New lower bounds for the local Hilbert number for cubics systems and piecewise systems [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Luiz F. S. Gouveia
Universitat Autònoma de Barcelona


Let $\mathcal{P}_n$ the class of polynomial differential systems of degree $n.$ In this class, we are interested in the isolated periodic orbits, the so called limit cycles, surrounding only one equilibrium point of monodromic type. For the unperturbed system, the origin is always an equilibrium point of nondegenerate center-focus type. We define $M(n)$ as the maximum number of small limit cycles bifurcating from the origin via a degenerate Hopf bifurcation. We will prove that $M(3)\geq 12$. We will also consider this problem in the class of piecewise polynomial systems defined in two zones. Here, we are interested in the small crossing limit cycles surrounding only one equilibrium point or an sliding segment. When the separation curve is a straight line, we provide a piecewise cubic system exhibiting at least $24$ small crossing limit cycles. All of them nested surrounding only one equilibrium point, in fact an sliding segment. The computations use a parallelization algorithm.

This is a joint work with Joan Torregrosa.

The seminar will be presented in Spanish.