LAUSR

A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems.