Upper bound in Tonnelier’s conjecture for continuous piecewise-linear Liénard systems

Prove that a piecewise-linear Liénard system \(\dot{x}=F(x)-y,\ \dot{y}=x\), with \(F\) continuous and piecewise linear on \(n+1\) intervals, has at most \(n\) limit cycles for every \(n\geq3\).

Background

Tonnelier conjectured that the number of limit cycles is at most the number of fold points when the piecewise-linear function FF is continuous. The paper establishes the corresponding lower bound but does not prove the upper bound in the general case. The authors explicitly identify the conjecture as unresolved for three or more fold points.

References

Therefore, part~$(i)$ of Tonnelier's conjecture has been positively resolved for $n=1$ and $n=2$. However, up to now, the conjecture remains open for $n\ge3$.

— The number of limit cycles of piecewise linear Liénard systems  (2608.19542 - Chen et al., 20 Aug 2026) in Section 1, Introduction

Motivated by Tonnelier's conjecture and the existing results for systems involving only fold points or only jump points , together with the evidence obtained from our constructions, we propose the following conjecture concerning the general case in which $F(x)$ contains both jump and fold points. To the best of our knowledge, this problem remains completely open.

\begin{con} \label{con:m} System ls, with $F(x)$ piecewise linear on $n+1$ intervals and having $m$ jump points and $n-m$ fold points, has up to $n+m$ limit cycles, where $0\le m\le n$. \end{con}

— The number of limit cycles of piecewise linear Liénard systems  (2608.19542 - Chen et al., 20 Aug 2026) in Section 1, Introduction, immediately before Conjecture \ref{con:m}