Two limit cycles for a class of discontinuous piecewise linear differential systems with two pieces

Abstract

This paper is a survey on the study of the maximum number of limit cycles of planar continuous and discontinuous piecewise differential systems formed by two linear centers and defined in two pieces separated by

\begin{eqnarray*}\Sigma =\left\{ (x,y)\in%TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion^{2}:x=ly,l\in 

%TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} 

%EndExpansion \text{ and }y\geq 0\right\} \\ \cup\left\{ (x,y)\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} 

%EndExpansion ^{2}:y=0\text{ and }x\geq 0\right\} . \end{eqnarray*}% 

We restrict our attention to the crossing limit cycles, i.e. to the limit cycles having exactly two or four points on $\Sigma $. We prove that such discontinuous piecewise linear differential systems can have 1 or 2 limit cycles. The limit cycles having two intersection points with $\Sigma $ can reach the maximum number 2. The limit cycles having four intersection points with $\Sigma $ are at most 1, and if it exists, the systems could simultaneously have 1 limit cycle intersecting $\Sigma $ in three points.