The "dimension of phase space" is not quite the right way to characterize systems that can be chaotic. The proper criterion involves writing the equations in first-order autonomous form: for variables $\xi_1, \dots, \xi_N$, the equations of motion are of the form
$$
\dot{\xi}_i = f_i(\xi_1, \dots, \xi_N)
$$
where the functions $f_i$ do not depend explicitly on $t$. "First-order" means that the left-hand sides of the $N$ equations are the first derivatives of the configuration; "autonomous" means that the right-hand sides don't depend explicitly on $t$. $N$ is (I think) what you're thinking of as "the dimension of phase space."
Any system of ODEs can be put into this first-order autonomous form. For the system to exhibit chaotic behavior, it is necessary that when the equations are written in this form, the functions $f_i$ are non-linear, and that $N \geq 3$. As you note, the Lorenz equations have $N = 3$, but they are also autonomous; their equations of motion do not depend explicitly on $t$. For the double pendulum, we have $N = 4$, though the fact that we also have a conserved quantity means that the accessible phase space is effectively 3-dimensional instead.
So what about the Duffing oscillator? We can try putting it in first-order form as follows:
\begin{align*}
\dot{v} &= -Av - Bx - C x^3 - D \cos (\omega t) \\
\dot{x} &= v
\end{align*}
This is a first-order non-linear set of equations. But it is not autonomous, since $t$ appears on the right-hand side of the first equation. Not to fret, though: we can define an equivalent autonomous system with three variables as follows:
\begin{align*}
\dot{v} &= -Av - Bx - C x^3 - D \cos (\omega z) \\
\dot{x} &= v \\
\dot{z} &= 1
\end{align*}
This system is autonomous, and since it has 3 degrees of freedom it can (and does) exhibit chaotic behavior. The variable $z$ is, of course, secretly equal to $t$.
This trick is quite general, by the way: any non-autonomous first-order system of $N$ equations can be recast as an autonomous first-order system of $N+1$ equations, simply by treating $t$ as one of the variables of the system.
