Topologically Transitive Flow on $R^n$

Question

I have been looking but cannot find the following: an example of a topologically transitive map on $\mathbb{S}^n$ or on $\mathbb{R}^n$ (for arbitrary $n$). Here, by a topologically transitive flow, I mean a smooth map $$ \begin{aligned} f:[0,1]\times \mathbb{S}^n \rightarrow \mathbb{S}^n \end{aligned} $$ such that $\{f(t,x):t \in [0,1]\}$ is dense in $\mathbb{S}^n$ for some $x \in \mathbb{S}^n$.

Answer 1

You can have a look at the paper

V. López and G. López, Transitive flows on manifolds, Rev. Mat. Iberoam. 20 (2004), 107-130

for the general solution.