Vanishing set of inverse Jacobi multipliers and attractor/repeller sets

Abstract

In this paper we study conditions under which the zero-set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink singularities, orbitally asymptotic limit cycles and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repeller sets with different topologies, several examples are constructed illustrating the results presented.

I.A.G., J.G., and S.M. are partially supported by Ministerio de Ciencia, Innovación y Universidades under Grant No. MTM2017-84383-P and by AGAUR under Grant No. 2017SGR-1276. J.L. is partially supported by Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación under Grant No. MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca under Grant No. 2017SGR1617, and the H2020 European Research Council under Grant No. MSCA-RISE-2017-777911.