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On a constructive approach to the solution of dynamical systems

par Brenig, Léon
Référence From Combinatorics to Dynamical Systems, De Gruyter, page (179-193)
Publication Publié, 2008-01

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Résumé :

A series of constructive results on dynamical systems governed by systems of nonlinear ODEs are presented. We use the fact that a large class of dynamical systems can be reshaped in a form involving only polynomial, or more generally, quasi-polynomial (QP) nonlinearities in their vector fields. Such QP-systems can be cast into a standard representation that is form invariant under a certain group of transformations. Among these transformations, there always exists one particular that transforms a given QP-system into a canonical form involving only homogeneous quadratic nonlinearities and entirely characterized by a fundamental matrix. The use of that canonical form leads us to several results on the integrability conditions of these systems and on the Taylor expansion of the general solution. We present some arguments for a conjecture relating the asymptotic solutions of the canonical form to another linear equation in an infinite vector space. A perturbative scheme is described for solving the latter. Its convergence will be studied in further works.