Hamiltonian delay equations -- examples and a lower bound for the number of periodic solutions

TL;DR

This paper introduces a variational approach to Hamiltonian delay equations on symplectically aspherical manifolds, establishing a lower bound on the number of periodic solutions that extends Arnold's conjecture to delayed systems.

Contribution

It develops a new variational framework for Hamiltonian delay equations and proves a lower bound on periodic solutions, extending classical results to systems with delay.

Findings

Lower bound matches the sum of Betti numbers of the manifold

Extension of Arnold's conjecture to Hamiltonian delay equations

Examples illustrating the variational approach

Abstract

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.