In this paper we derive a representation of the ¶ map for a periodically driven differential equation. We then compare the mapping in this form to results obtained by traditional numerical integration of the differential equation.
For the purposes of illustration, we confine ourselves to the well-known Duffing equation in the form
although our conclusions are valid for a wide class of differential equations. The period of the drive T = 1 and its amplitude is A. This differential equation has been studied intensively in the past because, despite its apparent simplicity, it can display coexisting solutions, and periodic and chaotic behaviour as A is varied.
Throughout what follows, it is convenient to re-write equation (1) as two coupled differential equations
where 
is the time derivative of x. The state vector, which
describes the state of the system at a time t, is then 
.
Of particular interest is the ¶ map for Duffing's equation, defined as
the function 
such that
or, splitting into its components,
It can be shown [1] that the mapping so defined is unique, single valued
and differentiable. The ¶ map expresses all the essential information about
solutions to Duffing's equation; for instance, for a value of A for
which the differential equation has a period-n/chaotic solution, iterating the
¶ map will also produce a sequence of points that repeats every n/is chaotic.
Hence, if the ¶ map were available in closed form, this
could be iterated for an arbitrary initial condition 
to unravel
the behaviour of solutions of Duffing's equation (with the same initial
condition) without resorting to numerical solutions.
What does 
actually look like? Figures (1)
and (2) show, respectively, 
and 
. Figure (3) is a section through these maps
at y(0) = 2. The figures suggest that has a reasonably complex
structure with large first- and higher-order derivatives. This in turn
suggests the impracticality of representing the map by standard surface
fitting methods, for
example two-dimensional splines or two-variable polynomial curve fitting. In
the case of splines, a large number would be required to reproduce
the mapping faithfully, and in the polynomial case, a very high order
polynomial would be needed. This is contrary to our stated objective
to represent the ¶ map in as succinct a form as possible.
Figure: The ¶ map, , obtained numerically. In this
and the following two figures, A = 13, corresponding to a chaotic solution.
Figure: Sections through the ¶ map, 
(left) and
(right).
