A generalization of Vinograd's theorem for dynamical systems

Abstract

The theory of dynamical systems has been expanded by the introduction of local dynamical systems [10, 4, 9] and local semidynamical systems [1]. Using integral curves of autonomous ordinary differential equations to illustrate these generalizations, we find that, roughly, the integral curves form a local dynamical system if solutions exist and are unique without requiring existence for all time, and the integral curves form a local semidynamical system if solutions exist and are unique in the positive sense but need not exist for all positive time. In addition to autonomous ordinary differential equations, the enlarged theory of dynamical systems has applications to nonautonomous ordinary differential equations, certain partial differential equations, functional differential equations, and Volterra Integral equations [9, 1, 2, 8], respectively. All of these have metric phase spaces. Since many dynamic considerations are invariant to reparameterizations, it is of interest to known when a local dynamical (or semidynamical) system can be reparameterized to yield a “global” dynamical (or semidynamical) system. For autonomous ordinary differential equations, Vinograd [7] has shown that the local dynamical system on an open subset ofR n formed by integral curves is isomorphic (in the sense of Nemytskii and Stepanov) to a global dynamical system. In an extensive study of isomorphisms, Ura [12] has expanded the Gottschalk-Hedlund notion of an isomorphism and restated Vinograd's result in terms of a reparameterization. In this paper we study the problem of finding a global dynamical (or semidynamical) system which is isomorphic to a given local system. A necessary and sufficient condition is found which is then used to show that the Vinograd result holds on metric spaces.