An iterated graph construction and periodic orbits of Hamiltonian delay equations

TL;DR

This paper extends lower bounds on the number of periodic solutions from classical Hamiltonian systems to certain Hamiltonian delay equations using an iterated graph construction and Lagrangian Floer homology.

Contribution

It introduces a novel approach combining iterated graph construction and Floer homology to establish periodic orbit bounds for Hamiltonian delay equations.

Findings

Lower bounds on periodic solutions are valid for specific Hamiltonian delay equations.

The method adapts Floer homology techniques to delay differential equations.

Provides a new link between symplectic topology and delay equations.

Abstract

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.