Convergence of Taylor Transfer Map for Duffing Equation

Abstract

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.