Section: New Results

Higher-order averaging, formal series and numerical integration

Participant : Philippe Chartier.

The paper  considers non-autonomous oscillatory systems of ordinary differential equations with $d=1$ non-resonant constant frequencies. Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all. The new method may be applied to obtain a variety of averaged systems. In particular we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (resp. of a divergence-free) system results in a canonical (resp. in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly $d$ formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of coupled harmonic oscillators with one slow frequency and four resonant fast frequencies.

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems $\stackrel{˙}{y}=f\left(y,t\right)$ with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system $\stackrel{˙}{Y}=F\left(Y\right)$ whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of $F$; all information on $F$ required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper  provides an analytic and experimental study of SAM and two related techniques: the LISP algorithms of Kirchgraber and multirevolution methods.