Maslov index for homoclinic orbits of Hamiltonian systems. (English) Zbl 1202.37081
Summary: A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper, we use a geometric approach to set up the Maslov index for homoclinic orbits of Hamiltonian systems. Also, a relative Morse index for homoclinic orbits is derived through Fredholm index theory. It will be shown that these two indices coincide.
MSC:
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 53D12 | Lagrangian submanifolds; Maslov index |
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