Steering Through Chaos and Instability: How Geometry Designs Unstable Trajectories in the Spatial 3-Body Problem

Abstract
Understanding global instability and how trajectories traverse phase space remains a classical challenge in Hamiltonian dynamics. This talk presents a general semi-analytical and geometric framework to construct and control globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center equilibrium. By utilizing a Birkhoff normal form near the NHIM, we gain precise control over both the hyperbolic and center components of the flow. We construct transition maps on the NHIM that encode the dynamics along transverse homoclinic and heteroclinic channels, proving that these geometric pathways can be efficiently shadowed by true trajectories of the full system. We illustrate the high efficiency of this methodology on a benchmark problem from celestial mechanics: the libration point L₁ of the Sun–Earth Spatial Restricted Three Body Problem (SR3BP). Combining the analytical normal form with a specialized multiple shooting algorithm, we show how complex, multi-iteration transition chains with massive initial discontinuities can be automatically corrected down to millimeter-level numerical precision, significantly outperforming standard astrodynamics correction tools.

Biography Amadeu Delshams is Professor Emeritus in the Department of Mathematics at the Universitat Politècnica de Catalunya (UPC) in Barcelona, where he is a founder of the Laboratory of Geometry and Dynamical Systems and a key member of the Institute of Mathematics of the UPC (IMTech). Professor Delshams has made substantial contributions to the field of dynamical systems, particularly in the study of Hamiltonian and reversible systems, effective stability, whiskered tori, hyperbolic invariant objects, splitting of separatrices, and Arnold diffusion. Over his career, he has led numerous research projects, mentored 13 Ph.D. students, organized major international mathematical meetings, and served as Editor-in-Chief for the ICIAM 2019 SEMA SIMAI Springer Series. He currently sits on the editorial boards of several leading journals, including the J. Differential Equations, J. Nonlinear Sci., Discrete Contin. Dyn. Syst., Qual. Theory Dyn. Syst. Systems, and Electron. Res. Arch..