On the cyclicity of Kolmogorov polycycles

Abstract

In this paper we study planar polynomial Kolmogorov's differential systems \[ X_μ\quad\sist{xf(x,y;μ),}{yg(x,y;μ),} \] with the parameter $\mu$ varying in an open subset $\Lambda \subset {\mathbb{R}}^N$. Compactifying $X_{\mu }$ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle $\Gamma$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $\mu \in \Lambda .$ We are interested in the cyclicity of $\Gamma$ inside the family $\{X_{\mu }{\}}_{\mu \in \Lambda },$ i.e., the number of limit cycles that bifurcate from $\Gamma$ 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 $\mu \in \Lambda ,$ including those parameters for which the return map along $\Gamma$ is the identity.