Trigonometric Abel differential equations II

Definitions and mathematical statement

Fix integers $p>q\ge2$ and nonnegative integers $m,n$. A real trigonometric polynomial of degree $m$ is a function $A(t)=a_0+\sum_{j=1}^m(a_j\cos jt+\widetilde a_j\sin jt)$, with $(a_m,\widetilde a_m)\ne(0,0)$ when $m>0$ and $a_0\ne0$ when $m=0$. Define $B$ similarly with degree $n$, and consider \[ x'=A(t)x^p+B(t)x^q. \] The degrees are exactly $m$ and $n$, as in the source question; the leading harmonics therefore remain present throughout this fixed-degree family.

A periodic solution here is a real-valued solution that exists without blow-up on all of $\mathbb R$ and satisfies $x(t+2\pi)=x(t)$. A limit cycle means that its value at $t=0$ is an isolated fixed point of the period return map. The question asks for the maximum number of these cycles as a function of $p,q,m,n$, with all coefficients otherwise unrestricted.

The constant zero solution is included if isolated. Nonzero solutions can be positive or negative and both signs must be counted; their parity behavior depends on the exponents. Distinct initial values at the fixed forcing phase represent distinct periodic solutions. A period which divides $2\pi$ is allowed. Continuous families of periodic solutions and solutions that become infinite in finite time do not contribute isolated cycles. The target is a uniform sharp bound and examples attaining it, or a proof that no finite bound exists for some fixed degree and exponent data.

Short English statement

Determine the maximum number of periodic limit cycles from the two exponents and the trigonometric coefficient degrees.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700007 (AMR-046-0007), 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 7: Trigonometric Abel differential equations II. Original source indexed by AMR-046-0007. https://arxiv.org/abs/2012.02524.