Lyapunov functions and bounded solutions of linear systems of differential equations
In problems involving the existence of invariant manifolds of dynamical systems one has to consider families of equations \(dh/dt=(H(t)\cdot h)\in {\mathbb{R}}^ n.\) The present article continues the research concerning such systems in Ukr. Math. J. 36, 720-729 (1984; Zbl 0555.34026). The system is imbedded in the 2n-dimensional system \(\dot h=H\cdot h\), \(\dot y=-h- H^ Ty\). The main theorem concerns the implications following from the existence of a Lyapunov function V (of a certain type) such that \(\dot V\geq -\| h\|^ 2\). In particular, given an arbitrary continuous vector function f the authors obtain a representation \(h={\mathcal M}f\), \({\mathcal M}={\mathcal M}^ 2\) and show that it represents all invariant manifolds of a 2n-dimensional system of linear differential equations with bounded coefficient matrices.
invariant manifolds of dynamical systems
scientific article; zbMATH DE number 3989837
scientific article