Some properties of the Dulac functions set

Osvaldo Osuna,Gabriel Villaseñor-Aguilar

doi:10.14232/ejqtde.2011.1.72

Abstract

In order to rule out the existence of periodic orbits in the plane for a given system of differential equations, we discuss the feature of the set of Dulac functions, establishing some of its properties as well as some results for special cases where this set of functions is not empty. We give some examples to illustrate applications of these results.

Highlights

Many problems of the qualitative theory of differential equations in the plane refer to the existence of periodic orbits, for example in mechanical or electrical engineering, biological models and many others
There are some criteria that allow us to rule out the existence of periodic orbits in the plane such as Poincare-Bendixson, the index theory and special systems such as the system gradient, among others, see ([1],[9],[8] and [5])
To rule out the existence of periodic orbits of the system (1) in a connected region D, we need to find a function h(x1, x2) that satisfies the conditions of the theorem of Bendixson-Dulac

Summary

Introduction

Many problems of the qualitative theory of differential equations in the plane refer to the existence of periodic orbits, for example in mechanical or electrical engineering, biological models and many others. A classical criterion to discard the existence of periodic orbits (or limiting the number of these) in a given region is the Bendixson-Dulac theorem. To rule out the existence of periodic orbits of the system (1) in a connected region D, we need to find a function h(x1, x2) that satisfies the conditions of the theorem of Bendixson-Dulac.

Results

Conclusion