Analysis of dimensional and recursive properties for almost periodic solutions of nonlinear partial differential equations

近年来，在动力系统的研究中，对复杂性或混沌行为进行了大量的研究。本文研究了拟周期动力系统的分形维数和轨道的递推性质，并将抽象结果应用于非线性偏微分方程的概周期解或拟周期解。在[1]（of 11.REF.）中，我们利用由非合理频率的一些代数性质导出的参数估计离散准周期轨道的相关维数。另一方面，在[2]中，我们通过定义循环维数来研究准周期轨道的递推性质，并给出了相关维数与循环维数之间的不等式关系。为了估计这些维数，我们引入了一类新的无理数，即拟罗斯数，拟或弱刘维尔数，它们分别根据有理数近似的不良近似性质或（极）好的性质进行分类。此外，在[2]和[3]中，我们研究了具有准周期扰动的非线性偏微分方程的拟周期解，并估计了吸引子的这些维数。分形维数是最重要的，因为它们显示了复杂性、自相似性或随机性的水平。另一方面，众所周知，在各种混沌路径中，周期或准周期状态作为主要门户占据着重要地位。在接下来的论文（11.REF.）中，我们已经展示了各种基本结果，这些结果将对研究非线性动力学模型的混沌行为起重要和必不可少的作用。

In recent years great efforts have been made to analyze complexity or chaotic behaviors in the study of dynamical systems. In this research we investigate fractal dimensions and recursive properties of orbits for quasi-periodic dynamical systems and then, we apply the abstract results to almost or quasi-periodic solutions for nonlinear partial differential equations. In [1] (of 11.REF.) we estimate correlation dimensions of discrete quasi-periodic orbits by using the parameters derived from some algebraic properties of the irrational frequencies. On the other hand, in [2], we study recursive properties of the quasi-periodic orbits by defining recurrent dimensions and show inequality relations between the correlation dimensions and the recurrent dimensions. To estimate these dimensions we introduce new class of irrational numbers, quasi Roth numbers, quasi or weak Liouville numbers, which are classified according to badly approximable properties or (extremely) good properties for the rational approximations, respectively.Furthermore, in [2] and [3] we investigate quasi-periodic solutions of nonlinear partial differential equations with quasi periodic perturbations and estimate these dimensions of the attractors.Fractal dimensions are most essential in the sense that they show the level of complexity, or selfsimilarity or randomness. On the other hand, it is well known that periodic or almost periodic states occupy the important positions as main gateways in various routes to chaos. In the following papers (of 11.REF.) by the head and co-investigators we have shown various fundamental results, which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.