On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set (Q2083758)

Let \(n\) be a positive integer. An algebraic curve of degree \(n\) in \(\mathbb{R}^2\) is the zero set of a polynomial of degree \(n\) in two variables. The second part of the Hilbert's sixteenth problem is to find a uniform upper bound \(\mathcal{H}(n)\) of limit cycles of a planar polynomial system of degree \(n\). The number \(\mathcal{H}(n)\) is called the Hilbert number. A similar problem for planar piecewise smooth systems is to find a uniform upper bound \(\mathcal{L}(n)\) of limit cycles of a planar piecewise linear system with two zones separated by a branch of an algebraic curve of degree \(n\). In this paper, the author proved that for any positive integer \(k\), \(\mathcal{L}(2k)\geq k^2+2k+1\), \(\mathcal{L}(2k+1)\geq k^2+2k+3\). In terms of \(n\), this is equivalent to: \[\mathcal{L}(n)\geq \frac{n^2+4n+4}{4}\] if \(n\) is even, and \[\mathcal{L}(n)\geq \frac{n^2+2n+9}{4}\] if \(n\) is odd. Thus for any positive integer, \(\mathcal{L}(n)>\frac{1}{4}n^2\), which means that \(\mathcal{L}(n)\) grows as fast as \(n^2\). This is a significant improvement on the previous lower bound of \(\mathcal{L}(n)\), \(\mathcal{L}(n)\geq \left[\frac{n}{2}\right]\).