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

Abstract: 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\ge3) remained open. In this paper, we show that the maximal number of limit cycles is at least (|3n-4|) for (n\in\mathbb N^+), thereby disproving Tonnelier's conjecture for (n\ge3). 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}).