On the cyclicity of Kolmogorov polycycles

TL;DR

This paper investigates the cyclicity of hyperbolic polycycles in planar polynomial Kolmogorov systems, introducing functions analogous to Lyapunov quantities to determine the number of bifurcating limit cycles.

Contribution

It defines three functions that determine the cyclicity of Kolmogorov polycycles, extending the understanding of bifurcations in these systems beyond classical methods.

Findings

Determined cyclicity for specific cubic Kolmogorov families.

Extended cyclicity analysis to parameter values with identity return map.

Introduced Lyapunov-like functions for polycycle cyclicity analysis.

Abstract

In this paper we study planar polynomial Kolmogorov's differential systems \[ X_\mu\quad\sist{xf(x,y;\mu),}{yg(x,y;\mu),} \] with the parameter $\mu$ varying in an open subset $\Lambda\subset\R^N$. Compactifying $X_\mu$ to the Poincar\'e 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.