Reducible Ordinary Differential Equations

Abstract

The class of reducible differential equations under consideration here includes the class of symmetric systems, and examples show that the inclusion is proper. We first discuss reducibility, as well as the stronger concept of complete reducibility, from the viewpoint of Lie algebras of vector fields and their invariants, and find Lie algebra conditions for reducibility which generalize the conditions in the symmetric case. Completely reducible equations are shown to correspond to a special class of abelian Lie algebras. Then we consider the inverse problem of determining all vector fields which are reducible by some given map. We find conditions imposed on the vector fields by the map, and present an algorithmic access for a given polynomial or local analytic map to \({\mathbb R}.\) Next, reducibility of polynomial systems is discussed, with applications to local reducibility near a stationary point. We find necessary conditions for reducibility, including restrictions for possible reduction maps to a one-dimensional equation.