American Institute of Mathematical Sciences

November  2014, 34(11): 4827-4854. doi: 10.3934/dcds.2014.34.4827

Linearised higher variational equations

 1 Department of Mathematics, University of Portsmouth, Lion Gate Bldg, Lion Terrace, Portsmouth PO1 3HF, United Kingdom

Received  March 2013 Revised  April 2014 Published  May 2014

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations $\mathrm{LVE}_{\psi}^k$ of a generic autonomous system along a particular solution $\psi$. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.
Citation: Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
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