On the cyclicity of Kolmogorov polycycles

Abstract

In this paper we study planar polynomial Kolmogorov’s differential systems Xµ {ẋ = x f (x, y; µ), ẏ = yg(x, y; µ), with the parameter µ varying in an open subset Λ ⊂ RN. Compactifying Xµ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all µ ∈ Λ. We are interested in the cyclicity of Γ inside the family {Xµ}µ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb µ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N = 3 and N = 5, and in both cases we are able to determine the cyclicity of the polycycle for all µ ∈ Λ, including those parameters for which the return map along Γ is the identity. © 2022, University of Szeged. All rights reserved.