The Fixed Point Index of the Poincaré Translation Operator on Differentiable Manifolds

Abstract

The fixed point index of the Poincare translation operator associated to an ordinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclidean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincare translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation operator associated with non-autonomous equations will be deduced as consequences. We emphasize that other results, unrelated to our approach, but still involving the fixed point index of the Poincare translation operator have been successfully exploited, for instance, by Srzednicki (see e.g. [Srz2, Srz3]).