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| 摘要: |
| 经典LS(Liapunov-Schmidt)方法将奇异点的某个邻域内高维代数方程组平衡点的求解转化为低维代数方程组的求解问题,低维系统包含了原代数方程组奇异点附近特性的全部信息。改进LS方法,降维常微分方程组得到一组低维的常微分方程,它能够反映原系统在受扰后的动态变化过程。利用广义矩阵的理论求解LS降维过程所需的雅可比矩阵值域及其正交补空间的正交投影算子,将系统投影到2个空间上;再通过多元Taylor级数展开的方法将状态变量表示成为仅与雅可比矩阵零空间向量和参数有关的函数,将其代入投影到雅可比矩阵值域正交补空间的方程中,得到微分形式的约化方程;通过分析对比3节点系统与其降维系统在鞍结分岔点受扰后状态变量的变化,验证该方法能够准确地反映原系统在奇异点附近的特征信息,并且有效节省计算时间。 |
| 关键词: LS方法,常微分方程,正交投影算子,鞍结分岔 |
| DOI： |
| 分类号:TM744 |
| 基金项目:国家自然科学基金 |
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| Improve LS method to reduce differential equation dimensions |
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WANG Chao ZHANG Yao WU Zhigang
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| Abstract: |
| Classical LS(Liapunov-Schmidt) method transforms the equilibrium point computation of high-dimension equations at the neighborhood of singular point into that of reduced-dimension equations,which contains all information of the original equations. The classical LS method is modified to reduce the dimension of ordinary differential equations,which can dynamically reflect the variation of original system after disturbance. The generalized matrix theory is used to compute the orthogonal projection operators of Jacobian's range and its complementary space respectively and the original system is projected to the two spaces. The state variables are then simply expressed as the function of Jacobian's null space vectors and the parameters by Taylor expansion,which is substituted for the state variables in the differential equations of the orthogonal complementary space to reduce the dimensions. The change of state variables are compared between a 3-bus system and its reduced system disturbed at SNB(Saddle Node Bifurcation) point. Results show the accuracy and effective timesaving of this method. |
| Key words: Liapunov-Schmidt method,ordinary differential equation,orthogonal projection operator,SNB |
