论文标题
具有任意非线性和频率的强烈耗散式迫切强制系统的响应解决方案
Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies
论文作者
论文摘要
我们考虑在存在耗散的情况下进行准周期系统，并研究响应溶液的存在，即具有与强迫项相同的频率矢量的准周期溶液。当耗散足够大并且涉及强迫的合适函数具有简单的零时，已知响应溶液在不假设频率向量上任何非共振条件的情况下存在。我们分析了非简单零的情况，为了解决小型除数问题，我们将自己局限于二维频率向量，以便使用持续分数的属性。我们表明，如果零的顺序是奇数（如果是奇数，则一般不存在响应解决方案），则响应解决方案仍然存在，前提是测量耗散的参数的倒数属于无限间隔的集合，该集合取决于频率矢量两个分量比率的收敛。间隔可能是不相交的，因此我们在带有“孔”的集合中获得了响应解决方案的存在。如果我们希望该集合连接，我们必须在频率上需要一些非共振条件：实际上，我们需要一个小于通常在小型分裂问题中考虑的Bryuno条件弱的条件。
We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors problem, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the frequency vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with "holes". If we want the set to be connected we have to require some non-resonance condition on the frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.
