Existence of piecewise linear differential systems with exactly \(n\) limit cycles for all \(n\in \mathbb N\)
The authors consider the piecewise linear feedback control system \[ \begin{pmatrix} \dot{x}\\ \dot{y}\end{pmatrix}= \begin{pmatrix} 0,&-1\\1,&\phantom{-}0\end{pmatrix} \begin{pmatrix} x\\ y\end{pmatrix} -\phi(x)\begin{pmatrix} \varepsilon\\0\end{pmatrix}, \tag{1} \] where \(\phi:\mathbb{R}\to\mathbb{R}\) is an odd piecewise linear periodic function of period \(4\), defined by \[ \phi(x)=\begin{cases} x, & x\in[0,1],\\ 2-x, & x\in[1,2]. \end{cases} \] System (1) can be written as a piecewise linear Liénard system \[ \dot{x}= -y-\varepsilon\phi(x),\qquad \dot{y}=x, \tag{2} \] which is not differentiable at the points \((2m+1,y)\), \(m\in\mathbb{Z}\). The main result of the paper is that, for \(\varepsilon\neq0\), system (2) has exactly \(n\) limit cycles in the strip \(| x| \leq2n+2\), \(n\in\mathbb{N}\), where stable cycles alternate with unstable ones; especially system (2) has infinitely many limit cycles in the real plane.
scientific article; zbMATH DE number 1963674