Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

Abstract

We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from L2 (the collinear libration point between the primaries) up to half the distance to the smaller primary in the Jupiter-Sun planar circular restricted three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

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

“…Rigorous numerical investigation of periodic orbits to ODEs became quite standard [2,9,11,24,25,41,45,48,49,50]. To the best of our knowledge, there are very few results regarding computer-assisted verification of bifurcations of periodic orbits of ODEs, and the field remains widely open.…”

On the construction of Lyapunov functions with computer assistance

“…Now we refer to the concept of cones in, e.g., [1,18]. In these references a cone is defined as a quadratic form Q(z) = Q 1 (z) − Q 2 (z) such that Q 1 and Q 2 are positive definite.…”

“…Although Algorithm 7.2 also makes use of the integral form of A I , we should be very careful of estimation criteria for obtaining rigorous enclosures of matrices. 1 A procedure such as Algorithm 7.2 works to prove the existence of Lyapunov functions as long as the above algorithm passes successfully, even if domains does not contain fixed points. An immediate benefit of Algorithm 7.2 as well as Corollary 5 is that we can use better enclosure of matrices associated with B in (4.4).…”

“…[2,9] There are several preceding works for validating functionals like Lyapunov functions within explicit domains (e.g. [1,5,7]), many of which apply functionals called cones satisfying cone conditions (e.g. [18]) to understanding asymptotic behavior around invariant sets .…”

Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem

“…The method has also been adapted by Zgliczyński, Simó and Capiński for proofs of normally hyperbolic invariant manifolds [6,8,9]. The above methods have been used and applied to a number of systems including the restricted three body problem [7,11,26,27] rotating Hénon map [6,9], driven logistic map [8], forced damped pendulum [29], and proofs of slow manifolds [19]. All these results rely on suitable definitions of covering relations and cone conditions.…”