Communicated by Arieh Iserles.

Abstract We show how B-series may be used to derive in a systematic way the an-alytical expressions of the high-order stroboscopic averaged equations that approxi-mate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with O(j) errors, j = 1,2,3 (2π de-notes the period of the fast oscillations). For second-order systems with large O(−1) forces, we give the explicit form of the averaged systems with O(j) errors, j = 1,2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adi-abatic invariance of the oscillatory energy. Finally we use B-series to analyze mul-tiscale numerical integrators that implement the method of averaging. We construc