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Jacky Cresson

Publications and source records attributed to Jacky Cresson.

At least 37 records · Page 2Linked to original sources
arXiv2016

Comments on various extensions of the Riemann-Liouville fractional derivatives : about the Leibniz and chain rule properties

Starting from the Riemann-Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated analogue of the Leibniz relation. Discussing in full generality the existence of such operator over continuous functions, we derive some obstruction Lemma which can be used to prove the triviality of some operators as long as the linearity and the Leibniz property are preserved. As an application, we discuss some properties of the Jumarie's fractional derivative as well as the local fractional derivative. We also discuss the chain rule property in the same perspective.

math.CA↗
arXiv2016

Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, wa are able to describe a very large class of nonisochronous Hamiltonian system of even degree of degree arbitrary large.

math.DS↗
arXiv2016

Discrete and continuous fractional persistence problems - the positivity property and applications

In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical ($α=1$) differential equation in the fractional case (here using fractional Caputo's derivatives) and the numerical scheme which are associated (here with discrete Grünwald-Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Grünwald-Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey-predator model introduced by Javidi and al.

math.NA↗
arXiv2016

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

This 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.

math.NA↗
arXiv2016

A continuous/discrete fractional Noether's theorem

We prove a fractional Noether's theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.

math.DS↗
arXiv2016

Helmholtz's inverse problem of the discrete calculus of variations

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

math.DS↗
arXiv2015

Multiscale functions, Scale dynamics and Applications to partial differential equations

Modeling phenomena from experimental data, always begin with a \emph{choice of hypothesis} on the observed dynamics such as \emph{determinism}, \emph{randomness}, \emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \emph{"With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \emph{multi-scale functions}, \emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale calculus} (see \cite{bohn}). These definitions will be illustrated on the \emph{multi-scale Okamoto's functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \emph{scale regimes} whereas the equation is \emph{scale invariant}. A typical example of such an equation, is the \emph{Euler-Lagrange equation} and particularly the \emph{Newton's equation} which will be discussed. Notably, we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton's equation} and also the \emph{non-linear Schrödinger equation} via the \emph{scale Newton's equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schrödinger equation}.

math-ph↗
arXiv2015

Dynamics of a rotating ellipsoid with a stochastic flattening

Experimental data suggest that the Earth short time dynamics is related to stochastic fluctuation of its shape. As a first approach to this problem, we derive a toy-model for the motion of a rotating ellipsoid in the framework of stochastic differential equations. Precisely, we assume that the fluctuations of the geometric flattening can be modeled by an admissible class of diffusion processes respecting some invariance properties. This model allows us to determine an explicit drift component in the dynamical flattening and the second zonal harmonic whose origin comes from the stochastic term and is responsible for short term effects. Using appropriate numerical scheme, we perform numerical simulations showing the role of the stochastic perturbation on the short term dynamics. Our toy-model exhibits behaviors which look like the experimental one. This suggests to extend our strategy with a more elaborated model for the deterministic part.

math-ph↗
arXiv2014

Continuous versus discrete structures I -- Discrete embeddings and ordinary differential equations

We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit three main discrete associate : the differential, integral or variational discrete embeddings which corresponds to classical numerical scheme including variational integrators.

math.NA↗
arXiv2014

Non standard finite difference scheme preserving dynamical properties

We study the construction of a non-standard finite differences numerical scheme for a general class of two dimensional differential equations including several models in population dynamics using the idea of non-local approximation introduced by R. Mickens. We prove the convergence of the scheme, the unconditional, with respect to the discretisation parameter, preservation of the fixed points of the continuous system and the preservation of their stability nature. Several numerical examples are given and comparison with usual numerical scheme (Euler, Runge-Kutta of order 2 or 4) are detailed.

math.NA↗
arXiv2014

The Sharma-Parthasarathy stochastic two-body problem

We study the Sharma-Parthasarathy stochastic two-body problem introduced by N. Sharma and H. Parthasarathy in \cite{sharma}. In particular, we focus on the preservation of some fundamental features of the classical two-body problem like the Hamiltonian structure and first integrals in the stochastic case. Numerical simulations are performed which illustrate the dynamical behaviour of the osculating elements as the semi-major axis, the eccentricity and the pericenter. We also derive a stochastic version of Gauss's equations in the planar case.

math-ph↗
arXiv2012

Validating Stochastic Models: Invariance Criteria for Systems of Stochastic Differential Equations and the Selection of a Stochastic Hodgkin-Huxley Type Model

In recent years, many difficulties appeared when taking into account the inherent stochastic behavior of neurons and voltage-dependent ion channels in Hodgking-Huxley type models. In particular, an open problem for a stochastic model of cerebellar granule cell excitability was to ensure that the values of the gating variables remain within the unit interval. In this paper, we provide an answer to this modeling issue and obtain a class of viable stochastic models. We select the stochastic models thanks to a general criterion for the flow invariance of rectangular subsets under systems of stochastic differential equations. We formulate explicit necessary and sufficient conditions, that are valid for both, Itô's and Stratonovich's interpretation of stochastic differential equations, improving a previous result obtained by A. Milian [A.Milian, Coll. Math. 1995] in the Itô case. These invariance criteria allow to validate stochastic models in many applications. To illustrate our results we present numerical simulations for a stochastic Hodgkin-Huxley model.

math.DS↗
arXiv2012

Differential, integral, and variational delta-embeddings of Lagrangian systems

We introduce the differential, integral, and variational delta-embeddings. We prove that the integral delta-embedding of the Euler-Lagrange equations and the variational delta-embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least action principle is obtained.

math.OC↗
arXiv2011

Lagrangian for the Convection-Diffusion Equation

Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients.

math.AP↗
arXiv2009

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.

math-ph↗
arXiv2008

Lagrangian structures for the Stokes, Navier-Stokes and Euler equations

We prove that the Navier-Stokes, the Euler and the Stokes equations admit a Lagrangian structure using the stochastic embedding of Lagrangian systems. These equations coincide with extremals of an explicit stochastic Lagrangian functional, i.e. they are stochastic Lagrangian systems in the sense of [Cresson-Darses, J. Math. Phys. 48, 072703 (2007]

math.AP↗