Section:
New Results
Higher-order averaging, formal series and numerical integration
Participant :
Philippe Chartier.
The paper [42] considers non-autonomous oscillatory systems of ordinary differential equations with 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 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 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 whose solutions Y interpolate the sought highly
oscillatory solutions y. SAM integrates numerically the averaged system without
using the analytic expression of ; all information on 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 [15] provides an analytic
and experimental study of SAM and two related techniques: the LISP algorithms
of Kirchgraber and multirevolution methods.