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Étude de réseaux complexes de systèmes dynamiques dissipatifs ou conservatifs en dimension finie ou infinie. Application à l'analyse des comportements humains en situation de catastrophe.

Abstract : This thesis is devoted to the study of the dynamics of complex systems. We consider coupled networks built with multiple instances of deterministicdynamical systems, defined by ordinary differential equations or partial differential equations of parabolic type, which describe an evolution problem.We study the link between the internal dynamics of each node in the network, its topology, and its global dynamics. We analyze the coupling conditions which favor a particular dynamics at the network's scale, and study the impact of the interactions on the bifurcations identified on each node. In particular, we consider coupled networks of reaction-diffusion systems; we analyze their asymptotic behavior by searching positively invariant regions, and proving the existence of exponential attractors of finite fractal dimension, derived from energy estimates which suggest the dissipative nature of those networks of reaction-diffusion systems.Our framework includes the study of multiple applications. Among them, we consider a mathematical model for the geographical analysis of behavioral reactions of individuals facing a catastrophic event. We present the modeling choices that led to the study of this evolution problem, and its mathematical study, with a stability and bifurcation analysis of the equilibria. We highlight the decisive role of evacuation paths in coupled networks built from this model, in order to reach the expected equilibrium corresponding to a global return of all individuals to the daily behavior, avoiding a propagation of panic. Furthermore, the research of emergent periodic solutions in complex networks of oscillators brings us to consider coupled networks of hamiltonian systems, for which we construct polynomial perturbationswhich provoke the emergence of limit cycles, question which is related to the sixteenth Hilbert's problem.
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https://tel.archives-ouvertes.fr/tel-01945759
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Submitted on : Wednesday, December 5, 2018 - 3:01:08 PM
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Guillaume Cantin. Étude de réseaux complexes de systèmes dynamiques dissipatifs ou conservatifs en dimension finie ou infinie. Application à l'analyse des comportements humains en situation de catastrophe.. Mathématiques générales [math.GM]. Normandie Université, 2018. Français. ⟨NNT : 2018NORMLH16⟩. ⟨tel-01945759⟩

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