I am looking for examples of dynamical systems
- in discrete time,
- with state space $\mathbb{R}^n$ (for some arbitraty $n$),
- with a chaotic attractor $A \subset \mathbb{R}^n$ with a global basin of attraction $\text{basin}(A)= \mathbb{R}^n$.
For many classical examples I have found, such as the $2$-dimensional Henon-map, the basin of attraction of the chaotic attractor $A$ is a proper subset of $\mathbb{R}^n$ and thus the attractor is not global. However, I would explictly be interested in global attractors.
Since whatever happens outside the basin of attraction does not affect the attractor itself, you can just change the dynamics to map those points to the basin of attraction (and thus include them in it).
For example, take the logistic map: $$ x_{t+1} = r x_t (1-x_t) $$ Its basin of attraction is the interval $(0,1)$. You can extend that basin to all of $ℝ$ like this:
$$ x_{t+1} = \begin{cases} r x_t (1-x_t) & \text{if}\quad 0<x_t<1\\ \frac{1}{π} & \text{else} \end{cases}.$$ This is admittedly very crude and it may not fulfil your desires on continuity, but doing so is a plainly technical exercise. For example the following map is a crude continuous approximation of the logistic map at $r=4$ seems chaotic at first glance:
$$ x_{t+1} = \exp \left(-8 \left(x_t-\tfrac{1}{2}\right)^2 \right) .$$