It is necessary to look on the solution x(t) as a function of complex time. This function has poles which are a short distance off the real t-axis, and their positions can be calculated [4] using, for instance, the ratio-like test of [5]. It is the existence of these poles that prevents us from writing the ¶ map simply as a two variable power series in x(0), y(0) -- they are close enough to the real t-axis to prevent a single such series from converging over the whole range t = 0 to t = T.
Figure 4: Analytical continuation is used to derive the
mapping. The circles 
, 
...with radii 
, 
..., are
the regions of convergence of the series solution to Duffing's equation
about the points 
, 
.... The crosses (X) represent
the position of the poles in the complex plane, and occur in
conjugate pairs.
Analytical continuation is a technique that allows us to work round this problem.
We assume that about a point 
, there exists a series solution
to a second-order differential equation with initial conditions
, in the form
with 
and 
.
The series (5) converges inside the circle 
, centre 
, of radius 
;
is the distance to the nearest pole in the complex time plane, as shown in
figure (4). The calculation of the mapping then consists of the
following steps:
, at 
, calculate
the series for 
as described below.
. A practical value of 
usually has to be found experimentally, since the radius of convergence of the series
is rarely known in advance. ,
..., 
, where 
, the number of circles used, is chosen
so that an adequate representation of the map
is obtained. Again, this value generally has to be obtained experimentally.
Hence, defining the function 
as
the ¶ map can be expressed as
We now explain how to derive the series for the , before
demonstrating how the result in equation (6) works in practice.
