Hopf Algebra Techniques to Handle Dynamical Systems and Numerical Integrators

Abstract

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, universal version of it which is solved algebraically; then, the results are transferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.

High-order stroboscopic averaging methods for highly oscillatory delay problems

“…Errors for Ω = 1024π are not reported because, with our facilities, the computation of the reference solution with dde23 would take several days. The error bounds are different from those for (3) because for this tougher problem the microintegration bound is as in (21) so that the impact Ωµ of the microintegration is now Ω 4 h 4 or N −4 ; this impact is then Ω independent. In fact, the main difference observed when comparing Tables 3 and 4, is that in Table 4 errors along each row stay constant while in Table 3 they decrease as Ω increases as discussed above.…”

“…Let us prove the error bound (21); the proof of (22) follows the same pattern and will not be given. We start by noting that the solution of the initial value problem given by…”

Word series high-order averaging of highly oscillatory differential equations with delay

“…The difficulties in obtaining high-order averaged systems are compounded if the system to be averaged has delays. In this paper we show that, for periodically forced differential systems with constant delay, it is possible to obtain high-order averaged systems by an application of the word-series results in [6,7,8,9,15,10,18,19]. The simple treatment presented here is only possible when the forcing period is a submultiple of the delay, a hypothesis whose scope is discussed in Section 3.…”

“…For order ≥ 4 it is not practical to find the general expression of the averaged system and then to apply it to the specific instance of (1) at hand. One should rather find recursively the word basis functions corresponding to the g of interest, perhaps with the help of a symbolic manipulator, see [15,19] for additional details.…”

Word combinatorics for stochastic differential equations: Splitting integrators

“…An alternative methodology, patterned after Butcher's treatment of the Runge-Kutta case was introduced in [32] (a summary may be seen in [21,section III.3]). A third possibility is the use of word series expansions [31,14,15,33,34,35,36,37]. Word series are patterned after B-series; rather than combining elementary differentials they combine word basis functions.…”