Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case

This paper concerns the planar Liénard system $\dot{x}=y-F(x),\dot{y}=-x$, where $F(x)$ is a piecewise linear function with exactly $m$ jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is $2m$. The conjecture was confirmed for $m=1$ in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of $2m$ for arbitrary $m$ was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least $4m-2$ hyperbolic crossing limit cycles for every positive integer $m$, thereby disproving Tonnelier’s conjecture for $m\ge 2$. We also establish the uniform upper bound $2^{224(m+1{)}^2}$ for the total number of crossing, grazing, sliding, and composite limit cycles.