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According to a theorem of Poincare, the solutions to differential equationsare analytic functions of (and therefore have Taylor expansions in) the initialconditions and various parameters providing the right sides of the differentialequations are analytic in the variables, the time, and the parameters. Wedescribe how these Taylor expansions may be obtained, to any desired order, byintegration of what we call the complete variational equations. As illustratedin 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 cascadesand strange attractors) of the solutions of the underlying differentialequations.

Poincare Analyticity and the Complete Variational Equations