Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case

This paper concerns the planar Liénard system $\dot{x}=y-F(x),\dot{y}=-x$, where $F(x)$ is a continuous piecewise linear function with exactly $n$ fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is $n$ when $F$ has exactly $n$ fold points. The conjecture was confirmed for $n=1$ and $n=2$ in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case $n\ge 3$ remained open. In this paper, we show that the maximal number of limit cycles is at least $\mathrm{\mid }3n-4\mathrm{\mid }$ for $n\in {\mathbb{N}}^{+}$, thereby disproving Tonnelier’s conjecture for $n\ge 3$. The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound $2^{28(n+1{)}^2}$.