Georgia Institute of Technology College of Sciences

My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems. It consists of four part. At first, I will introduce the results in integrable Hamiltonian systems since what we focus on is nearly-integrable Hamiltonian system. The second part is the definition of nearly-integrable Hamiltonian system and the classical KAM theorem. After then, I will introduce that what is Poincar\'e problem and some interesting results corresponding to this problem. The last part, which is also the main part, I will talk about the definition and the background of nearly-integrable Hamiltonian system, then the persistence of lower dimensional tori on resonant surface, which is our recent result. I will also simply introduce the Technical ingredients of our work.