Periodic limit for non-autonomous Lagrangian systems and applications to a Kuramoto type model

TL;DR

This paper studies the long-term behavior of non-autonomous Lagrangian systems with periodic limits, establishing convergence of solutions and applying results to prove the existence of invariant tori in a Kuramoto-type model.

Contribution

It introduces a method to analyze the asymptotic behavior of non-autonomous Lagrangian systems and applies it to demonstrate invariant tori in a Kuramoto model.

Findings

Convergence of the Lax-Oleinik semigroup to a periodic solution.

Graph of the gradient converges in Hausdorff distance.

Existence of an invariant torus in the Kuramoto model.

Abstract

This paper explores the asymptotic properties of non-autonomous Lagrangian systems, assuming that the associated Tonelli Lagrangian converges to a time-periodic function. Specifically, given a continuous initial condition, we provide a suitable construction of a Lax-Oleinik semigroup such that it converges toward a periodic solution of the equation. Moreover, the graph of its gradient converges as time tends to infinity to the graph of the gradient of the periodic limit function with respect to the Hausdorff distance. Finally, we apply this result to a Kuramoto-type model, proving the existence of an invariant torus given by the graph of the gradient of the limiting periodic solution of the Hamilton-Jacobi equation.