一类分段光滑不连续映射的动力学研究

发布时间：2018-03-17 01:23

本文选题：分段映射边界碰撞分岔　切入点：混沌边界　出处：《广西大学》2017年硕士论文　论文类型：学位论文

【摘要】：近年来,学者们对分段光滑映射进行了深入的分析和研究,这类映射被广泛的应用于多个科学领域,如电力电学、生物学、医学和经济学等领域.现在对分段光滑映射系统理论的研究还不完善,许多遗留的问题还要继续研究.本文在分析和总结分段光滑映射已有研究成果的基础上,对一类双侧增加的分段线性不连续映射的周期轨道,混沌边界、混沌控制及应用进行研究.具体结果如下:(1)研究一类左右两侧都增加的分段线性不连续映射的动力学行为.该模型可能应用到物理科学和工程方面,也有助于研究一些经济模型.通过调节系统的重要参数,借助理论分析和数值仿真发现映射存在周期数成等差数列增长的加周期现象,也存在混沌和发散现象.通过推导周期轨道的边界碰撞分岔参数曲线,确定了稳定周期轨道区域.又根据高复杂度水平周期轨道的边界碰撞分岔曲线,结合双参数分岔图解释了加周期和周期叠加现象.(2)针对左右两侧都增加的分段线性不连续映射动力系统,以重要参数为分岔参数得到系统的分岔图,发现在系统的不变吸引区间内,周期轨道的每个周期点都有一定的存在范围,使得分岔结构中出现迭代禁区现象.通过理论推导确定了周期轨道周期点的存在范围和禁区边界,进一步通过禁区边界得到了混沌区域与周期n轨道区域边界的解析表达式,应用Lyapunov指数对分析结果进行了验证.(3)主要基于分段不连续映射产生的混沌达到两个目标,第一个是混沌控制.首先,根据分段映射具有多分支的特点,对分段映射的双侧都添加线性控制器.然后,由分段不连续映射周期轨道稳定性判别方法确定控制器的参数.最后,仿真结果表明,双侧添加线性控制器的方法可以实现将混沌控制到任意周期2轨道上.对控制系统添加高斯白噪声后,发现该控制方法具有一定的抗噪声能力.第二个是利用混沌进行图像加密.基于双侧增加的线性不连续映射产生的混沌序列,设计了双混沌序列置乱,双混沌异或的图像加密算法.对实验结果进行安全性分析表明,该加密算法密钥空间大,灵敏度高,具有很高的安全性.
[Abstract]:In recent years, scholars have carried out in-depth analysis and research on piecewise smooth mapping, which has been widely used in many scientific fields, such as electric power, biology, electricity and electricity. Medicine and economics. The theory of piecewise smooth mapping system is not perfect, and many remaining problems still need to be studied. In this paper, based on the analysis and summary of the existing research results of piecewise smooth mapping, For the periodic orbits of a class of bilaterally increased piecewise linear discontinuous mappings, the chaotic boundary, Chaotic control and its application are studied. The results are as follows: 1) the dynamical behavior of a class of piecewise linear discontinuous mappings with increasing left and right sides is studied. The model may be applied to physical science and engineering. It is also helpful to study some economic models. By adjusting the important parameters of the system, by means of theoretical analysis and numerical simulation, it is found that the mapping exists the phenomenon of increasing the number of periods in the sequence of equal difference. Chaos and divergence also exist. By deducing the boundary collision bifurcation parameter curve of periodic orbit, the stable periodic orbit region is determined, and according to the boundary collision bifurcation curve of high complexity horizontal periodic orbit, Combined with the two-parameter bifurcation diagram, the phenomenon of adding period and periodic superposition is explained. The bifurcation diagram of the system is obtained by taking important parameters as bifurcation parameters for the dynamic system of piecewise linear discontinuous mapping, which is increased on both sides of the left and right. It is found that in the invariant attraction interval of the system, every periodic point of the periodic orbit has a certain range of existence, which makes the iterative exclusion zone appear in the bifurcation structure. The existence range and the forbidden zone boundary of the periodic point of the periodic orbit are determined by theoretical derivation. Furthermore, the analytic expressions of chaotic region and periodic n-orbit region are obtained through the restricted area boundary. The results are verified by using Lyapunov exponent. The results are mainly based on the chaos generated by piecewise discontinuous mapping to achieve two objectives. The first is chaos control. Firstly, linear controllers are added to both sides of the piecewise mapping according to the multi-branch characteristic of piecewise mapping. The parameters of the controller are determined by the method of judging the stability of periodic orbits of piecewise discontinuous mapping. Finally, the simulation results show that, The method of adding linear controller on both sides can control chaos to any period 2 orbit. After adding Gao Si white noise to the control system, It is found that this control method has a certain anti-noise capability. The second is to encrypt the image by using chaos. Based on the chaotic sequence generated by linear discontinuous mapping on both sides, a double chaotic sequence scrambling is designed. The security analysis of the experimental results shows that the encryption algorithm has the advantages of large key space, high sensitivity and high security.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP309.7;O231

【相似文献】