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Etude d'une classe d'équations différentielles affines par morceaux modélisant des réseaux de régulation biologique

Abstract : This thesis deals with a class of biological networks' models, more specifically proposed for gene regulatory networks. These models are formulated as systems of piecewise affine differential equations. The phase space is euclidian, with the number of interacting biological elements as dimension. It is partitioned into rectangular pieces, in which the dynamics is affine. Each coordinate represents the activity level of an element in the net. The thesis is structured in three parts. First, a short biological introduction is proposed, followed by a presentation of the most widely used mathematical models. Piecewise affine models are defined in details, and compared with purely discrete models, and smooth models with sigmoids. A detailed state of the art is given. In a second part, theoretical results are presented. The analysis of periodic orbits, previously known for piecewise linear systems, is extended to the piecewise affine case. Then, a geometric and combinatorial approach is used to analyse the local dynamics, in boxes where the dynamics is affine. Global consequences of this local study are described in terms of symbolic dynamics. Especially, it is shown that the topological entropy of piecewise affine systems is strictly lower than that of purely discrete models, for a large class of systems. The third part deals with numerical analysis of the studied systems. The implemented algorithms are described, and some simulation data of systems in 4 dimensions are analysed. The dynamics of a specific system in 3 dimensions is also investigated.
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Contributor : Etienne Farcot <>
Submitted on : Thursday, October 6, 2005 - 7:19:59 PM
Last modification on : Friday, November 6, 2020 - 3:52:44 AM
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  • HAL Id : tel-00010463, version 1



Etienne Farcot. Etude d'une classe d'équations différentielles affines par morceaux modélisant des réseaux de régulation biologique. Modélisation et simulation. Institut National Polytechnique de Grenoble - INPG, 2005. Français. ⟨tel-00010463⟩



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