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.
The fixed point index of the Poincaré translation operator on differentiable manifolds / M. FURI; M. PERA; M. SPADINI. - STAMPA. - (2005), pp. 741-782.
The fixed point index of the Poincaré translation operator on differentiable manifolds
FURI, MASSIMO;PERA, MARIA PATRIZIA;SPADINI, MARCO
2005
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.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.