Mar Giralt Miron defended her PhD thesis Homoclinic and chaotic phenomena to L3 in the Restricted 3-Body Problem on November 25th, 2022. The thesis was produced within the UPC doctoral program in Applied Mathematics and was supervised by Inma Baldomá Barraca and Marcel Guàrdia Munárriz. Currently she is a postdoc at the Università degli Studi di Milano.
Thesis summary
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom autonomous Hamiltonian system with five critical points, L1, . . . , L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest one.
In this thesis we study some of the homoclinic and chaotic phenomena occurring around L3 and its stable and unstable manifolds when the ratio between the masses of the primaries μ is small.
In Part I and II of the thesis, we obtain an asymptotic formula for the distance between the unstable and stable manifolds of L3, see [1, 2]. When μ is small the eigenvalues associated with L3 have different scales, with the modulus of the hyperbolic eigenvalues smaller than the elliptic ones. Due to this rapidly rotating dynamics, the invariant manifolds of L3 are exponentially close to each other with respect to √μ and, therefore, the classical perturbative techniques (i.e. the Poincaré-Melnikov method) cannot be applied. Then, one infers that there do not exist 1-round homoclinic orbits, i.e. homoclinic connections that approach the critical point only once.
In Part III of the thesis, we apply the asymptotic formula obtained in the previous parts to prove the existence of homoclinic and chaotic phenomena around L3 and its invariant manifolds, see [3]. Firstly, we study the existence of 2-round homoclinic orbits to L3, i.e. connections that approach the critical point twice. More concretely, we prove that there exist 2-round connections for a specific sequence of values of the mass ratio parameters. We also obtain an asymptotic expression for this sequence.
Moreover, we prove the existence of Lyapunov periodic orbits whose two dimensional unstable and stable manifolds intersect transversally. The family of Lyapunov periodic orbits of L3 has Hamiltonian energy level exponentially close to that of the critical point L3. Then, by the Smale-Birkhoff homoclinic theorem, this implies the existence of chaotic motions (Smale horseshoe) in a neighborhood exponentially close to L3 and its invariant manifolds.
In addition, we also prove the existence of a generic unfolding of a quadratic homoclinic tangency between the unstable and stable manifolds of a specific Lyapunov periodic orbit, also with Hamiltonian energy level exponentially close to that of L3.
Highlighted publication: [1]
References
[1] I. Baldomá, M. Giralt, and M. Guardia. Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation, Advances in Mathematics 408 (2022), Part A, 108562 (64 pages).
[2] I. Baldomá, M. Giralt, and M. Guardia. Breakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula, arXiv (preprint 2021, 66 pages).
[3] I. Baldomá, M. Giralt, and M. Guardia. Chaotic coorbital motions to L3 in the RPC3BP, In preparation (2023).