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In this chapter we study the limit behaviour of families of periodic orbits in Hamiltonian systems. We first introduce basic notions for hypersurfaces in Hamiltonian systems, and then we show that the limit set of a smooth family of parametrised periodic orbits on a homotopy of stable regular energy surfaces is nonempty, compact and connected. In particular, there do not occur blue sky catastrophes in this setup. Finally, we discuss implications for the analysis of noncontractible periodic orbits on lens spaces.

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The Limit Set of a Family of Periodic Orbits

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  • 发表日期：

    2026年

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  • Yannis Bähni

摘要

In this chapter we study the limit behaviour of families of periodic orbits in Hamiltonian systems. We first introduce basic notions for hypersurfaces in Hamiltonian systems, and then we show that the limit set of a smooth family of parametrised periodic orbits on a homotopy of stable regular energy surfaces is nonempty, compact and connected. In particular, there do not occur blue sky catastrophes in this setup. Finally, we discuss implications for the analysis of noncontractible periodic orbits on lens spaces.