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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.
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with m
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in cite{P} in a sharp way.
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the
M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid oscillations are supp
Quasi-periodic solutions with Liouvillean frequency of forced nonlinear Schrodinger equation are constructed. This is based on an infinite dimensional KAM theory for Liouvillean frequency.