Remarks on Equivalence and Linearization of Nonlinear Systems

Abstract

Abstract In this note we consider a problem of equivalence of nonlinear systems described by a finite set of real variables satisfying a system of ordinary differential equations of finite order. Two systems are called equivalent (resp. weakly equivalent) if there exist nonlinear transformations so that the variables of one system are functions of the variables (resp. the variables and their derivatives) of the second system and vice versa, so that the sets of formal trajectories are maped onto the sets of formal trajectories. To each system we associate an algebraic object which is a D-algebra (a differential algebra). We show that our systems are weakly equivalent if and only if the corresponding D-algebras are isomorphic. The same result holds with weak equivalence replaced bv equivalence if the corresponding algebraic object is a D-algebra with filtration. As corollaries we obtain algebraic criteria for equivalence (weak equivalence) of a system to a linear controllable system.