WIAS Preprint No. 2864, (2021)

Global algebraic Poincaré--Bendixson annulus for van der Pol systems


  • Grin, Alexander
  • Schneider, Klaus R.

2020 Mathematics Subject Classification

  • 34C05 34C07 34C23


  • Limit cycle, equivalent van der Pol systems, Dulac--Cherkas function, Poincaré--Bendixson annulus, singularly perturbed system




By means of planar polynomial systems topologically equivalent to the van der Pol system we demonstrate an approach to construct algebraic transversal ovals forming a parameter depending Poincaré-Bendixson annulus which contains a unique limit cycle for the full parameter domain. The inner boundary consists of the zero-level set of a special Dulac-Cherkas function which implies the uniqueness of the limit cycle. For the construction of the outer boundary we present a corresponding procedure

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