Advanced Mathematics from an Elementary Viewpoint: Chaos, Fractal Geometry, and Nonlinear Systems

大多数自然发生的过程本质上是非线性的，并且可以产生非常复杂的行为。即使是非常简单的数学模型也可以表现出非常复杂（通常非常漂亮）的分形形状。这一全新的数学领域的发现，在很大程度上依赖于计算密集型的图形方法，并催生了一个全新的数学研究范式：实验数学。虽然这些材料是目前研究的前沿，但高中生可以使用，通过调查产生的图形图像信息量很大，通常具有不寻常的视觉丰富性和美感。数学材料和相关的计算机软件，旨在激励和授权高中学生的平均数学能力进行实验数学调查将被设计，开发和评估。数学内容将包括分形，非线性动力学和数学混沌。基本的数学概念将被应用到各种各样的物理，生物和社会过程（如人口增长，流行病学问题，军备竞赛的经济学）。几何和非线性动力学之间的深刻联系将通过开发基于计算机的工具，使学生能够生成分形地图和引人注目的复杂性和美丽的图片明确。最后，通过他们在实验数学的工作，学生将获得数学和科学思维的更深入的理解。

Most naturally occurring processes are inherently nonlinear and can give rise to very complex behaviors. Even very simple mathematical models can exhibit behavior gives rise to extremely convoluted (and often very beautiful) fractal shapes. The discovery of this fundamentally new area of mathematics has been crucially dependent on computationally intensive graphic methods and has given birth to a radically new paradigm for mathematical research: experimental mathematics. Though this material is at the frontier of current research, it is accessible to high school students, and the graphic images that are generated through its investigation are highly informative and often of uncommon visual richness and beauty. Mathematical materials and associated computer software aimed at motivating and empowering high school students of average mathematical ability to perform experimental mathematical investigations will be designed, developed, and evaluated. The mathematical content will comprise fractals, nonlinear dynamics, and mathematical chaos. Fundamental mathematical concepts will be applied to a wide variety of physical, biological and social processes (e.g. population growth, problems in epidemiology, and the economics of the arms race). The deep connection between geometry and nonlinear dynamics will be made explicit by developing computer-based tools that will enable students to generate fractal maps and pictures of compelling complexity and beauty. Finally, through their work in experimental mathematics students will acquire a deeper understanding of mathematical and scientific thinking.