Analysis on the structure of quasi periodic attractors for nonlinear partial differential equations

近年来，人们对各个领域的复杂或混沌行为进行了大量的研究。在本研究中，我们引入了离散动力系统的递归维，并估计了离散拟周期轨道的上下递归维，以分析各类偏微分方程拟周期解的复杂性。我们还提出了上下递归维度之间的差距作为衡量轨道不可预测性程度的指标参数。我们证明了当无理频率为弱Liouville数时，拟周期轨道的递归维隙为正值，并且具有足够大的有理数逼近良阶。这些结果是由首席研究员在2003年NACA2003国际会议上宣布的([1]、[2])，并将在DISR上发表。康蒂。戴恩。系统([3])。计算吸引子的维度是为了测量它们的复杂性和随机性。在[5]、[6]、[7]中，合作者Y.Oshima利用概率论证明了随机性的一些相关结果。另一方面，合作者M.Misawa在[8]-[11]中给出了关于偏微分方程组的各种基本结果，这些结果将对研究非线性动力学模型的混沌行为起到至关重要的作用。

In recent years great efforts have been made to analyze complexity or chaotic behaviors in various fields. In this research we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits to analyze complexity of quasi-periodic solutions given by various types of partial differential equaitions. We also proposed the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. We show that the gaps of recurrent dimensions of quasi-periodic orbits take positive values when the irrational frequencies are weak Liouville numbers with sufficiently large orders of goodness levels of approximation by rational numbers. These results were announced by the head investigator in the international conference NACA2003 ([1], [2]) and will appear in Discr. Conti. Dyn. Systems ([3]).Calculating the dimensions of the attractors is to measure their level of complexity and randomness. In [5], [6], [7] the co-investigator Y. Oshima proved some related results for randomness, using probability theory. On the other hand, in [8] -[11] the co-investigator M. Misawa have shown various fundamental results on P.D.E., which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.