American Institute of Mathematical Sciences

March  2014, 34(3): 1021-1040. doi: 10.3934/dcds.2014.34.1021

The tridendriform structure of a discrete Magnus expansion

 1 ICMAT, Nicolás Cabrera 13-15, 28049 Madrid, Spain 2 Laboratoire de Mathématiques, UMR 6620, CNRS-Université Blaise Pascal, BP 80026, F63171 Aubière Cedex, France

Received  December 2012 Revised  May 2013 Published  August 2013

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a linear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik--Plebański--Strichartz formula for this logarithm.
Citation: Kurusch Ebrahimi-Fard, Dominique Manchon. The tridendriform structure of a discrete Magnus expansion. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1021-1040. doi: 10.3934/dcds.2014.34.1021
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