On the cyclicity of Kolmogorov polycycles

Abstract

In this paper we study planar polynomial Kolmogorov's differential systems X μ { x ˙ = f ( x , y ; μ ) , y ˙ = g ( x , y ; μ ) , with the parameter μ varying in an open subset Λ ⊂ R N . Compactifying X μ 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 { X μ } μ ∈ Λ , i.e., the number of limit cycles that bifurcate from Γ as we perturb $\mu.$ 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.