Uniform limit-cycle bounds for quadratic planar polynomial systems

Establish the maximum number and relative positions of limit cycles of planar polynomial differential systems of degree two, thereby resolving the quadratic case of the second part of Hilbert’s sixteenth problem.

Background

The paper places the study of piecewise-linear Liénard systems within the broader context of Hilbert’s sixteenth problem, which asks for bounds on the number and arrangement of limit cycles of planar polynomial systems of fixed degree. It explicitly notes that even the quadratic case remains unresolved. The paper does not address quadratic polynomial systems or resolve this general problem.

References

Despite extensive efforts over more than a century, this problem remains open even for quadratic polynomial systems.

— 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