Paper detail

On the cyclicity of Kolmogorov polycycles

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 $μ$ varying in an open subset $Λ\subset\R^N$. 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 $μ\inΛ.$ We are interested in the cyclicity of $Γ$ inside the family $\{X_μ\}_{μ\inΛ},$ 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 $μ\inΛ,$ including those parameters for which the return map along $Γ$ is the identity.