Existence and vanishing set of inverse integrating factors for analytic vector fields

Abstract

In this paper we address the problem of existence of inverse integrating factors for an analytic planar vector field in a neighborhood of its nonwandering sets. It is proved that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle, obtaining a necessary and sufficient condition for the existence of an analytic one. This condition is expressed in terms of the Ecalle–Voronin modulus of the associated Poincaré map. The existence of inverse integrating factors in a neighborhood of an elementary singularity is also established, and we give the first known examples of analytic vector fields in ℝ2 not admitting a Cω inverse integrating factor in any neighborhood of either a limit cycle or a weak focus. Moreover, it is shown that a C1 inverse integrating factor of a C1 planar vector field must vanish identically on the polycycles that are limit sets of its flow, thereby solving a problem posed by García and Shafer (‘Integral invariants and limit sets of planar vector fields’, J. Differential Equations 217 (2005) 363–376).