Volume 10, Issue 1
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Approximating Solutions of Linear Ordinary Differential Equations with Periodic Coefficients by Exact Picard Iterates
Systems of differential equations with periodic coefficients arise in several different applications in physics and engineering. Because of their applications, such systems also deserve attention from pure scientists. One important method for understanding them is linearization around an equilibrium solution. This justifies the attention we devote to systems of linear differential equations with periodic coefficients.
Despite being simpler than nonlinear systems, linearized systems of differential equations in general are not exactly solvable if their coefficients are not constant. It is thus natural to consider approximate solutions for these systems. Besides traditional numerical methods, other kinds of methods, such as perturbation theory  and averaging , are used. Whereas traditional numerical methods suffer from the problem of being difficult to apply in cases where the equations depend on parameters, perturbation and averaging usually perform well only in some limited regions in the space of parameters.
Picard iteration, described in Section 2, is a well-known method because of its theoretical importance in proving the existence-uniqueness theorem for differential equations . We were delighted and surprised to see it appear in  as a practical tool for approximating solutions of differential equations. Furthermore, it can also be used if the equations depend on parameters.
The authors of  expand the coefficients matrix of a linear system with periodic coefficients in a series of shifted Chebyshev polynomials. By cleverly using properties of the Chebyshev polynomials, they are able to trade the integrations occurring in Picard iteration by matrix multiplications. As a result, excellent approximations for the coefficients of the Chebyshev polynomial series of high-order Picard iterates can be obtained. Their method is thus approximate in a double sense.
First, the infinite sequence of Picard iterates converges to the exact solution, but the Picard iterate at any finite order is only an approximate solution for the differential equations. Second, they do not obtain exact Picard iterates but a truncation of a series converging to them.
While reading  we asked ourselves what the difficulty would be if, instead of approximate Picard iterates, we tried to calculate exact ones. Picard iterates are seldom used as an approximate solution method to general differential equations because functions that are impossible to integrate are likely to appear during the iteration. We will see that this is indeed what happens in a simple case, the nonlinear pendulum equation. However, this will not be the case for systems of linear ODEs with periodic coefficients. This assertion will be proved in Section 3. In order to approximately solve these equations, all we then need is a sufficiently fast integration function.
In , we showed that it is possible to obtain practical results by calculating exact Picard iterates for linear differential equations with periodic coefficients. The number of iterations performed were equal to the ones computed by Sinha and Butcher in  and . Although the computers and Mathematica versions used are different, the times needed are much shorter with our method. Moreover, exactness in our results leads to important differences in accuracy in the parameter regions considered. In this article we will concentrate on explaining the implementation in detail instead of comparing results.
Our method directly uses definition (2) of the Picard iterate and the specialized integration function we created, which calculates all necessary integrals much faster than by using the built-in function Integrate. We provide the package (described in Section 4) in Additional Material that carefully implements all these ideas.
Section 5 is devoted to the application of the package to the question of linear stability of a system of upside-down pendula driven by vertical periodic motion of its suspension point. This kind of system has been considered recently both by pure [7, 8, 9] and applied [1, 2] scientists. Besides being an application of the method devised, it is also a good illustration of the power of Mathematica. Not only is it used to approximately solve the differential equations, but also for deducing the equations of motion of the pendula, linearizing them, diagonalizing matrices, and graphing the results.
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