Theory of Index for Dynamical Systems of Order Higher Than Two

Abstract

This paper is concerned with the generalization of Poincaré’s theory of index to systems of order higher than two. The basic tool used in the generalization is the concept of the degree of a map. In topology this concept has been used to discuss the index of a vector field. In this paper we shall use the degree of a map concept to present a theory of index for higher-order systems in a form which might make it more accessible to engineers for applications. The theory utilizes the notion of the index of a hypersurface with respect to a given vector field. After presenting the theory, it is applied to dynamical systems governed by ordinary differential equations and also to dynamical systems governed by point mappings. Finally, in order to show how the abstract concept of the degree of a map, hence the index of a surface, may actually be evaluated, illustrative procedures of evaluation for two kinds of hypersurfaces are discussed in detail and an example of application is given.