Trajectory Geometry Near Analytic Vector-Field Singularities

Definitions and mathematical statement

Let $n\geq 1$, let $U\subseteq{\mathbb{R}}^n$ be an open neighborhood of $0$, and let $X:U\to{\mathbb{R}}^n$ be a real analytic vector field satisfying $X(0)=0$. Thus $0$ is a singular point; the source does not assume that it is isolated. A parametrized trajectory is a continuously differentiable map $\gamma:I\to U$, where $I\subseteq{\mathbb{R}}$ is an interval, satisfying \[ \gamma'(t)=X(\gamma(t))\qquad(t\in I). \] A trajectory converges to the singularity at an endpoint $\tau$ of its time interval if \[ \lim_{\substack{t\to\tau\\t\in I}}\gamma(t)=0, \] where an endpoint may be infinite. The supplied statement does not specify a direction of time.

The supplied research question is: what qualitative dynamics near $0$ can be described from geometric properties of trajectories converging to $0$, including their finiteness properties with respect to analytic sets and their behavior relative to neighboring trajectories?

This is a research program rather than a specified conjecture. The source does not identify the geometric hypotheses, the intended finiteness property, the class of analytic sets being tested, the meaning of neighboring trajectories, or a precise dynamical conclusion. Consequently, no definite implication or universally quantified finiteness assertion can be extracted without additional information. In particular, the statement does not assert that every trajectory converges to $0$.

Short English statement

For a real analytic vector field vanishing at the origin, investigate how the geometry of trajectories approaching the origin informs nearby dynamics, interactions with analytic sets, and relations between trajectories. The supplied question leaves the exact hypotheses and desired conclusions unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003503$, title: Trajectory Geometry Near Analytic Vector-Field Singularities. Collection: Oberwolfach Reports Open Problems. Exact source problem code: catalogue entry. The registry records proposed year $2017$ and names no individual proposer.
• Supplied report citation: O-Minimality and its Applications to Number Theory and Analysis (2018), Oberwolfach Reports, pp. 1349--1420, DOI 10.4171/owr/2017/22. https://doi.org/10.4171/owr/2017/22. The supplied background describes the reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Oberwolfach Reports 2018": https://doi.org/10.4171/owr/2017/22.
• Supplied literature source, labeled "Le Gal et al., interlaced analytic vector fields": https://arxiv.org/abs/1310.2225.
• Supplied literature source, labeled "Le Gal and Sanz, trajectories near invariant curves": https://arxiv.org/abs/2311.06821.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that special cases involving interlaced three-dimensional analytic vector fields and several asymptotic classes are understood, while general qualitative dynamics and finiteness questions around the singularity remain open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This records the supplied editorial assessment, not mathematical verification of this draft.