We have derived and verified a compact, iterative representation of the ¶ map for the Duffing equation. Our method can be regarded as a highly specialised method for solving that particular differential equation, and as a result of this, it is very efficient compared to standard, general numerical techniques. In practice, our method is around 45 times quicker. There is every reason to believe that the same technique can be applied to other periodically-driven differential equations.
We have verified that bifurcation diagrams and ¶ sections can both be faithfully reproduced by our mapping, whose basin of attraction, although finite, is large enough for most applications we can envisage.
Finally, our work raises several interesting questions, among them:
Can any analytical results be derived from our representation?
Can a useful bifurcation analysis be carried out using our representation?
The expansion points used, 
, were uniformly spaced; might there be a
better arrangement?
Is there an even more compact representation available? For instance, a
Padé approximation to the series (5) might have some advantages.