Dynamics of Differential Equations on Invariant Manifolds

The simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice. Appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics. This paper develops an approach which avoids the selection of coordinate systems on the manifold. Conditions are given in terms compound linear differential equations for the stability of equilibria and periodic orbits. Global results include criteria for the nonexistence of periodic orbits and a discussion of the nature of limit sets. As an application, a global stability criterion is established for the endemic equilibrium in an epidemiological model.