Grechko A.L.

On Lyapunov and Ricatti Monotone Differential Matrix Equation

The purpose of the paper is to generalize the Polacik–Terescak theorem for a monotone differential matrix equation of Lyapunov and Ricatti. Our goal is to study the existence of the one-dimensional invariant manifold (corresponding to Lyapunov and Ricatti monotone differential matrix equation). Using the method introduced by Hilbert Birkgoff in the projective contraction fixed point theorem, we determine conditions under which Lyapunov differential matrix equation has a one-dimensional invariant manifold in the cone of positive definite of quadratic form.

On smoothness of invariant sets of the monotone linear extensions

This paper considers the monotone linear extensions of dynamic systems. Moreover, we use the projection dynamics to prove the smoothness of invariant sets (manifolds) of linear extensions.