AI 中文总结
研究具有拟周期系数的可逆杜芬方程，通过有限正规形过程、对数傅里叶截断等方法，在\(n\geq2(m + 1)\)时构造余维一可逆KAM环面，证明所有解有界，得到拉格朗日稳定性和拟周期解的存在性。
AI 中文摘要
我们考虑方程\[ \ddot x+f(x,\omega t)\dot x+g(x,\omega t)=0, \]其中\[ f(x,\theta)=\sum_{j=0}^{m}a_j(\theta)x^{2j+1},\qquad g(x,\theta)=x^{2n+1}+\sum_{j=0}^{n-1}b_j(\theta)x^{2j+1}. \]系数函数在环面上实解析且为偶函数，频率向量\(\omega\)是丢番图的。若\(n\geq2(m + 1)\)，我们构造在无穷远处积累的余维一可逆KAM环面并证明所有解有界。主要方法是有限正规形过程，经可逆多项式约化后引入对数傅里叶截断，在第\(v\)步在非共振作用区间求解截断同调方程，新误差满足显式有限步递推，经有限步达到大作用的任意小负幂次，最后由可逆KAM定理证明拉格朗日稳定性和拟周期解的存在性。
英文摘要
We consider \[ \ddot x+f(x,ωt)\dot x+g(x,ωt)=0, \] where \[ f(x,θ)=\sum_{j=0}^{m}a_j(θ)x^{2j+1},\qquad g(x,θ)=x^{2n+1}+\sum_{j=0}^{n-1}b_j(θ)x^{2j+1}. \] The coefficient functions are real analytic and even on the torus and the frequency vector $ω$ is Diophantine. If $n\geq2(m+1)$, we construct codimension-one reversible KAM tori accumulating at infinity and prove that all solutions are bounded. The main point is a finite normal-form procedure. After the reversible polynomial reduction, a logarithmic Fourier cut-off is introduced. At the $v$-th step a truncated homological equation is solved on a non-resonant action interval and the new error satisfies an explicit finite-step recurrence. Thus an arbitrarily small negative power of the large action is reached after finitely many steps. Finally, the Largrangian stability and the existence of quasi-periodic solutions are proved by the reversible KAM theorem.
