Periodic Riccati differential equations

Definitions and mathematical statement

Fix $T>0$ and real continuous $T$-periodic functions $A_0,A_1,A_2$. For the Riccati equation \[ x'(t)=A_2(t)x(t)^2+A_1(t)x(t)+A_0(t), \] consider real solutions defined and finite for every $t\in\mathbb R$. A periodic solution satisfies $x(t+T)=x(t)$. It is a limit cycle of this periodic equation if its initial value is an isolated fixed point of the time-$T$ return map. Constant solutions count when they are isolated in this sense; solutions passing through infinity do not count.

Let $D$ be the set of initial values whose solutions exist on $[0,T]$, and write $R(x_0)=x(T;x_0)$ on $D$. The question asks for effective conditions on the coefficient functions distinguishing four possibilities: a continuum of periodic solutions, exactly two isolated periodic solutions, exactly one, or none. A return-map reformulation alone is not the requested effective criterion; the aim is to extract usable conditions from the coefficients.

One useful formulation writes $x=u/v$ and uses the linear system $u'=A_1u+A_0v$, $v'=-A_2u$. Its monodromy induces a fractional-linear map. However a fixed projective direction contributes an admissible real periodic solution only when the corresponding $v(t)$ never vanishes during the period. Any criterion based on monodromy must therefore also control this finite-solution condition, including degenerate and identity-monodromy cases.

Short English statement

Give usable coefficient criteria for the number of globally finite periodic solutions of a periodic Riccati equation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700005 (AMR-046-0005), 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 5: Periodic Riccati differential equations. Original source indexed by AMR-046-0005. https://arxiv.org/abs/2012.02524.