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.
For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of [ H(x, d_x u)=c(H) ] by a vanishing discount approach, where $c(H)$ is the Ma~ne critical value. We also dis
cuss the dynamical significance of such a special solution.
For the conformally symplectic system [ left{ begin{aligned} dot{q}&=H_p(q,p),quad(q,p)in T^*mathbb{T}^n dot p&=-H_q(q,p)-lambda p, quad lambda>0 end{aligned} right. ] with a positive definite Hamiltonian, we discuss the variational significance of i
nvariant Lagrangian graphs and explain how the presence of the KAM torus dominates the $C^1-$convergence speed of the Lax-Oleinik semigroup.
For strictly convex billiard maps of smooth boundaries, we get a Birkhoff normal form via a list of constructive generating functions. Based on this, we get an explicit formula for the beta function (locally), and explored the relation between the sp
ectral invariants of the billiard maps and the beta function.
For the discounted Hamilton-Jacobi equation,$$lambda u+H(x,d_x u)=0, x in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on the projected Aubry set. The smoothness of such subsolutions can be improved under additional hyperbolic
ity assumptions. As applications, we can use such subsolutions to identify the maximal global attractor of the associated conformally symplectic flow and to control the convergent speed of the Lax-Oleinik semigroups
The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem. A trajectory is
called {it oscillatory} if it goes arbitrarily faraway but returns infinitely many times to the same bounded region. We prove the existence of such type of trajectories provided the primaries evolve in suitable periodic orbits.
For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*Mtimesmathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_epsilon^-:Mrightarrowmathbb R$ be the viscosity soluti
on of the parametrized contact Hamilton-Jacobi equation [ H(x,partial_x u_epsilon^-(x),epsilon u_epsilon^-(x))=c(H) ] with $c(H)$ being the Ma~ne Critical Value. We prove that $u_epsilon^-$ converges uniformly, as $epsilonrightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation [ H(x,partial_x u_0^-(x),0)=c(H) ] which can be characterized as a minimal combination of associated Peierls barrier functions.
We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $cin H^1(mathbb T,mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(
x,t)$ parametrized by $c(sigma)in H^1(mathbb T,mathbb R)$ with $c(sigma)$ continuous and monotonic and [ partial_tu_c+H(x,partial_x u_c+c,t)=alpha(c),quad text{a.e. } (x,t)inmathbb T^2, ] such that sequence of weak KAM solutions ${u_c}_{cin H^1(mathbb T,mathbb R)}$ is $1/2-$Holder continuity of parameter $sigmain mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving [ left{ begin{aligned} &dot{x}(s)in text{co} Big[partial_pHBig(x(s),c+D^+u_cbig(x(s),s+tbig),s+tBig)Big], & &x(0)=x_0,quad (x_0,t)inmathbb T^2,& end{aligned} right. ] we evaluate it by a uniquely identified rotational number $omega(c)in H_1(mathbb T,mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set $mathcal U$ in phase space independent of fixed measure, where the set of initial points which lead to collision is $O(mu^frac{1}{20})$ dense as $murightarrow 0$.
For symmetrically analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of periodic $2,3$ caustics is that the deformation has to be an isometric transformation.
