Multiplicity of non-symmetric periodic solutions for Hamiltonian delay systems

In 2020, Albers et al. [2] introduced a variational framework for Hamiltonian delay systems on closed symplectic manifolds in which the delay is confined to the nonlinear term. Their construction, however, is not applicable to the standard Hamiltonian delay systems on manifolds, or even in Euclidean space, given by x ′ ( t ) = J ∇ H ( x ( t − τ ) ) . We address this gap through three complementary contributions. First, we devise a new variational principle for the above system in Euclidean space, embedding the delay in the kinetic (quadratic) term and thereby bypassing the limitations of the aforementioned framework. Second, assuming the Hamiltonian is asymptotically linear at both the origin and infinity, we combine critical point theory with an S 1 -index theory to obtain lower bounds on the number of periodic solutions; a fully worked example illustrates the computation. Third, we propose a Gaussian process with a delay kernel as a numerical method for finding periodic solutions of Hamiltonian delay systems. Experiments show that a composite kernel combining periodic and delay components reliably reconstructs periodic orbits from sparse data.