New lower bound for the Hilbert number in low degree Kolmogorov systems - Carvalho, Yagor Romano; Da Cruz, Leonardo P.C.; Gouveia, Luiz F.S. [UNESP] (Elsevier, 2023)
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 MK(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 MK(3)≥6. In addition, we obtain new lower bounds for MK(n) proving that MK(4)≥13 and MK(5)≥22.
Keywords: Center-focus; Cyclicity; Kolmogorov systems; Limit cycles; Lotka–Volterra systems; Lyapunov quantities; Weak-focus order