Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, EpiSciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Effectiveness of the Bendixson-Dulac theorem

Abstract : We illustrate with several new applications the power and elegance of the Bendixson–Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Li´enard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m ≥ 2, we provide a simple 1-parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval of length smaller that 3√2(3/m)m/2 that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.
Document type :
Preprints, Working Papers, ...
Complete list of metadata
Contributor : Hector Giacomini Connect in order to contact the contributor
Submitted on : Tuesday, January 12, 2021 - 10:59:57 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:35 PM
Long-term archiving on: : Tuesday, April 13, 2021 - 7:06:11 PM


Files produced by the author(s)


  • HAL Id : hal-03108070, version 1
  • ARXIV : 2101.03874



Gasull Armengol, Hector Giacomini. Effectiveness of the Bendixson-Dulac theorem. 2021. ⟨hal-03108070⟩



Record views


Files downloads