A Markus–Yamabe problem for differential equations

Definitions and mathematical statement

For a smooth map $X:\mathbb R^3\to\mathbb R^3$, consider the autonomous differential equation $\dot z=X(z)$. Assume that $X$ has an equilibrium, which after translation is $X(0)=0$, and that its Jacobian is Hurwitz at every point: \[ \operatorname{Re}\lambda<0\quad \text{for every eigenvalue }\lambda\text{ of }DX(z), \quad z\in\mathbb R^3. \] The question is whether some such vector field has a nonconstant periodic solution. Precisely, does there exist $T>0$ and a nonconstant differentiable map $\gamma:\mathbb R\to\mathbb R^3$ with $\gamma(t+T)=\gamma(t)$ and $\gamma'(t)=X(\gamma(t))$?

The spectral assumption is pointwise, with no uniform negative bound on the real parts required. Smooth means $C^\infty$; the vector field need not be polynomial. The periodic orbit need not be attracting, hyperbolic, or isolated. Its existence is the requested obstruction, so a counterexample to global stability caused solely by an escaping trajectory does not answer this particular question.

For context, global asymptotic stability of the equilibrium would mean Lyapunov stability together with existence for all positive time and convergence to zero for every initial condition. The general Markus–Yamabe implication from the stated Jacobian condition to that conclusion fails in dimension three. The source asks whether failure can specifically be witnessed by recurrent motion along a closed nonconstant trajectory under the same everywhere-Hurwitz hypothesis.

Short English statement

Can an everywhere-Hurwitz smooth vector field on three-dimensional space have a nonconstant periodic orbit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700019 (AMR-046-0019), 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 19: A Markus–Yamabe problem for differential equations. Original source indexed by AMR-046-0019. https://arxiv.org/abs/2012.02524.