Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line (Q1733223)

The purpose of this paper is to characterize the global dynamics of the following planar systems related to the Liénard equation \[ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = y- f(x),\\ \frac{\mathrm{d}y}{\mathrm{d}t} =a -x, \end{cases} \tag{1} \] where $a$ is a given constant and $f$ is the continuous piecewise linear function \[ f(x)=\begin{cases} k_1x& \text{ if }x\geq0,\\ k_2x& \text{ otherwise}\end{cases} \] with $k_1$ and $k_2$ nonzero real constants. \par In order to obtain the desired characterization, the possible phase portraits of system (1) in the Poincaré disc are completely classified. Also, necessary and sufficient conditions for existence and uniqueness of limit cycles are given.