This thesis is part of a project to automate asymptotic analysis. We study systems of functional equations, which, besides the usual algebraic operations, involve the asymptotic relations of equivalence and inequality. We develop an algebraic and effective theory for solving such systems.

In part A, we develop the algebraic formalism, using the language of transseries. This enables us to describe very violent asymptotic behaviours, which are for instance encountered in the study of non-linear differential equations.

In part B, we give algorithms for the asymptotic resolution of algebraic differential equations and systems of transseries in several variables. In a first stage, our results only apply to functions with strongly monotonic asymptotic behaviours, i.e. functions which do not present any direct or indirect oscillatory phenomena at infinity: from an analytical point of view, this corresponds to considering only Hardy field functions. But we have already extended some of our results to the weakly oscillatory case.

Keywords: asymptotic analysis, computer algebra, transseries, differential equation, functional equation, exp-log function, Hardy field

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