In pg. 10 of [1] they define the notion of a recurrent vector field as:
A vector field $f$ on a manifold, say $\mathbb{R}^n$, is recurrent if for every point $x_0\in\mathbb{R}^n$ and every neighbourhood $V$ of $x_0$, there exists some time $t^*$ greater than any given $t\in\mathbb{R}$ such that $\gamma(t^*)\in V$ where $\gamma$ is the flow of $f$ with $\gamma(0)=x_0$. The intuition being a recurrent vector field is one where its flow curves always return arbitrarily close to the initial point, infinitely many times. (Presumably $V$ is an open neighbourhood.)
In [2] (same authors) they also make mention of recurrent vector fields and cites [1] for the definition.
I can't find this concept defined anywhere else after searching Google. I'm wondering if there's an alternate name for this or if anyone has come across a similar concept before.
I ask this since in [1] they refer to this as "the classical definition of recurrent vector field", so it sounds like it might be standard terminology. There is a Wikipedia article on recurrent tensor fields but it doesn't seem related.
[1]. Ugo Boscain, Mario Sigalotti. Introduction to controllability of non-linear systems. Serena Dipierro. Contemporary Research in Elliptic PDEs and Related Topics, Springer, 2019, 10.1007/978-3-030-18921-1_4. hal-02421207. <Link: https://hal.inria.fr/hal-02421207/document >
[2]. Uho Boscain and Mario Sigalotti and Dominique Sugny. Introduction to the Foundations of Quantum Optimal Control. arXiv:2010.09368 [quant-ph]. <Link: https://arxiv.org/pdf/2010.09368.pdf >