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

Abstract

Consider a normally degenerate Hamiltonian system with the following Hamiltonian H(θ,I,x,y,ϵ)=ω,I+λxn+1n+1+y22+ϵP(θ,x,y),(θ,I,x,y)∈Td×Rd+2, which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where 0≠λ∈R 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 [∂P(θ,0,0)∂x] 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 Td. However, when [∂P(θ,0,0)∂x]=0, 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 2p < n, if P satisfies [∂jP(θ,0,0)∂xj]=0 for j = 1, 2, …, p and either λ−1[∂p+1P(θ,0,0)∂xp+1]<0 as n − p is even or λ−1[∂p+1P(θ,0,0)∂xp+1]≠0 as n − p is odd, we obtain the following results: (1) For λ̃<0 [see λ̃ in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjuno-type non-resonant condition. (2) For λ̃>0 and ϵ* sufficiently small, there exists a Cantor set E∈(0,ϵ*) with almost full Lebesgue measure such that response tori exist for each ϵ∈E if ω satisfies a Diophantine condition. In the case where λ−1[∂p+1P(θ,0,0)∂xp+1]>0 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)].