A survey on the inverse integrating factor

TL;DR

This survey comprehensively reviews the role of the inverse integrating factor in planar differential systems, covering its properties, relation to limit cycles, bifurcations, and various applications in system analysis.

Contribution

It provides an exhaustive, self-contained overview of all main results related to the inverse integrating factor and its significance in dynamical systems theory.

Findings

Summarizes key properties of the inverse integrating factor.

Details its relation to limit cycles and bifurcations.

Includes a comprehensive list of references and results.

Abstract

The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic $\omega$-limit set and some generalizations.