Background: I am a newbie to nonlinear dynamical systems. But I have taken a graduate course in linear dynamical system.
Question: Is it possible to construct an autonomous $C^1$ vector field $f:U\rightarrow \mathbb{R}^n$, where $U$ is a bounded open subset of $\mathbb{R}^n$, such that it has neither any stationary points nor any limit cycles in $U$? Another constraint is that if we start at any $x_0 \in V \subset U$, the solution of $\dot{x}=f(x)$ should always remain inside $U$.
It will be really helpful if I can get an example of such a vector field for dimension $n\ge 2$ along with the sets $U$ and $V$.
Let $U$ be the annulus $1<x^2+y^2<9$. Draw the spiral $$\gamma:\quad{\mathbb R}\to U,\qquad t\mapsto r(t)\bigl(\cos t,\sin t\bigr)\ ,$$ whereby $t\mapsto r(t):=2+\tanh t$ is monotonically increasing from $1$ to $3$ during the infinitely many turns at both ends. Draw unit tangent vectors along $\gamma$, and create a rotationally symmetric vector field $X$ on $U$ coinciding with the drawn tangent vectors along $\gamma$. Then all solution curves are spirals congruent to $\gamma$.