Symbolic Computation of High-Order Exact Picard Iterates for Systems of Linear Differential Equations with Time-Periodic Coefficients

In symbolic manipulation packages such as MATHEMATICA it is possible to substitute the built-in function for integration by a user-programmed specific integration function and symbolically evaluate exact high-order Picard iterates for systems of linear differential equations with time-periodic parameter-dependent coefficients. With this technique we get excellent approximations in feasible CPU times for the solutions of these differential equations explicitly dependent on the parameters. We compare our technique with the one described by Sinha and Butcher [9], in which instead of exact Picard iterates, only a certain number of terms in the expansion of these iterates in Chebyshev polynomials is obtained. Our technique is exemplified by application to the Mathieu equation and calculation of linear stability domains for simple and double inverted pendula subjected to vertical periodic motions of the pivot.