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。本文引入离散动力系统的循环维数，估计了离散准周期轨道的上、下循环维数，分析了不同类型偏微分方程拟周期解的复杂性。我们还提出了上下循环维之间的间隙作为衡量轨道不可预测性水平的指标参数。我们证明了当无理数频率是弱刘维尔数，且具有足够大阶的有理数近似良度时，拟周期轨道的循环维间隙取正值。这些结果由首席研究员在国际会议NACA2003（[1],[2]）上宣布，并将发表在Discr上。孔蒂。Dyn. Systems （[3]）计算吸引子的维度是为了衡量它们的复杂性和随机性。在b[5]，[6]，[7]中，合作研究者Y. Oshima用概率论证明了随机性的一些相关结果。另一方面，在[8]-[11]中，共同研究者M. Misawa在P.D.E上已经展示了各种基本的结果，这些结果将对研究非线性动力学模型的混沌行为起重要的作用。

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.