  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.

Dynamical Systems;Analysis of PDEs