泊松白噪声与宽带噪声共同激励下拟可积哈密顿系统动力学研究

In the past research of nonlinear stochastic dynamics, continuous and discrete excitations were considered separately. The combination of these two types of excitations was seldom considered. However, in nature、engineering and society, the coexistence of jump and continuous stochastic excitations is almost universal. Thus, further research the dynamical properties of nonlinear systems under the jump stochastic processes excitations together with continuous excitations has important theory significance and engineering application values. The dynamics of quasi integrable Hamiltonian systems under combined Poisson white and wide-band noises excitations are studied in the present project. Firstly, the stochastic averaging method for quasi integrable Hamiltonian systems under combined Poisson white and wide-band noises excitations are proposed, the averaged stochastic integral differential equations (SIDEs) of the original system, the corresponding FPK equations and the boundary conditions can be derived, the approximately stationary solution of system response can be obtained by solving above equations. Secondly, the stochastic stability of the original system can be studied by using the largest Lyapunov exponent method and Lyapunov function method based on the averaged SIDEs. Then, the stochastic bifurcation phenomenon of the system can be analyzed based on the above study. Finally, apply above methods to the typical nonlinear stochastic dynamical systems of science and engineering, and the theoretical results will be confirmed by comparing with those from numerical simulation. After three years of study, a dynamical theoretical method on the quasi integrable Hamiltonian systems under combined Poisson white and wide-band noises excitations will be established initially.

在以往非线性随机动力学研究中，往往只考虑连续的随机激励或只考虑离散的随机激励，很少将两种类型的激励联合考虑。而在自然界、工程及社会中却普遍存在跳跃与连续并存的随机激励，因此深入研究跳跃与连续共同激励下非线性系统的动力学具有重要的理论意义及工程应用价值。本项目拟研究泊松白噪声与宽带噪声共同激励下拟可积哈密顿系统的动力学。提出泊松白噪声与宽带噪声共同激励下拟可积哈密顿系统的随机平均法，推导系统的平均随机微分积分方程以及与之相应的FPK方程及其边界条件。求解上述方程得到系统响应的近似平稳解。基于所得平均随机微分积分方程，用最大Lyapunov指数与Lyapunov函数方法研究系统的随机稳定性。分析系统的随机分岔现象。将上述理论方法应用于科学与工程中的典型非线性系统，并用数字模拟验证上述理论方法的正确性。通过三年研究，初步建立起一套泊松白噪声与宽带噪声共同激励下拟可积哈密顿系统的动力学理论方法。

基于跳跃与连续共同激励的非线性动力系统在自然界，工程及社会中的广泛存在，本项目重点研究了泊松白噪声与宽带（谐和）噪声共同激励下非线性系统的动力学。首先对泊松白噪声与宽带（谐和）噪声共同激励下非线性系统的响应与分岔进行了研究。推导了振幅与相位差的平均随机微分方程及与之相应的广义FPK方程。然后给出其满足的边界条件，应用有限差分与超松弛迭代的方法求解该FPK方程，得出系统响应的近似解析解。最后将该方法应用于泊松白噪声与宽带（谐和）噪声共同激励下的Duffing振子，在主共振情形下，得出系统稳态响应的近似解析解，研究了随着系统参数（频率，泊松白噪声的激励强度）的变化引起的随机分岔现象。然后，基于前面发展的随机平均法，对泊松白噪声与宽带（谐和）噪声共同参激下Duffing振子及含有分数阶导数阻尼的Duffing振子的随机稳定性进行了研究。推导了系统的最大Lyapunov指数的解析表达式，给出了系统概率为1渐近稳定的条件。得出了最大Lyapunov指数随着泊松白噪声强度的增大而增大，随着分数阶导数的增大而减小，系统的稳定区域随着分数阶导数的增大而增大的结论。最后用Monte Carlo数值模拟验证了所提方法的有效性。本项目所得的成果为继续研究跳跃与连续共同激励的非线性系统的动力学奠定了坚实的工作基础，并为工程应用提供了理论依据。在项目资助下共完成11篇论文的写作。其中正式发表论文7篇，被SCI期刊收录2篇，被EI收录4篇，被核心期刊收录1篇；待发表论文4篇。培养硕士学位青年教师3名，项目组成员1人博士后出站。