Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping

TL;DR

This paper extends Aubry Mather theory to time-periodic dissipative Hamiltonian systems, analyzing viscosity solutions of related Hamilton-Jacobi equations and their implications for understanding global dynamics in physical models.

Contribution

It develops a novel framework for parameterized viscosity solutions in dissipative Hamiltonian systems with time periodic damping, linking them to global dynamics analysis.

Findings

Asymptotic behavior of viscosity solutions analyzed.

Connections established between solutions and global dynamics.

Potential applications to physical models discussed.

Abstract

In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation \[ \left\{ \begin{aligned} \dot x&=\partial_p H(x,p,t),\\ \dot p&=-\partial_x H(x,p,t)-f(t)p \end{aligned} \right. \] with $(x,p,t)\in T^*M\times\mathbb T$ (compact manifold $M$ without boundary). We discuss the asymptotic behaviors of viscosity solutions of associated Hamilton-Jacobi equation \[ \partial_t u+f(t)u+H(x,\partial_x u,t)=0,\quad(x,t)\in M\times\mathbb T \] w.r.t. certain parameters, and analyze the meanings in controlling the global dynamics. We also discuss the prospect of applying our conclusions to many physical models.