KAM theorem for degenerate generalized Hamiltonian systems with continuous parameters

Abstract

In this paper, we prove the persistence of invariant tori for degenerate generalized Hamiltonian systems with continuous parameters. We demonstrate that the persistent invariant tori retain the same frequency as the unperturbed tori under a certain transversality condition and a weak convexity condition for the frequency mapping. Generally speaking, at least Lipschitz continuity with respect to the parameter is needed in KAM-type results, while in this paper, we only require it to be continuous. Additionally, the system we consider is degenerate. Therefore, this paper can also be seen as an extension of KAM-type theory from non-degenerate generalized Hamiltonian systems to degenerate generalized Hamiltonian systems.