【作者】 刘威；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机应用技术， 2005， 硕士

【摘要】 非线性理论包含三个非常重要的概念:分形、混沌和孤子,同这三个概念相对应的理论共同构成了非线性这门学科的理论基础。本文讨论了分形学中具有重要的意义的牛顿(Newton)迭代M-J集的建模与表示、基于轨道技术的伪3D分形渲染,以及非线性理论中具有实用性的多混沌伪随机序列密码发生算法。 牛顿迭代是求解方程中一种非常重要的方法,它不仅仅在实空间和复空间都适用,而且在单根的附近还能达到较高的收敛速度。但是在多重根附近,牛顿迭代会退化到接近线性状态。在实际的应用过程中人们往往根据需要对该方法采取一些变形。文中对牛顿迭代及其相关方法的分形几何集进行构建,将动态参数和异步用于牛顿迭代,构造并研究了广义牛顿变换的J集。通过分形几何确立了误差影响的关系,动态参数对迭代的影响,发现三阶简化牛顿法的了集中含有经典M集结构,重根吸引域的敏感依赖性,松弛牛顿变换的了集中不存在单根吸引域,并由J集的对称性确立了变参对迭代的影响。对于在这一领域的深入研究起着一定的推动作用。这些成果被《Applied Mathematics and Computation》、《数学研究与评论》和《工程图学学报》录用。 接下来将轨道技术渲染技术与牛顿法结合起来,推广了Pickover和Carlson的陷阱技术,按轨道形态进行分类,构造了更具有广泛意义多项式的牛顿变换伪3D M集。得到了M集中都存在着由“坏点”组成的经典M集,M-J集中存在具有伪3D效果且与对应陷阱单元形状相近的大小不同的彩色元素并具有自相似特征等结果。此部分成果发表在《计算机辅助设计与图形学学报》上。 传统的单混沌动力学系统可能出现性能退化缺陷,多混沌已经成为引人注目的方法。本文结合现有的多混沌系统提出了开关混沌系统伪随机比特发生器来产生密钥流。利用这种方法可以获得更好的随机性,使系统更加安全,并且在硬件允许的环境下进行并行计算可以极大地提高加密速度。更多还原

【Abstract】 Non-linear theory contains three important concepts: Fractal, Chaos and Soliton. The theory of these concepts makes up of theoretical basis of non-line. This thesis discussed some important problems, modeling and showing M-J Set of Newton’s method, utilizing trap technique to construct virtual three-dimension Newton transform Mandelbrot-Julia sets, and designing the practical algorithm of switch control chaos pseudo random bit generator.Newton’s method for finding a real or complex root of function is very efficient near a simple root because the algorithm converges quadratically in the neighborhood of such a root. However, at a multiple root, it only converges linearly. Various modifications of Newton’s method have been proposed in practice. In this paper create and study the generalized Newton transform J-Set, dynamic parameter and asynchronous Newton’s method, analyze the effect of error and dynamic parameter. The Julia set of third-order simplified Newton’ method contains the structure of classical Mandelbrot set. The multiple root attraction regions of these kinds of Julia sets are sensitive. There is not simple root attraction region in relax methodIn the following, through combining the orbit render and Newton’s method, extend the Pickover and Calson trap technique, construct and study more generalized polynomial’s 3D M-Set, generalizes orbit render technique. It always is found that there is standard Mandelbrot structure in the 3D M-set, which is formed by "bad" points. In M-J set, there is various 3D self-similarity color cell that corresponds with the shape of trap unit. All of results of fractal above-mentioned have been published or accepted by the journal.The traditional single may exist potential insecurity caused by intelligent methods to extract useful information from chaotic orbits. This paper introduces a method that generate pseudo random key stream by switch multiple chaoses. By this way, the randomicity and system security will be better than before. And if use parallel compute hardware, the speed of encrypt should be improved sharply.更多还原