Averaging of time - periodic systems without a small parameter

In this article, we present a new approach to averaging in non-Hamiltoniansystems with periodic forcing. The results here do not depend on the existenceof a small parameter. In fact, we show that our averaging method fits into anappropriate nonlinear equivalence problem, and that this problem can be solvedformally by using the Lie transform framework to linearize it. According tothis approach, we derive formal coordinate transformations associated with bothfirst-order and higher-order averaging, which result in more manageableformulae than the classical ones. Using these transformations, it is possible to correct the solution of anaveraged system by recovering the oscillatory components of the originalnon-averaged system. In this framework, the inverse transformations are alsodefined explicitly by formal series; they allow the estimation of appropriateinitial data for each higher-order averaged system, respecting the equivalencerelation. Finally, we show how these methods can be used for identifying and computingperiodic solutions for a very large class of nonlinear systems withtime-periodic forcing. We test the validity of our approach by analyzing boththe first-order and the second-order averaged system for a problem inatmospheric chemistry.