相图在不变环面上的非线性微分方程组的渐近积分
Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori
摘要：
我们考虑一类自治系统$\dot x=f(x)$，其中$x \in {\bf R}^{2n}$，$f \in C^1({\bf R}^{2n})$，其相图是$n$二维{ \em中心}的笛卡尔积。我们也考虑这个系统的扰动，即$\dot x=f(x)+g(t,x)$，其中$g \in C^1({\bf R}\times{\bf R}^{2n})$和$g$是渐近小的，即$g\Rightarrow 0$作为$t\to +\infty$关于$x$一致。假设$g$的降低率为$t^{-p}$，其中$p>1$。在此条件下，我们证明了扰动系统有界解的存在性，并讨论了它们收敛于未扰动系统解的问题。这种收敛依赖于$p$。此外，我们表明，原来的未扰动系统可以减少到$\dot r=0$，$\dotθ=A(r)$的形式，并采取$r\in {\bf R}^m_{+}$，$θ\in {\bf T}^n$，其中${\bf T}^n$表示的$n$维环面，我们调查的更一般的情况下，系统的相图是叶状不变环面。我们注意到，可积哈密顿系统具有相同的性质。我们也给出了几个例子，表明我们的定理的条件不能得到改善。
英文摘要：
We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product of $n$ two-dimensional {\em centres}. We also consider perturbations of this system, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bf R}^{2n})$ and $g$ is asymptotically small, that is $g\Rightarrow 0$ as $t\to +\infty$ uniformly with respect to $x$. The rate of decrease of $g$ is assumed to be $t^{-p}$ where $p>1$. We prove under this conditions the existence of bounded solutions of the perturbed system and discuss their convergence to solutions of the unperturbed system. This convergence depends on $p$. Moreover, we show that the original unperturbed system may be reduced to the form $\dot r=0$, $\dotθ=A(r)$, and taking $r\in {\bf R}^m_{+}$, $θ\in {\bf T}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate the more general case of systems whose phase portrait is foliated on invariant tori. We notice that integrable Hamiltonian systems are of the same nature. We give also several examples, showing that the conditions of our theorems cannot be improved.
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