A Formal KAM Theorem for Hamiltonian Systems and Its Application to Hyperbolic Lower Dimensional Invariant Tori

In this paper we reformulate a formal KAM theorem for Hamiltonian systems with parameters under Bruno-Rüssmann condition. The proof is based on KAM iteration and the key is to adjust the parameters for small divisors after KAM iteration instead of in each KAM step. By this formal KAM theorem we can follow some well known KAM-type results for hyperbolic tori. Moreover, it can also be applied to the persistence of invariant tori with prescribed frequencies.