Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping

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 $(x,p,t)in T^*Mtimesmathbb T$ (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 Mtimesmathbb 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.