Aubry-Mather Sets Of Duffing-type Equations

Duffing-type equation is one of important models in nonlinear Hamilton systen, its dynamical behaviour has been widely investigated in the literature because of its significance for the applications as well as for its mathematical fascination. In this thesis, we consider the existence of the Aubry-Mather sets for Duffing-type equation.In section 1, we first introduce a new coordinate transformation which is similar to polor coordinate transformation. With careful estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property , then based on the Aubry-Mather theorem generalized by Pei, we get the existence of the Aubry-Mather sets for a class of semilinear Duffing equations.In section 2, we extend the coordinate transformation in section 1 which is suitable to sublinear Duffing-type equation. With careful estimates after action-angle variable transformation, we prove that the Poincare map is a monotone map. then applying the Aubry-Mather theorem generalized by Pei, we obtain the existence of the Aubry-Mather sets for a class of sublinear Duffing-type equations.In section 3, at the beginning, we introduce a new coordinate transformation which changes the semilinear Duffing-type equations with bouncing from right half plane into the whole plane except the origin. With some delicate estimates after action-angle variable transformation, we show that the Poincare map satisfies the monotone twist property. Then we obtain the existence of the Aubry-Mather sets of semilinear Duffing equation with bouncing via the the Aubry-Mather theorem generalized generalized by Qian.