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I will present computational methods for symplectic invariants and their applications to the restricted three-body problem. The talk will focus on two applications based on joint works with Otto van Koert and Dayung Koh. First, I will discuss the Birkhoff conjecture on the existence of a disk-like global surface of section. Using numerical continuation and rigorous numerics, we validate a smooth family of Hopf links formed by retrograde and direct periodic orbits, and compute their transverse Conley–Zehnder and mean indices. These results have global dynamical consequences, including existence of infinitely many periodic orbits. Second, I will study bifurcations of spatial periodic orbits, with emphasis on bifurcation types and invariance of local Floer homology. As a key example, I will describe a period-tripling bifurcation whose type changes under symmetry breaking in the perturbative regime of Hill’s lunar problem. These methods provide insight into bifurcation networks of polar orbits, including near-rectilinear halo orbits, relevant to mission design.
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