Lyapunov functions and bounded solutions of linear systems of differential equations (Q1088048)

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.