Bifurcations of symmetric periodic orbits via Floer homology
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Calculus of Variations
Bifurcations of symmetric periodic orbits via Floer homology Joontae Kim1 · Seongchan Kim2 · Myeonggi Kwon3 Received: 16 September 2019 / Accepted: 31 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct circular orbits in the rotating Kepler problem and observe bifurcations of torus-type orbits. Our setup is motivated by numerical work of Hénon on Hill’s lunar problem. Keywords Bifurcation · Families of symmetric periodic orbits · Reversible Hamiltonian systems Mathematics Subject Classification Primary 37J20 · Secondary 53D40
Contents 1 Introduction . . . . . . . . . . . . . . . . 2 Statement of the results . . . . . . . . . . 2.1 Setup . . . . . . . . . . . . . . . . . . 2.2 Bifurcations . . . . . . . . . . . . . . 2.3 Local eLRFH . . . . . . . . . . . . . 2.4 Outline of the proofs of the theorems . 2.5 Example: the rotating Kepler problem
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Communicated by P. Rabinowitz.
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Seongchan Kim [email protected] Joontae Kim [email protected] Myeonggi Kwon [email protected]
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School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
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Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland
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Fakultät Für Mathematik, Ruhr-Universität Bochum, IB 3/85, Universitätsstrasse 150, 44801 Bochum, Germany 0123456789().: V,-vol
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3 Rabinowitz Floer theory . . . . . . . . . . . . . . . . . . . . . . . 3.1 Equivariant Lagrangian Rabinowitz Floer homology . . . . . . 3.1.1 The case of cotangent bundles . . . . . . . . . . . . . . 3.2 Local non-equivariant Lagrangian Rabinowitz Floer homology 3.3 Local equivariant Lagrangian Rabinowitz Floer homology . . . 4 Proof of Theorems 2.1, 2.6, and 2.7 and of Corollary 2.4 . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J. Kim et al. . . . . . . .
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1 Introduction The study of periodic orbits gives crucial insight into dynamical systems as Poincaré regarded them as the “skeleton” of dynamical sys
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