Fractional embeddings and stochastic time

As a model problem for the study of chaotic Hamiltonian systems, we look for the effects of a long-tail distribution of recurrence times on a fixed Hamiltonian dynamics. We follow Stanislavsky's approach of Hamiltonian formalism for fractional systems. We prove that his formalism can be retrieved from the fractional embedding theory. We deduce that the fractional Hamiltonian systems of Stanislavsky stem from a particular least action principle, said causal. In this case, the fractional embedding becomes coherent.Comment: 11 page

Irreversibility, least action principle and causality

The least action principle, through its variational formulation, possesses a finalist aspect. It explicitly appears in the fractional calculus framework, where Euler-Lagrange equations obtained so far violate the causality principle. In order to clarify the relation between those two principles, we firstly remark that the derivatives used to described causal physical phenomena are in fact left ones. This leads to a formal approach of irreversible dynamics, where forward and backward temporal evolutions are decoupled. This formalism is then integrated to the Lagrangian systems, through a particular embedding procedure. In this set-up, the application of the least action principle leads to distinguishing trajectories and variations dynamical status. More precisely, when trajectories and variations time arrows are opposed, we prove that the least action principle provides causal Euler-Lagrange equations, even in the fractional case. Furthermore, the embedding developped is coherent.Comment: 14 page

Fractal traps and fractional dynamics

Anomalous diffusion may arise in typical chaotic Hamiltonian systems. According to G.M. Zaslavsky's analysis, a description can be done by means of fractional kinetics equations. However, the dynamical origin of those fractional derivatives is still unclear. In this talk we study a general Hamiltonian dynamics restricted to a subset of the phase space. Starting from R. Hilfer's work, an expression for the average infinitesimal evolution of trajectories sets is given by using Poincar\'{e} recurrence times. The fractal traps within the phase space which are described by G.M. Zaslavsky are then taken into account, and it is shown that in this case, the derivative associated to this evolution may become fractional, with order equal to the transport exponent of the diffusion process

Homogeneous fractional embeddings

Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retain homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.Comment: 14 page

Variational integrators of fractional Lagrangian systems in the framework of discrete embeddings

International audienceThis paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a variational structure called Lagrangian structure. We are specially interested in the conservation at the discrete level of this Lagrangian structure by discrete embeddings. We then replace in this framework the variational integrators developed in [10, Chapter VI.6] and in [12]. Finally, we extend the notion of discrete embeddings and variational integrators to fractional Lagrangian systems

Variational integrator for fractional Euler–Lagrange equations

International audienceWe extend the notion of variational integrator for classical Euler-Lagrange equations to the fractional ones. As in the classical case, we prove that the variational integrator allows to preserve Noether-type results at the discrete level

Dynamique fractionnaire pour le chaos hamiltonien

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.De nombreuses caractéristiques des systèmes hamiltoniens chaotiques, notamment mises en évidence à l'aide de simulations numériques, restent encore mal comprises. Parmi différentes pistes de recherches, Zaslavsky propose une analyse de ces systèmes à l'aide de dérivées fractionnaires. Même si son travail n'est pas complètement formalisé, ses résultats semblent prometteurs. Le calcul fractionnaire permet de généraliser des équations différentielles afin de prendre en compte certains phénomènes complexes. Pour les systèmes lagrangiens et hamiltoniens, le plongement fractionnaire développé par Cresson fournit une procédure basée sur le principe de moindre action pour construire des équations fractionnaires de la dynamique. L'objectif principal de cette thèse est d'utiliser ce formalisme afin de consolider le travail de Zaslavsky. Après avoir présenté quelques éléments sur le calcul fractionnaire, nous enrichissons le plongement fractionnaire en le conciliant avec le principe de causalité et en le rendant dimensionnellement homogène. Nous tentons ensuite de comprendre comment peut émerger une dynamique fractionnaire dans les systèmes hamiltoniens chaotiques, à travers deux pistes respectivement basées sur les travaux de Stanislavsky et Hilfer. Si la première reste problématique, la seconde se concrétise en un modèle simple de dynamique où une dérivée fractionnaire apparaît lorsqu'est prise en compte l'analyse de Zaslavsky. Enfin, en marge de l'étude de ces systèmes, nous montrons que la formulation causale du plongement permet de doter certaines équations dissipatives de structures lagrangiennes fractionnaires, ouvrant ainsi la voie à de nouvelles modélisations numériques

Dynamique hamiltonienne fractionnaire

International audienc

Dynamique hamiltonienne fractionnaire

International audienc