# 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.

## Key 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.