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