有界噪声扰动下非线性动力系统的研究

【作者】 杨晓丽；

【导师】 徐伟；

【作者基本信息】 西北工业大学 ， 应用数学， 2005， 硕士

【摘要】 本论文研究了典型强非线性随机系统的响应、稳定性及谐和激励与有界噪声扰动下具有三势井的非线性系统的混沌运动。论文的主要内容如下: 第一章简要地介绍了弱非线性随机动力学及强非线性随机动力学的研究现状及目前存在的问题,给出论文中用到的基本知识和论文的主要内容。 第二章研究了窄带随机噪声参激下强非线性Van der Pol-Duffing系统的1/2亚谐共振响应。首先由MLP方法引入变换参数,然后用多尺度法推导了系统的振幅和相位的控制方程,求出了最大Lyapunov指数的解析表达式,分析了系统在1/2亚谐共振区的性态。数值模拟的结果表明MLP方法结合多尺度法研究窄带随机噪声参激下强非线性系统的响应、稳定性和分叉问题是有效的。 第三章研究了窄带随机噪声外激下强非线性Duffing-Rayleigh振子的主共振响应。先借助于参数变换技术引入小参数,然后用多尺度法推导了系统的振幅和相位的控制方程,并由摄动法和矩方程法得到了系统的稳态响应。利用Routh-Hurwitz准则得到了稳态解稳定的充要条件。数值模拟的结果表明运用参数变换法结合多尺度法研究窄带随机噪声外激下强非线性系统的响应、稳定性等问题的有效性。 第四章研究了具有同宿轨道、异宿轨道和三势井的Duffing振子在谐和激励与有界噪声扰动下的混沌运动。由随机Melnikov方法推导了系统存在混沌运动的必要条件及出现分形域边界的充分条件。结果表明:当Wiener过程的强度参数较大时,噪声增大了出现混沌的有界噪声的临界幅值,缩小了参数空间的混沌域,而且出现混沌的临界幅值随着噪声强度的增大而增大。数值计算系统的最大Lyapunov指数也得到基本一致的结论。进一步用Poincare截面分析了有界噪声对系统混沌运动的影响,结果表明Wiener过程的强度参数越大,混沌吸引子扩散的面积越大。 第五章给出了全文的工作总结及有待进一步展开的研究。更多还原

【Abstract】 In this dissertation, the response and stability in several typical strongly stochastic nonlinear systems perturbed by random narrow-band noise and the chaotic motion of a nonlinear system with triple well under harmonic and bounded noise excitations are investigated. The main contents of the dissertation are as follows:Chapter one reviews briefly the current developments and problems of weakly/strongly nonlinear stochastic dynamical system, and introduces the main contents of this dissertation, together with the preparations including the method of MLP, the method of multiple scales, the Routh-Hurwitz criterion and the models of bounded narrow-band noise.Chapter two investigates the 1/2 subharmonic resonant response of a strongly nonlinear Van der Pol-Duffing oscillator subject to parametric random narrow-band excitation. The technique of MLP method is used to introduce a new expansion parameter, and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response. The maximum Lyapunov exponent is obtained analytically and the dynamics near the resonant domain is analyzed. Numerical simulation is carried out to verify the analytical results and the excellent agreement between theoretical results and numerical ones can be found immediately. Thus the present method combining the MLP method with the multiple scales method is applicable to solve strongly nonlinear problems to parametric random narrow-band excitation.Chapter three studies the prinple resonant response of a strongly nonlinear Duffing-Rayleigh oscillator to additive random narrow-band excitation. Firstly, a new small expansion parameter is introduced by the parameter transformation technique. Then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response.The steady state mean square response is obtained by the moment method and perturbation method and its local stability is checked by Routh-Hurwitz criterion. Analytical results are verified by numerical simulations, which indicates that the present method combining the parameter transformation technique with the multiple scales method is adapt to solve strongly nonlinear problems to additive random narrow-band excitation.Chapter four investigates the chaotic behaviors of a Duffing oscillator with triple well possessing both homoclinic and heteroclinic orbits subject to harmonic and bounded noise excitations. From Melnikov theory, the semi-analytical criteria for the occurrence of transverse intersection on the surface of homoclinic and heteroclinic orbits are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that for larger noise intensity the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps of the system responses is also discussed which indicates the chaotic attractor is diffused as the noise intensity increases, and the larger the noise intensity results in the more diffused attractor.Chapter five concludes the work and innovation of this dissertation, and points out some aspects to be further studied on stochastic nonlinear system.更多还原