Low-degree classical Liénard systems

Definitions and mathematical statement

Let $F\in\mathbb R[x]$ have degree $n$ and normalize $F(0)=0$; subtracting its constant term merely translates the $y$ coordinate. Consider the classical Liénard system \[ \dot x=y-F(x),\qquad \dot y=-x. \] Write $\operatorname{Lie}(n)$ for the supremum of the number of its isolated nonconstant periodic orbits over all such $F$. The equalities $\operatorname{Lie}(3)=\operatorname{Lie}(4)=1$ are known. The question asks for proofs of these two equalities using the Bendixson–Dulac method.

For a vector field $X=(P,Q)$, a Dulac multiplier is a differentiable scalar function $B$ on an appropriate region such that \[ \operatorname{div}(BX)=\partial_x(BP)+\partial_y(BQ) \] has a controlled sign. On a simply connected region, a suitable strict-sign condition rules out periodic orbits by Green's theorem. For uniqueness one may partition the plane along a zero or singular set of $B$, allowing a region with one hole and controlling any closed trajectories on its boundary.

The task is to construct suitable multipliers, or equivalent auxiliary functions, and verify the sign and topology conditions uniformly for every cubic and quartic $F$. Cases where the origin is a center must be handled: their non-isolated periodic orbits do not count as limit cycles. This is a request for a particular kind of proof of known numerical bounds, together with the usual examples showing the bound one is attained.

Short English statement

Find Bendixson–Dulac proofs that cubic and quartic classical Liénard systems have at most one limit cycle.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700004 (AMR-046-0004), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 4: Low-degree classical Liénard systems. Original source indexed by AMR-046-0004. https://arxiv.org/abs/2012.02524.