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Dynamique fractionnaire pour le chaos hamiltonien

Abstract : Many properties of chaotic Hamiltonian systems have been exhibited by numerical simulations but still remain not properly understood. Among various directions of research, Zaslavsky carries on an analysis which involves fractional derivatives. Even if his work is not fully formalized, his results seem promising. Fractional calculus, also used in several other fields, generalizes differential equations in order to take into account some complex phenomena. Concerning Lagrangian and Hamiltonian systems, the fractional embedding developped by Cresson provides a procedure based on the least action principle to build fractional dynamical equations. The main goal of the thesis consists in using this formalism to consolidate Zaslavsky's work. After a presentation of the fractional calculus adapted to our work, we enhance the fractional embedding by reconciling it with the causality principle and by making it dimensionally homogeneous. Once this formal framework is established we try to understand how a fractional dynamics can emerge in chaotic Hamiltonian systems, through two tracks respectively based on Stanislavsky's and Hilfer's works. The first one faces two difficulties, but the second leads to a simple dynamical model, where a fractional derivative appears when Zaslavsky's analysis is taken into account. We finally leave chaotic systems to show that thanks to the causal formulation of the fractional embedding, some classical dissipative equations reveal fractional Lagrangian structures, which could be of numerical interest.
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  • HAL Id : tel-01958537, version 1

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Pierre Inizan. Dynamique fractionnaire pour le chaos hamiltonien. Astrophysique [astro-ph]. Observatoire de Paris, 2010. Français. ⟨tel-01958537⟩

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