The Restricted Planar Circular 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries, which perform circular orbits coplanar with that of the massless body. In rotating coordinates, it can be modelled by a two degrees of freedom Hamiltonian system, which has five critical points called the Lagrange points. Among them, the point $ L_3 $ is a saddle-center which is collinear with the primaries and beyond the largest of the two. The papers [3,4] provide an asymptotic formula for the distance between the one dimensional stable and unstable manifolds of $ L_3 $ in a transverse section for small values of the mass ratio $ 0 < \mu\ll 1 $. This distance is exponentially small with respect to $ \mu $ and its first order depends on what is usually called a Stokes constant. The non-vanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. In this paper, we prove that the Stokes constant is non-zero. The proof is computer assisted.
Citation: |
Figure 5. A closeup of the crossing of the trajectory through the section $ \{\text{Re}\; U = 0\} $ projected onto $ U $ on the left (compare with Figure 4), and onto coordinates $ (\text{Re}\; U, \text{Re}\; Y $ on the right. In black we have the interval arithmetic bounds. In green, we have singled out the bounds on the trajectory for two disjoint time intervals, to demonstrate that it indeed does cross $ \{\text{Re}\; U = 0\} $. In blue we have a non-rigorous plot of the trajectory, which is added to the figure as a point of reference
[1] | I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems, Nonlinearity, 19 (2006), 1415-1445. doi: 10.1088/0951-7715/19/6/011. |
[2] | I. Baldomá, M. Capiński, M. Guardia and T. M. Seara, Breakdown of heteroclinic connections in the analytic Hopf-Zero singularity: Rigorous computation of the Stokes constant, Journal of Nonlinear Science, 33 (2023), 28. |
[3] | I. Baldomá, M. Giralt and M. Guardia, Breakdown of homoclinic orbits to $L_3$ in the RPC3BP (Ⅰ). Complex singularities and the inner equation, Adv. Math., 408 (2022), 108562, 64 pp. doi: 10.1016/j.aim.2022.108562. |
[4] | I. Baldomá, M. Giralt and M. Guardia, Breakdown of homoclinic orbits to $L_3$ in the RPC3BP (Ⅱ). An asymptotic formula, Adv. Math., 430 (2023), 109218, 72 pp. doi: 10.1016/j.aim.2023.109218. |
[5] | I. Baldomá, M. Giralt and M. Guardia, Coorbital homoclinic and chaotic dynamics in the Restricted 3-Body Problem, Preprint arXiv, (2023), 2312. |
[6] | I. Baldomá and T. M. Seara, The inner equation for generic analytic unfoldings of the Hopf-zero singularity, Discrete Continuous Dynamical Systems - B, 10 (2008), 323. |
[7] | J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Revised and corrected reprint of the 1983 original Appl. Math. Sci., 42 Springer-Verlag, New York, 1990. |
[8] | G. Gómez, A. Jorba, J. J. Masdemont and C. Simó. Dynamics And Mission Design Near Libration Points - Volume 4: Advanced Methods For Triangular Points, volume 5. World Scientific, 2001. |
[9] | X. Hou, J. Tang and L. Liu, Transfer to the collinear libration point L3 in the Sun–Earth+Moon system, Nasa Technical Report, 2007. |
[10] | À. Jorba and B. Nicolás, Transport and invariant manifolds near L3 in the Earth-Moon Bicircular model, Communications in Nonlinear Science and Numerical Simulation, 89 (2020), 105327. |
[11] | À. Jorba and B. Nicolás, Using invariant manifolds to capture an asteroid near the L3 point of the Earth-Moon Bicircular model, Communications in Nonlinear Science and Numerical Simulation, (2021), page 105948. |
[12] | T. Kapela, M. Mrozek, D. Wilczak and P. Zgliczyński, CAPD: : DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 101 (2021), Paper No. 105578. doi: 10.1016/j.cnsns.2020.105578. |
[13] | T. Kapela, D. Wilczak and P. Zgliczyński, Recent advances in a rigorous computation of Poincaré maps, Commun. Nonlinear Sci. Numer. Simul., 110 (2022), Paper No. 106366. doi: 10.1016/j.cnsns.2022.106366. |
[14] | V. F. Lazutkin, Splitting of Separatrices for the Chirikov Standard Map, Journal of Mathematical Sciences, 128 (2005), 2687-2705. doi: 10.1007/s10958-005-0219-7. |
[15] | K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, volume 90, 3$^{rd}$ edition, Springer, Cham, 2017. |
[16] | C. Simó, P. A. Sousa-Silva and M. O. Terra, Practical Stability Domains Near L4, 5 in the Restricted Three-Body Problem: Some Preliminary Facts, In Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics, (2013), 367-382. |
[17] | V. G. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, New York [etc.], 1967. |
[18] | M. Tantardini, E. Fantino, Y. Ren, P. Pergola, G. Gómez and J. J. Masdemont, Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem, Celestial Mechanics and Dynamical Astronomy, 108 (2010), 215-232. doi: 10.1007/s10569-010-9299-x. |
[19] | Wolfram Research, Inc, Mathematica, Version 13.3, Champaign, IL, 2023. |
Projection onto the
Projection onto the
The inner domain,
The bound on the domain on
A closeup of the crossing of the trajectory through the section