Degenerate response tori in Hamiltonian systems with higher zero-average perturbation

Abstract

Consider a normally degenerate Hamiltonian system with the following Hamiltonian [Formula: see text] which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where [Formula: see text] and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [Formula: see text] satisfies some non-zero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on [Formula: see text]. However, when [Formula: see text], no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2 p < n, if P satisfies [Formula: see text] for j = 1, 2, …, p and either [Formula: see text] as n − p is even or [Formula: see text] as n − p is odd, we obtain the following results: (1) For [Formula: see text] [see [Formula: see text] in ( 2.1 )] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjuno-type non-resonant condition. (2) For [Formula: see text] and [Formula: see text] sufficiently small, there exists a Cantor set [Formula: see text] with almost full Lebesgue measure such that response tori exist for each [Formula: see text] if ω satisfies a Diophantine condition. In the case where [Formula: see text] and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)].