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)\geq 6$. In addition, we obtain new lower bounds for $\mathcal M_{K}(n)$ proving that $\mathcal M_{K}(4)\geq 13$ and $\mathcal M_{K}(5)\geq 22$.