【作者】 郭云松；

【导师】 甘春标；

【作者基本信息】 浙江大学 ， 一般力学与力学基础， 2005， 硕士

【摘要】 本论文主要讨论有界噪声激励对软弹簧杜芬振子的倍周期分岔至混沌道路的影响及对应的统计特性。利用动力学理论、随机振动理论及数值模拟方法等,讨论了此类系统的安全盆的侵蚀、运动的相图和倍周期分岔图等复杂动力学行为的变化过程,由系统响应的庞卡莱映射图与最大里亚普诺夫指数等模拟结果分析了噪声诱发混沌这一新动力学现象。基于本论文所讨论系统的强非线性与随机特性,作者应用Chen和Cheung等提出的方法,推导出了此系统的随机平均方程,并初步给出了一些关于此系统响应的平稳概率密度模拟结果。对此类系统的研究表明,外加的有界噪声激励的作用往往会掩盖原确定性系统内在的规则运动,对原确定性系统的动力学行为有典型的分散效应,并可延缓系统的分岔。此外,有界随机激励的作用还可使得系统的内在随机行为提前发生,即可使得系统更容易出现混沌运动。本论文安排如下: 第一章简要介绍混沌的定义、研究概况以及近期发展情况,并介绍了一些国内外学者对噪声诱发系统内在混沌的讨论结果;第二章给出本论文的研究模型——有界噪声激励下的软弹簧杜芬振子,详细讨论了有界噪声激励对此系统的安全盆的侵蚀的影响;第三章利用第二章关于系统安全盆侵蚀的模拟结果,选定对系统响应进行模拟时的初值点,通过蒙特-卡罗模拟方法,给出了系统参数变化时的分岔图、相图、时间历程等,并进一步从庞卡莱映射图以及最大里亚普诺夫指数等计算结果讨论了噪声诱发的混沌响应这一动力学行为;第四章利用随机振动理论,对系统进行了一些理论分析,推导了此类系统的随机平均方程与FPK方程,并初步给出了一些关于系统响应的平稳概率密度的模拟结果。在本论文的最后,作者给出了结论与展望。更多还原

【Abstract】 This thesis discussed the effects of the bounded noise excitation on the period-doubling bifurcation and chaotic responses of the softening Duffing oscillator, and the corresponding statistical characteristics were also analyzed. By the dynamical theory, the stochastic oscillation theory and the Monte-Carlo method, the author studied the complicated dynamical behavior of the softening Duffing oscillator with the bounded noise excitation. The noise-induced chaotic responses were studied based on the numerical results for the Poincare map and the maximum Lyapunov exponent. Due to the difficulty from the strongly nonlinear and stochastic nature of the system, the method from Chen and Cheung was employed to obtain the stochastic averaging equation and the FPK equation, from which the stationary probability density function was simulated. It was shown that, the regular motions in the original deterministic system would be masked by the external bounded noise excitation, and the stochastic excitation played a dispersive role to the dynamical behavior and could delay the bifurcation of the system. In addition, the bounded noise excitation could bring forward the internal chaos, e.g., chaotic responses would arise more readily. The thesis was arranged as follows:The first chapter briefly surveyed the studies on chaos, some discussions on the noise-induced chaos by others were also introduced. In Chapter Ⅱ, the softening Duffing oscillator under the bounded noise excitation was presented, the erosion of the safe basin of the system was discussed in detail. Based on the simulation results from Chapter Ⅱ, the initial point for simulating the dynamical behavior was chosen, and the bifurcation scenario, the phase portrait and the time histories, etc., were presented. The noise-induced chaos was testified by the Poincare map and the maximum Lyapunov exponent. These results were given in Chapter Ⅲ. In the fourth chapter, theoretical analysis was performed by using the stochastic oscillation theory, the stochastic averaging equation and the FPK equation were deduced and some simulation results about the stationary probability density function of such system were given. Chapter Ⅴ presented the summarization and discussion.更多还原