拟周期扰动系统退化时的共振运动
Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems
摘要：
考虑一维系统在存在拟周期扰动的情况下，在解析的背景下，研究了与扰动频率向量共振的拟周期解的存在性问题.我们假设未扰动系统是局部可积的、各向异性的，并且扰动的频率向量满足Bryuno条件。共振解的存在性与一个合适的函数的零点有关，称为Melnikov函数-通过与周期情况类比。我们证明，如果Melnikov函数有一个奇数阶的零点，并且在扰动参数的符号上有进一步的条件，那么至少存在一个共振解，它是未扰动解的延续.如果Melnikov函数恒为零，那么我们可以将扰动理论推到Melnikov函数的对应函数出现且不恒为零的阶数：如果这样的函数有奇数阶的零，并且满足适当的正性条件，则再次获得相同的持久性结果。如果系统是哈密尔顿的，那么这个过程可以无限地迭代，并且不需要任何正性条件：作为副产品，结果如下：至少一个共振拟周期解总是存在，而不需要对扰动的假设。这样的解可以被解释为一个（抛物线）低维环面。
英文摘要：
We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function - by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.
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