# American Institute of Mathematical Sciences

October  2006, 14(4): 753-782. doi: 10.3934/dcds.2006.14.753

## Averaging of time - periodic systems without a small parameter

 1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France 2 Département Terre-Atmosphère-Océan and, Laboratoire de Météorologie Dynamique du CNRS/IPSL, École Normale Supérieure, Paris, France 3 Laboratoire de Météorologie Dynamique du CNRS/IPSL, École Normale Supérieure, Paris, France 4 Institute of Geophysics and Planetary Physics, University of California, Los Angeles, United States

Received  January 2005 Revised  May 2005 Published  January 2006

In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones.
Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation.
Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.
Citation: Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
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