脉冲系统Melnikov函数和最大Lyapunov指数的构建方法及其应用

本申请将采用理论分析和数值模拟结合的方法探索脉冲系统Melnikov 函数和最大Lyapunov 指数的构造方法，并用这些解析工具研究脉冲系统的全局动力学行为。1.利用正交多项式逼近，结合摄动法和光滑系统Melnikov 理论，通过衡量同宿轨稳定流形与不稳定流形之间距离，得到脉冲系统Melnikov 函数近似表达式。2. 构造不连续的切换映射，刻划脉冲所产生的不连续特性，利用Khasminskii 公式并结合Gram-Schmidt 标准正交化法则，给出一种计算脉冲系统最大Lyapunov 指数的方法。在上述基础上，考察脉冲信号作用下非线性系统全局动力学行为，结合相图、Poincaré截面图、分岔图等数值方法来验证理论结果的正确性。在上述问题解决后，可更进一步考虑有界噪声和白噪声作用下脉冲系统随机Melnikov 函数和最大Lyapunov 指数的计算方法。

Melnikov function and the largest Lyapunov exponent in impulsive system were studied using theory analysis and numerical simulation, and which would be used to study the dynamical action of impulsive system.1. The nonsmooth part in impulsve system would be approched using orthotropic polynomial, and Melnikov function would be deduced basing on perturbation method and Melnikov method in smooth system. 2. The nonsmooth character was depicted by constructing switch map in impulsive system, and one method about the largest Lyapunov exponent would be given using Khasminskii formula and Gram-Schmidt shandard orthogonalization principle. When all these analytic method in the studying of impulsive system were delivered, we would study global dynamics in nonlinear system with impulsive signals, analytic results would be proved by numerical results, such as phase diagram, Poincaré sectional view, branched graph. When the problem mentioned above were resolved, the construction of Melnikov function and the largest Lyapunov exponent in impulsive system with bounded noise and white noise would be took into account in the next step.