Numerical analysis of the chaotic bahavior of the free boundary

项目实施三年来，取得了许多成果。主要研究成果如下：1.本文的主要研究成果。分析混沌现象的数值方法的发展。德岛大学研究小组。提出了一种分析自由边界问题中混沌现象的方法，该方法基于数值方法，具有一定的通用性。此外，它还具有一个令人惊讶的特征，即无限维解空间中的吸引子可以任意逼近。这意味着该方法将理论分析和数值分析结合在一起，从而促进了理论研究和数值研究的发展。一个带吸引子的自由边界问题的建议。德岛大学集团。提出了一个带参数的一维自由边界问题。对于参数的特定值，它有精确的解，便于方法和程序的检验。在数值上找到了各种吸引子。模式分析。西浦解决了耗散系统的自复制动力学。他还揭示了描述两嵌段共聚物花纹形成的体系的极小值具有细观尺度的结构。Sakaguchi提出了一个通过细菌细胞群形成菌落模式的模型。数值模拟结果表明，尽管模型比较简单，但仍观察到了各种花样。快速数值计算及其应用的发展。德岛大学、池田和冈本等人在PVM提供的并行计算环境下提出了快速数值计算的方法。这些方法使自由边界问题或流体力学的大规模数值模拟成为可能。发现了一些新的现象。

In the term of the project (three years) many results are obtained. Important results are shown as follows.1. Development of numerical methods for analysis of chaotic phenomena.The group of Tokushima Univ. developed a method for analysis of chaotic phenomena in free boundary problems.It is based on numerical methods, so it is general. Moreover, it has a surprising feature that attractors in the infinite-dimensional solution space can be approximated arbitrarily. This means the method connects theoretical analysis and numerical analysis, therefore it enhances development of theoretical and numerical researches.2. A proposal of a free boundary problem with attractors.The group of Tokushima Univ. proposed a one-dimensional free boundary problem with some parameters. It has exact solutions for special values of the parameters, so it is covenient for the check of the method and programs. Various attractors are found numerically.3. Analysis of the pattern.Nishiura solved self-replicating dynamics of the dissipative system. He also revealed the minimizer of the system which describes the pattern formation of the diblock copolymer has the fine structure with the meso-scale. Sakaguchi proposed a model to formation of colony patterns by a bacterial cell population. From numerical simulation various patterns are observed in spite of the simplicity of the model.4. Development of fast numerical computation and its application.The group of Tokushima Univ., Ikeda and Okamoto developed methods for fast numerical computation in the environment of parallel computing offered by PVM. These methods enabled large-scaled numerical simulation of free boundary problems or fluid mechanics. Some new phenomena were found.