On Conley–Zehnder indices and bifurcation graphs in the spatial restricted three-body problem

• Applicant: Dr. Cengiz Aydin
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541062288

The goal of this project is to promote the interaction between modern symplectic geometry and highly involved engineering problems in the restricted three-body problem. To study the poles of planets or of moons, which are of interest, one needs spatial periodic orbits. One approach to finding spatial periodic orbits is to study spatial bifurcations from families of planar periodic orbits. This occurs when the spatial Floquet multipliers pass through a root of unity. Since these families can bifurcate again and meet each other, this procedure can get complicated. Therefore, a natural question to ponder is: ''Is there a well-organized global picture of the interconnectedness of such families?'' During my PhD I was able to take advantage of modern methods of symplectic geometry to address this question for the spatial Hill three-body problem, a limiting case of the restricted three-body problem. The novelty is to look at the Conley–Zehnder index and its interaction with bifurcation points. At bifurcation points the index jumps. Fortunately, when working locally near a family of non-degenerate periodic orbits, there is a fascinating bifurcation-invariant: the local Floer homology and thus its Euler characteristic. Since the index leads to a grading on local Floer homology, the index provides important information how different families are related to each other before and after bifurcation. The objective of this project is to extend my developed techniques and results from the Hill three-body problem to the restricted three-body problem, with a partical focus on the Earth–Moon system.

• DFG Programme: WBP Position