Topologically Transitive Flow on $R^n$

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 $x in mathbb{S}^n$.