  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 providing the right sides of the differential
equations are analytic in the variables, the time, and the parameters. We
describe how these Taylor expansions may be obtained, to any desired order, by
integration of what we call the complete variational equations. As illustrated
in a Duffing equation stroboscopic map example, these Taylor expansions,
truncated at an appropriate order thereby providing polynomial approximations,
can well reproduce the behavior (including infinite period doubling cascades
and strange attractors) of the solutions of the underlying differential
equations.
