耗散非自治系统的伪同步解
Pseudo-synchronous solutions for dissipative non-autonomous systems
摘要：
在KAM理论的框架下，通过假设频率上的非共振条件，如标准丢番图条件或较温和的Bryuno条件，证明了拟可积系统中不变环面的持久性.当存在耗散时，大多数的拟周期解消失了，最多只有少数解能与周期吸引子一起存在。然而，为了证明拟周期解确实存在，通常假设频率仍然满足丢番图条件，并且系统的一些外部参数与它们适当地调谐。本文考虑一维环面上一类具有耗散的周期扰动系统，证明了：无论耗散如何小，如果扰动是关于角度的三角多项式，且未扰动频率满足有穷级非共振条件，则系统的稳定性取决于耗散的大小，则存在具有轻微扰动频率的拟周期解，只要扰动的大小足够小。如果一方面扰动的最大大小在三角多项式的次数上是不均匀的，另一方面允许所有的频率，但是不允许有许多频率，并且不存在由外部参数的调谐引起的限制。一个物理上相关的情况，其中的结果适用，是自旋轨道模型，它描述了旋转的卫星围绕自己的轴，而在开普勒轨道上围绕行星旋转，在这种情况下，耗散是考虑到通过麦克唐纳扭矩。
英文摘要：
In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameters of the system are suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation, and show that, however small the dissipation, if the perturbation is a trigonometric polynomial in the angles and the unperturbed frequencies satisfy a non-resonance condition of finite order, depending on the size of the dissipation, then a quasi-periodic solution exists with slightly perturbed frequencies provided the size of the perturbation is small enough. If on the one hand the maximal size of the perturbation is not uniform in the degree of the trigonometric polynomial, on the other hand all but finitely many frequencies are allowed and there is no restriction arising from the tuning of the external parameters. A physically relevant case, where the result applies, is the spin-orbit model, which describes the rotation of a satellite around its own axis, while revolving on a Keplerian orbit around a planet, in the case in which the dissipation is taken into account through the MacDonald torque.
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