Maximum limit-cycle count for polynomial Liénard systems of degree at least five

Determine the maximum number of limit cycles of the polynomial Liénard system \(\dot{x}=F(x)-y,\ \dot{y}=x\) when \(F(x)\) is a polynomial of degree \(n\geq 5\).

Background

The paper reviews the Lins–de Melo–Pugh conjecture for polynomial Liénard systems, which proposed an upper bound of [n−12]\left[\frac{n-1}{2}\right] limit cycles for a polynomial FF of degree nn. Although several low-degree cases and constructions for higher degrees are known, the authors state that the exact maximum remains unknown for degrees n≥5n\geq5.

References

Regardless, Lins--de Melo--Pugh's conjecture is incorrect for degree n \ge 6. Thus, the maximum number of limit cycles of system~ls remains unknown for n\geq5.

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

Thus, although substantial progress has been achieved, the exact maximum number of limit cycles is still unknown in general, in particular for degrees n\geq5.

— Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case  (2610.10050 - Cai et al., 7 Oct 2026) in Section 1, Introduction and main results