Time-varying vector fields on a compact Riemannian manifold. I
The author is concerned with the local property of a critical point \(x_ 0\) of a time-varying vector field f on a compact Riemannian manifold M. The following results are proved via exponential dichotomy. If a critical point \(x_ 0\) admits an exponential dichotomy, then there are two families of manifolds, stable manifold family \(\{W^+_ t(x_ 0)| \quad t\in {\mathbb{R}}\}\) and unstable manifold family \(\{W^-_ t(x_ 0)| \quad t\in {\mathbb{R}}\}\) of f through \(x_ 0\) in some neighborhood V of \(x_ 0\), moreover, the critical point \(x_ 0\) is isolated. The solution curve family of any time-varying vector field g is qualitatively the same as that of f whenever \(\| g-f\|\) is small enough.
scientific article; zbMATH DE number 4080447