  According to a theorem of Poincare, the solutions to differential equations
are analytic functions of (and therefore have Taylor expansions in) the initial
conditions and various parameters provided that the right sides of the
differential equations are analytic in the variables, the time, and the
parameters. These Taylor expansions, which provide a transfer map M between
initial and final conditions, may be obtained, to any desired order, by
integration of the complete variational equations. As an example of this
approach, the convergence of such an expansion is investigated for the Duffing
equation stroboscopic map in the vicinity of a infinite period doubling cascade
and resulting strange attractor.
