Numerical approximations of strong (un)stable manifolds
Numerical approximations of strong (un)stable manifolds
复制标题
强（不稳定）流形的数值近似
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
S. Townley
中科院分区：
文献类型：
--
作者：
H. Osinga;G. R. Lamooki;S. Townley
计算光滑向量场的双曲平衡的全局一维稳定或不稳定流形的方法是众所周知的。这样的流形只由两条轨迹组成，任意大的碎片可以，例如，使用流形线性近似上接近平衡的初始点来生成。局部流形的吸引特性（向前或向后时间）确保计算误差（取决于计算块的弧长）保持有界。本文讨论了当平衡失去双曲性时，或者当一维不稳定流形实际上是包含在高维不稳定流形中的“强”不稳定流形时，这些误差界是如何变化的。在这些情况下，局部流形在向前或向后的时间中都不具有局部吸引，标准误差界不起作用。我们用数值计算来说明理论分析，使用一个可以明确找到全局流形的例子，以及不知道真正流形的更一般的向量场。
The method of computing global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field is well known. Such manifolds consist only of two trajectories and arbitrarily large pieces can, for example, be generated using an initial point close to the equilibrium on the linear approximation of the manifold. The attraction properties (in forward or backward time) of the local manifolds ensure that the computational error, which depends on the arclength of the computed piece, remains bounded. This paper discusses how these error bounds change as the equilibrium loses its hyperbolicity, or when the one-dimensional, say, unstable manifold is, in fact, a ‘strong’ unstable manifold that is contained in a higher-dimensional unstable manifold. For these cases, the local manifolds are not locally attracting either in forward or in backward time and the standard error bound does not work. We illustrate the theoretical analysis with numerical computations, using an example for which the global manifolds can be found explicitly, as well as more general vector fields where the true manifolds are not known.