New lower bound for the Hilbert number in low degree Kolmogorov systems

Abstract

Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by ${\mathcal{M}}_K(n)$ the maximum number of limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov vector field of degree $n$. In this work, we obtain another example such that ${\mathcal{M}}_K(3)\ge 6$. In addition, we obtain new lower bounds for ${\mathcal{M}}_K(n)$ proving that ${\mathcal{M}}_K(4)\ge 13$ and ${\mathcal{M}}_K(5)\ge 22$.