【作者】 李佼瑞；

【导师】 徐伟；

【作者基本信息】 西北工业大学 ， 应用数学， 2006， 博士

【摘要】 随机非线性动力系统的随机响应、随机稳定性、随机分岔、随机混沌、首次穿越及其最优控制问题，无论是理论上还是应用上都是值得研究和非常重要的问题。近些年，对于这些问题的研究，在理论上已取得一些初步的研究成果，但是对于一些较为复杂的典型动力学系统在不同随机激励下的动力学行为的研究仍处于起步阶段。另外，利用随机非线性动力学方程来研究经济、金融领域的问题还很不成熟。鉴于此，本文将主要研究两类典型的非线性动力系统—Mathieu-Duffing振子和Rayleigh振子在随机噪声的激励下的动力学行为及其一类系统在物价模型中的应用。第一部分较为系统的研究了随机Mathieu-Duffing系统的动力学行为。利用多尺度法研究了Gaussian白噪声激励下该系统的最大Lyapunov指数、零解和非零解的稳定性，研究结果表明：在参数主共振Ω=2ω₀附近邻域，最大Lyapunov指数λ在调谐参数σ=0时取得最大值，而随着σ的绝对值的增大，最大Lvapunov指数λ将趋于零。这说明，零解的稳定性随着调谐参数σ的绝对值的增大而趋于稳定；而对于系统的非零稳态响应，研究结果表明：非零解稳定的充要条件是-3βαa₀ cosη₀／2ω₀＞0。第二部分研究了有界噪声参激下带有Van der Pol阻尼项的Mathieu-Duffing系统的混沌运动。首先，利用随机Melnikov方法推导了随机Van der PolMathieu-Duffing方程的随机Melnikov过程，由广义过程在均方意义上出现简单零点给出了可能出现混沌运动的临界激励幅值，从而预测该系统可能存在混沌运动的参数域。然后，用数值方法计算该系统的最大Lyapunov指数、Poincare截面、时间历程图和相轨图，并用最大Lyapunov指数和Poincare截面验证解析结果及研究有界噪声的强度对该系统产生随机混沌的临界幅值的影响。结果表明：在ω₀：Ω₁：Ω₂=1：2：2和ω₀：Ω₁：Ω₂=1：1：2两种情形下，随着噪声强度的增大，随机系统的混沌吸引子不断扩散，随机激励的强度δ越大混沌吸引子的区域越大。说明混沌响应加强，即随机噪声会诱导混沌的产生。第三部分主要对确定性物价模型引入了噪声，建立了非线性随机动力学模型—随机Rayleigh方程，在此基础上利用动力学方法对该模型进行了系统研究。首先，对于非线性随机物价模型进行了最优化控制，通过对系统采用合理的控制策略，使得物价系统能在尽可能长的时间周期内处于平稳状态，避免出现通货膨胀或通货紧缩，这对于国家的经济良性发展也具有重要的意义。接着，对物价模型——Rayleigh方程在不同噪声激励下的稳态响应、首次穿越和随机Hopf分岔问题进行了进一步的讨论。具体的讲，研究了Rayleigh方程在Gaussian白噪声外激下的平稳响应首次穿越，利用随机平均法得出系统的随机平均It（?）微分方程，对平均方程建立条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程，通过这两个方程研究该系统的非线性阻尼项、随机激励项对系统平稳响应与首次穿越的影响；利用拟Hamilton系统随机平均法及扩散过程的奇异点理论研究了Gaussian白噪声参激下Rayleigh方程的随机Hopf分岔，并利用最大Lyapunov指数方法和数值方法对结论进行的验证。第四部分是全文的总结和展望。主要介绍文章的创新点和存在的不足，同时，提出在本研究工作的基础上，还有待进一步研究的问题以及所研究的方法和有可能碰到的困难。更多还原

【Abstract】 It is very important and worthwhile to study random responses, random stabilities,random bifurcations, first-passage failures and random optimal control of stochasticnonlinear dynamical system in theory and its applications. In recent years, there aremany preliminary research results in theory about stochastic nonlinear science.However, the results of some complicated classical dynamical systems are notcompletely, and some results can not agree with others. In addition, there is sufficientwork to do as how to applying stochastic nonlinear dynamical methods to solve realproblems of economics and finance. Considering that, in the present paper, the authorhas studied the behaviors of two classical nonlinear dynamical systems, which areMathieu-Duffing system and Rayleigh oscillator, under different kinds of stochasticnoise excitations. At the same time, the author has studied a price model by usingstochastic nonlinear dynamical equations.In the first section, the dynamical behaviors of stochastic Mathieu-Duffing systemhave been studied. Firstly, the author has studied the maximal Lyapunov exponentsand almost-sure stability of Mathieu-Duffing equation under Gaussian white noiseexcitation by multiple scales method. The results show that near the parametericresonance at excitation frequencyΩ=2ω₀, the system become more unstable with theincreasing of the maximal Lyapunov exponentλ, and the maximal Lyapunovexponents reache its maximum value whenσ=0. That is to say, the stability of trivialsolution of stochastic Mathieu-Duffing equation is more stable whileσtends tozero. But for the stability of nontrivial solution, the author has got the sufficient andnecessary condition is-3βαa₀ cosη₀/2ω₀＞0.In the second section, the author has studied the onset of chaotic motion of Van der Pol-Mathieu-Duffing system under bounded noise excitation. By using randomMelnikov technique, a mean square criterion is used to detect the necessary conditionfor chaotic motion of this stochastic system. Then, the Poincare map and Lyapunovexponents have been calculated by numerical method. The results show that when thenoise is presented in the system, the shapes of Poincare maps begin to diffuse intolarger area and the threshold of bounded noise amplitude for the onset of chaos in thissystem increases as the intensity of the bounder noise increases, which is furtherverified by the maximal Lyapunov exponents of the system, i.e.,whenω₀:Ω₁:Ω₂=1:2:2 orω₀:Ω₁:Ω₂=1:1:2, the chaotic motions of thesystem can be enhanced by the bound noise.In the third section, considering the government macro-control for price system asexternal control force, a dynamically controlled nonlinear stochastic price model isproposed based on deterministic model. By using stochastic averaging method andnonlinear stochastic control strategy, the stochastic model can been stabilized. Thecontrolling aim is to enable the system becoming more stable. Through comparingthe controlled Lyapunov exponents with uncontrolled Lyapunov exponents, it isshown that the controlled strategy is effective. This is very important to ourgovernment. Then, the author has not only studied the steady-state response andfirst-passage failure but also the applications in price modeling of Rayleigh oscillatorunder external and parametric excitations. Firstly, steady-state response and the firstpassage failure of Rayleigh oscillator under external Gaussian white noise excitationhave been studied. The equation of motion of the system is first reduced to averagedIto stochastic differential equations by using the stochastic averaging method. Abackward Kolmogorov equation governing the conditional reliability function and ageneralized Pontrayagin equation governing the conditional monments offirst-passage time are established. Statistical properties of the stationary response andfirst-passage failure are analyzed by using numerical results in three groups of varying parameters. Secondly, the stochastic Hopf bifurcation of Rayleigh oscillator subject toGaussian white noise parametric excitation is studied. The equation of motion of thesystem is reduced to an averaged It（?） stochastic differential equations by using thestochastic averaging method. Then, the critical D-bifurcation parameterc_D=-D/2μis determined approximately by using lyapunov exponents of theinvariable measureδ_c is trivial, and the critical P-bifurcation parameter valuec_P=0 is determined approximately by using the theory of singular points ofdiffusion processes. Finally, the procedure has been verified with the numericalsolution of the joint probability density of the stochastic system.In the last section, the author has introduced the creations and the problems thatwill be further studied.更多还原