【作者】 石其江；

【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】 非线性理论由三大理论构成:混沌理论、分形理论、孤立子理论,它们是非线性这门学科的理论基础。基于非线性理论,本文研究了混沌、分形领域中的若干问题及其在信息安全等领域的应用,具体研究内容如下: 推广了Baker、Devaney和Romera等人的工作,并构造出一系列复指数映射的广义Mandelbrot-Julia集,给出了复数阶的广义Julia集发生突变的理论依据,从理论上分析了广义Mandelbrot-Julia集的对称性和周期性。 研究了二维Logstic映射不动点的性质,给出了在参数空间中二维Logistic映射发生第一次分岔的边界方程。采用相图、分岔图、功率谱、Lyapunov指数计算和分维数计算方法,描述了二维Logistic映射从规则运动转化到混沌运动所具有的普适特征。 给出了二种数字水印算法。一种是基于图像特征和超混沌迭代的图像认证算法,另一种是结合迭代函数系(Iterated Function System,IFS)理论,利用IFS吸引子抗几何失真的特性,实现了非对称数字水印算法。研究结果表明:二种算法对噪声、滤波、压缩、旋转等图像处理方法具有较好的鲁棒性。 基于离散线性系统的稳定性理论,对Rssler混沌系统进行控制,使之追踪连续参考信号,并证明该控制系统指数收敛到参考信号,数值仿真进一步表明该方法的有效性。 推广了Peitgen、Pickover和Carlson的方法,构造了一系列拟3D广义Mandelbrot-Julia集,研究了拟3D广义Mandelbrot-Julia集的结构拓扑不变性和裂变演化规律。 以上的部分研究工作已发表在《自然科学进展》、《计算机研究与发展》和《应用力学学报》等刊物上。更多还原

【Abstract】 Non-linear theory contains three important parts: Fractal, Chaos and Soliton theory. They are theoretical foundation of non-linear theory.Based on non-linear theoretical analysis, the research has studied some chaotic and fractal problems and application in information security as follows:The paper develops Baker, Devaney and Romera’s studies, constructs a series of generalized Mandelbrot-Julia sets, presents the theoretic proof in the exploration of the generalized J set for the complex exponent, theoretically analyzes the nature of symmetry and period of the generalized Mandelbrot-Julia sets.The nature of the fixed points of the coupled Logistic map is studied analytically, and the boundary equation of the first bifurcation of the map in the parameter space is derived. By phase plot, bifurcation plot, power spectra, Lyapunov exponent and fractal dimension, the paper reveals the general features of coupled Logistic map transforming from regularity to chaos.In this paper two kinds of watermarking algorithm is proposed.One is an image authentication algorithm based on feature of original image and hyperchaotic iteration, the other is a asymmetric digital watermark scheme using the robust character against geometric distortion of IFS. The experimental results show they are robust to noise adding, filtering, compression, rotation.Based on the theory of stability of the linear system, a strategy of the Rossler’s chaotic system tracking control is presented. This strategy can track all kinds of reference signals. It is also proved that the strategy can make the system approach to any desired smooth orbit at an exponent rate. Numerical simulations have shown the proposed strategy is effective.This paper develops Peitgen, Pickover and Carlson’s technique. A series of pseudo-3D generalized Mandelbrot-Julia sets are constructed. The paper studies the structure topological inflexibility and the fission of evolution law for pseudo-3D generalized Mandelbrot-Julia sets.Some studies have been published in Progress in Natural Science, Journal of Computer Researchand Development and Chinese Journal of Applied Mechanics.更多还原