New lower bound for the Hilbert number in piecewise quadratic differential systems

TL;DR

This paper establishes a new lower bound of 16 for the number of crossing limit cycles in piecewise quadratic differential systems, improving the known bounds and providing insights into their bifurcation structure.

Contribution

The paper proves the existence of at least 16 crossing limit cycles in piecewise quadratic systems, setting the best known lower bound for degree 2 systems.

Findings

16 crossing limit cycles proven to exist

Limit cycles appear in a nested bifurcation from isochronous centers

Improves the lower bound for quadratic piecewise systems

Abstract

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)\geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.