Hamiltonian delay equations – examples and a lower bound for the number of periodic solutions. (English) Zbl 1451.34088
Summary: We describe a variational approach to a notion of Hamiltonian delay equations. Our delay Hamiltonians are of product form. We consider several examples. For closed symplectically aspherical symplectic manifolds \((M, \omega)\) we prove that for generic delay Hamiltonians the number of 1-periodic solutions of the Hamiltonian delay equation is at least the sum of the Betti numbers of \(M\), extending the proof of the Arnold conjecture to the case with delay.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 37J46 | Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems |
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