Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems
Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems
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DOI：
10.1007/s00332-001-0400-1
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发表时间：
2002-03-01
影响因子：
3
通讯作者：
Marquardt, W
中科院分区：
文献类型：
--
作者：
Mönnigmann, M;Marquardt, W
参数化常微分方程组x = f（x，alpha），x <$R-n，alpha <$R-m的平衡解可以用它们到临界解流形的参数距离来表征，在临界解流形处系统的行为发生定性变化。关键点是分叉点和违反状态变量约束或输出约束的点。我们使用临界点流形上的法向量来度量这些流形与平衡解之间的距离，如I。多布森[J.非线性科学，3：307-327，19931，其中提出了计算余维1分叉上的法向量的方程组。本文提出了一种方法来导出计算临界点流形上法向量的方程组，它（i）推广到任意余维的分支，（ii）可应用于状态变量约束和输出约束，（iii）意味着定义方程组的法向量的大小为c（1）n + c（2）m + c（3），c（i）≠ R，即，没有双线性项nm或高阶项的出现，（iv）减少了Hopf分支流形上的法向量方程的数量相比，以前的工作，和（v）简化了法向量系统的正则性证明。作为该方案的应用，我们提出了系统的方程的正常向量流形的输出/状态变量的约束，流形的鞍结，霍普夫，尖点，和孤立分支，我们给出了说明性的例子，他们在工程应用中的使用。
Equilibrium solutions of systems of parameterized ordinary differential equations x = f(x, alpha), x epsilon R-n, alpha epsilon R-m can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 19931, where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that he normal vector defining system of equations is of size c(1)n + c(2)m + c(3), c(i) epsilon R, i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.