Substituting the series (5) into an m-th order differential equation
results in a
recursion formula for the coefficients 
in terms of the
m initial conditions. For a linear/nonlinear differential equation this recursion
formula will be linear/nonlinear respectively.
In the case of the Duffing equation, the recursion formula is
where the 
are defined by
Nonlinearity enters the recursion formula (7) solely through the
.
We can now calculate the series for 
, from which
the ¶ map for Duffing's equation can be built up recursively.
