Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension
Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension
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DOI：
10.1007/978-3-642-56589-2_3
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
W. Beyn;Winfried Kleß;V. Thümmler
中科院分区：
文献类型：
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作者：
W. Beyn;Winfried Kleß;V. Thümmler
给出了一类参数化实大型稀疏阵的低维不变子空间的延拓方法。这种矩阵通常是在对动力学系统中的定态分支进行线性化时出现的，这些分支是通过时间相关的偏微分方程组的空间离散化获得的。主要感兴趣的是靠近虚轴的谱集的子空间。我们的连续过程提供了平稳地依赖于参数的不变子空间的基，只要连续的谱子集不与另一本征值冲突。推广了文献[32]的结果，我们发现，当来自连续谱集中的一个实本征值与来自外部的另一个本征值相遇时，通常会发生这种碰撞，从而形成复共轭对。这种情况与子空间问题的一个转折点有关，我们发展了一种在这一点上膨胀子空间的方法。我们证明了在延拓过程中的预报器和校正器步骤将导致Sylvester类型的有边值矩阵方程。对于这些方程，我们发展了一个边界版本的Bartels-Stewart算法，它允许将线性代数归结为求解一系列有边界的线性方程组。通过对抛物系统中行波稳定性问题的研究，特别是对Ginzburg-Landau方程和Fitzhugh-Nagumo方程的研究，说明了数值技巧。
We present a continuation method for low-dimensional invariant subspaces of a parameterized family of large and sparse real matrices. Such matrices typically occur when linearizing about branches of steady states in dynamical systems that are obtained by spatial discretization of time-dependent PDE’s. The main interest is in subspaces that belong to spectral sets close the imaginary axis. Our continuation procedure provides bases of the invariant subspaces that depend smoothly on the parameter as long as the continued spectral subset does not collide with another eigenvalue. Generalizing results from [32] we show that this collision generically occurs when a real eigenvalue from the continued spectral set meets another eigenvalue from outside to form a complex conjugate pair. Such a situation relates to a turning point of the subspace problem and and we develop a method to inflate the subspace at such points.We show that the predictor and the corrector step during continuation lead to bordered matrix equations of Sylvester type. For these equations we develop a bordered version of the Bartels-Stewart algorithm which allows to reduce the linear algebra to solving a sequence of bordered linear systems.The numerical techniques are illustrated by studies of the stability problem for traveling waves in parabolic systems, in particular for the Ginzburg-Landau and the FitzHugh-Nagumo equation.