My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems.

It consists of three parts. At first, I will introduce the KAM type results in nearly integrable Hamiltonian systems. In the second part, I will introduce the back ground and corresponding definitions of multi-scale nearly integrable Hamiltonian systems. After then, I will introduce that what is Poincare mechanics and certain interesting results corresponding to this problem. I will also simply introduce the technical ingredients of our work.The last part, I will talk about the applications of the persistence result to the nonlinear oscillator, which is our recent result. A joint with Prof. Y.Yi and Prof. Y.Li.