Upper bound in Tonnelier’s conjecture for piecewise-linear Liénard systems with jump points

Prove that a piecewise-linear Liénard system \(\dot{x}=F(x)-y,\ \dot{y}=x\), with \(F\) defined on \(n+1\) intervals and having \(n\ge2\) jump discontinuities and no fold points, has at most \(2n\) limit cycles.

Background

Tonnelier’s second conjecture predicts an upper bound of twice the number of jump points for piecewise-linear Liénard systems with jump discontinuities and no fold points. The paper notes that only the one-jump case had been resolved before its lower-bound constructions, leaving the upper-bound problem unresolved for two or more jumps.