolds [31,49], Melnikov theory [50], Arnold diffusion and transport [51,52,53], and existence/continuation/bifurcation of Halo orbits [32,54] -all in gravitational n-body problems and all using computer assisted arguments. Especially relevant to the present work are the computer assisted existence and KAM stability proofs for n-body choreographies in [13,14,15,16].…”