Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.