Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Veronica Danesi, Cristian Mendico, Xuan Tao, Kaizhi Wang

TL;DR

The paper analyzes non-autonomous Tonelli Lagrangian systems $(L_1)$ that converge to a time-periodic limit $(\overline{L_1})$, and shows that a suitably defined Lax–Oleinik operator $(\mathcal{T}_t)$ converges to a time-periodic viscosity solution $w$ of the limit Hamilton–Jacobi equation with exponential rate. It proves that the adherences of the gradient graphs converge in the Hausdorff sense to the graph of $dw$, providing a geometric weak KAM interpretation for the time-periodic limit. The results are then applied to a Kuramoto-type model with time-dependent coupling, yielding the existence of a weak KAM invariant torus given by the gradient of the limiting periodic solution, and, under hyperbolicity, an explicit exponential convergence to this torus. Overall, the work extends weak KAM theory to non-autonomous, time-varying Lagrangians with periodic limits and connects the asymptotics to invariant geometric structures in oscillator networks.

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.

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

TL;DR

The paper analyzes non-autonomous Tonelli Lagrangian systems $(L_1)$ that converge to a time-periodic limit $(\overline{L_1})$, and shows that a suitably defined Lax–Oleinik operator $(\mathcal{T}_t)$ converges to a time-periodic viscosity solution $w$ of the limit Hamilton–Jacobi equation with exponential rate. It proves that the adherences of the gradient graphs converge in the Hausdorff sense to the graph of $dw$, providing a geometric weak KAM interpretation for the time-periodic limit. The results are then applied to a Kuramoto-type model with time-dependent coupling, yielding the existence of a weak KAM invariant torus given by the gradient of the limiting periodic solution, and, under hyperbolicity, an explicit exponential convergence to this torus. Overall, the work extends weak KAM theory to non-autonomous, time-varying Lagrangians with periodic limits and connects the asymptotics to invariant geometric structures in oscillator networks.

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.
Paper Structure (14 sections, 6 theorems, 114 equations)

This paper contains 14 sections, 6 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. On weak KAM theory
  3. Weak KAM theory for time-periodic Lagrangians
  4. More on the Lax-Oleinik semigroup
  5. Weak KAM solutions
  6. Remarks on Bernard1 and Bernard2
  7. Convergence of adherences in time-periodic case
  8. Preliminary lemmas
  9. Proof of \ref{['thm_1']}
  10. Proof of \ref{['thm_2_intro']}
  11. Convergence to periodic solutions
  12. Proof of ($i$) in \ref{['thm_2_intro']}
  13. Convergence of adherences
  14. Second-order coupled oscillators

Key Result

Lemma 3.2

Given $u\in C(M,\mathbb R)$, let $\bar{u}={\lim_{n\to+\infty}}U^u_n$. Then $\|d_x\bar{u}\|$ and $\|d_\tau\bar{u}\|$ are bounded. Moreover, $\|d_xU^u_n\|$ and $\|d_{\tau}U^u_n\|$ are bounded by a constant independent of $n\in\mathbf{N}\backslash\{0\}$.

Theorems & Definitions (8)

  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.3
  • Proposition 4.4