Poincaré analyticity and the complete variational equations

According to a theorem of Poincaré, 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. 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.