Maximum number of limit cycles of the piecewise linear Liénard system with three zones

Abstract For the piecewise linear Liénard system , , where is a continuous piecewise linear function with fold points, the question of how many limit cycles that such a system can have has been a classical and open problem in differential equations and dynamical systems. Nowadays, we only know the answer for the case . For the cases , the problem still remains open. This paper aims to give an affirmative answer for the case of this open problem, that is, the maximum number of limit cycles of the continuous piecewise linear function with 2 fold points is 2. This system exhibits abundantly interesting and rich dynamics, including the generalized Hopf bifurcation, boundary equilibrium bifurcation, grazing limit cycle bifurcation, and double limit cycle bifurcation, which may have potential interdisciplinary applications.