Parabolic invariant tori in quasi-periodically forced skew-product maps
Parabolic invariant tori in quasi-periodically forced skew-product maps
复制标题
DOI：
10.1016/j.jde.2020.12.032
复制
发表时间：
2021-03
影响因子：
2.4
通讯作者：
Xinyu Guan;Jianguo Si;Wen Si
中科院分区：
文献类型：
--
作者：
Xinyu Guan;Jianguo Si;Wen Si
研究了一类拟周期强迫解析斜积映射φ：Rn × Td → Rn × Td：φ（z θ）=（z+ ψ（z）+ h（z，θ）+ ψ f（z，θ）θ+ ω）的抛物不变环面的存在性，其中ψ：Rn → Rn是l次齐次函数，l≥ 2，h= O（|z| l+ 1）。（a）当n= 1，l为奇数且λ充分小时，若ω满足Brjuno-Rüssmann非共振条件，则存在抛物不变环面. (b)当n= 1，且ω足够小时，若满足下列条件之一，则抛物不变环面也存在：（i）一阶平均值非零，一阶非平均部分足够小，且强迫频率ω不需要任何算术条件。(ii)（iii）l= 2，一阶平均为零，一阶和二阶非平均部分足够小，ω满足Brjuno型弱非共振条件;（iv）l> 2，一阶平均值为零，二阶平均值不为零，一阶和二阶非平均部分都足够小，ω满足Brjuno型弱非共振条件. (c)当n> 1时，若一阶平均值在S_p ∈ c（D_p）∈ i R= S_p的范围内，一阶非平均部分足够小，且强迫频率ω不需要任何算术条件，则上述拟周期强迫斜积映射在S_p足够小时存在抛物不变环面.本文的主要方法是KAM理论和不动点定理，最后证明了它可以直接应用于简并谐振子的拟周期响应解的存在性问题。
We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ: R n× T d→ R n× T d: φ (z θ)=(z+ ϕ (z)+ h (z, θ)+ ϵ f (z, θ) θ+ ω), where ϕ: R n→ R n is a homogeneous function of degree l with l≥ 2 and h= O (| z| l+ 1). We obtain the following results:(a) For n= 1, l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant condition.(b) For n= 1, and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds:(i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition.(ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition;(iii) l= 2, first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition;(iv) l> 2, first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition.(c) In the case n> 1, if first order average belongs to the interior of the range of ϕ, S p e c (D ϕ)∩ i R=∅, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.