Piecewise linear differential system with a center-saddle type singularity (Q1684724)

The authors prove that for every natural number \(n\) there exists a piecewise linear system of ordinary differential equations on the plane with two zones (separated by an analytic curve) such that the phase portrait contains exactly \(n\) hyperbolic limit cycles. The system is constructed explicitly in the form \(\dot {\mathbf x} = U {\mathbf x} - C\) if \(y > b \sin x\) and \(\dot {\mathbf x} = L {\mathbf x}\) if \(y < b \sin x\), where \(U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), \(L = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\), \(C = \begin{pmatrix} a \\ 0 \end{pmatrix}\), and \(a\) and \(b\) satisfy \(n \pi < a < (n+1) \pi\) and \(0 < b < \frac12(\sqrt{a^2 + 4} - a)\).