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Jacky Cresson

Publications and source records attributed to Jacky Cresson.

At least 19 recordsLinked to original sources
arXiv2026

Dynamics of Stochastic Hamiltonian systems I - Wong-Zakai Variational integrators

We construct variational integrators to study the dynamics of stochastic Hamiltonian systems using their Wong-Zakai approximation. This approach can be used to interpret and justify previous work of Wang et al (L. Wang, J. Hong, R. Scherer, F. Bai. Dynamics and variational Integrators of stochastic Hamiltonian systems. International Journal of Numerical Analysis and Modeling, 6(4), 2009) on stochastic variational integrators. Numerical examples are provided.

math.DS↗
arXiv2026

Variance of vector fields -- Definition and properties

We give a self contained presentation of the notion of variance of a vector field introduced by Jean Ecalle and Bruno Vallet in \cite{ev} following a previous work of Jean Ecalle and Dana Schlomiuk in \cite{es}. We give complete proofs and definitions of various results stated in these articles. Following J. Ecalle and D. Schlomiuk, We illustrate the interest of the variance by giving a complete proof of the formulas for the mould defining the nilpotent part of a resonant vector field.

math.DS↗
arXiv2023

From fractal R-L ladder networks to the diffusion equation

We give a self-contained presentation of fractal R-L ladder networks as well as a detailed computation of the admittance of these systems. We also discuss the conditions under which such systems display a fractional behavior. Finally, we give a full discussion of the connection existing between fractal R-L network and the diffusion equation.

math-ph↗
arXiv2022

Mid-point embedding of Hamiltonian systems and variational integrators

Following the discrete embedding formalism, we give a new derivation of the mid-point variational integrators as developed by J.M. Wendlandt and J.E. Marsden by defining an adapted order two discrete differential and integral calculus. This allows us to obtain a clearer correspondence between the discrete and continuous case. We also discuss the corresponding definition of a discrete Hamiltonian system. A complete comparaison with the results of J.M. Wendlandt and J.E. Marsden is provided.

math.DS↗
arXiv2022

On the equality of periods of Kontsevich-Zagier

Effective periods were defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of $\mathbb{Q}$-rational functions over $\mathbb{Q}$-semi-algebraic domains in $\mathbb{R}^d$. The Kontsevich-Zagier period conjecture states that any two different integral expressions of a period are related by a finite sequence of transformations only using three rules respecting the rationality of functions and domains: integral addition by integrands or domains, change of variables and Stokes' formula. In this paper, we introduce two geometric interpretations of this conjecture, seen as a generalization of Hilbert's third problem involving either compact semi-algebraic sets or rational polyhedra equipped with piece-wise algebraic forms. Based on known partial results for analogous Hilbert's third problems, we study possible geometric schemes to prove this conjecture and their potential obstructions.

math.NT↗
arXiv2021

Generation of Helical states -- breaking of symmetries, Curie's principle, and excited states

Following previous work of M.H. Garner, R. Hoffmann, S. Rettrup and G.C. Solomon, we discuss the generation of helical molecular orbitals (MOs) for linear chains of atoms. We first give a definition of helical MOs and we provide an index measuring how far a given helical states is from a perfect helical distribution. Structural properties of helical distribution for twisted $[n]$-cumulene and cumulene version of Möbius systems are given. We then give some simple structural assumptions as well as symmetry requirements ensuring the existence of helical MOs. Considering molecules which do not admit helical MOs, we provide a first way to induce helical states by the breaking of symmetries. We also explore an alternative way using excited conformations of given molecules as well as different electronic multiplicities. Several examples are given.

physics.chem-ph↗
arXiv2020

Stochastic modication of Newtonian dynamics and Induced potential -application to spiral galaxies and the dark potential

Using the formalism of stochastic embedding developed by [J. Cresson, D. Darses, J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two denitions of a stochastic Newton's equation called dierential and variational. We rst prove a stochastic virial theorem which is a natural generalization of the classical case. The stochasticity modies the virial relation by adding a potential term called the induced potential which corresponds in quantum mechanics to the Bohm potential. Moreover, the dierential stochastic Newton equation naturally provides an action functional which sat-ises a stochastic Hamilton-Jacobi equation. The real part of this equation corresponds to the classical Hamilton-Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic processes solutions of the stochastic Newton equation. It is proved that this density satises a nonlinear Schr{ö}dinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad-hoc ''dark potential'' used to recover a at rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by [D. Da Rocha and L. Nottale, Chaos, Solitons and Fractals, 16(4):565-595, 2003] where the emergence of the ''dark potential'' is seen as a consequence of the fractality of space in the context of the Scale relativity theory.

math.DS↗
arXiv2018

Selection of a Stochastic Landau-Lifshitz Equation and the Stochastic Persistence Problem

In this article, we study the persistence of properties of a given classical deter-ministic dierential equation under a stochastic perturbation of two distinct forms: external and internal. The rst case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic dierential equations. The second case corresponds to consider a parameters dependent dierential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary dierential equations. Our main concerns for the preservation of properties is stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically dierent. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss in particular previous results obtain by Etore and al. in [P. 'Etore, S.Labbe, J. Lelong, Long time behaviour of a stochastic nanoparticle, J. Differential Equations 257 (2014), 2115-2135] and we nally propose a new family of stochastic Landau-Lifshitz equations.

math.DS↗
arXiv2018

A stochastic invariantization method for Itô stochastic perturbations of differential equations

In general, adding a stochastic perturbation to a differential equation possessing an invariant manifold destroys the invariance as far as the Itô formalism is used. In this article, we propose an invariantization method for perturbations in the Itô case which can be used to restore invariance. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss in particular previous results obtained by Etore and al. in [P. Étoré, S.Labbé , J. Lelong, Long time behaviour of a stochastic nanoparticle, J. Differential Equations 257 (2014), 2115-2135].

math-ph↗
arXiv2018

About the Noether's theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in Appl. Math. Comp. 217,3,2010 was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in JMAA 429, 2, 2015 using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular(JMAA 334, 2007,Theorem 32). However, the counterexample does not explain why and where the proof given in Appl. Math. Comp. 217,3,2010 does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in JMAA 334, 2007 which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in Appl. Math. Comp. 217,3,2010. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost's method, in particular in the discrete setting. We then derive a fractional Noether's Theorem following this strategy, correcting the initial statement of Frederico and Torres in JMAA 334, 2007 and obtaining an alternative proof of the main result of Atanackovic and al. in Nonlinear Analysis 71, 2009.

math.OC↗
arXiv2018

Discrete embeddings for Lagrangian and Hamiltonian systems

The general topic of the present paper is to study the conservation for some structural property of a given problem when discretising this problem. Precisely we are interested with Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Considering a partial differential equation (PDE) deriving from such a variational principle, a natural question is to know whether this structure at the continuous level is preserved at the discrete level when discretising the PDE. To address this question a concept of \textit{coherence} is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. In case these procedures finally provide the same discrete problem we will say that the discretisation is \textit{coherent}. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations (finite elements, finite differences and finite volumes). Hamiltonian structures are shown to provide coherence between a discrete Hamiltonian structure and the discretisation of the mixed formulation of the PDE, both for mixed finite elements and mimetic finite differences methods.

math.NA↗
arXiv2017

Lie algebras and geometric complexity of an isochronous center condition

Using the mould formalism introduced by Jean Ecalle, we define and study the geometric complexity of an isochronous center condition. The role played by several Lie ideals is discussed coming from the interplay between the universal mould of the correction and the different Lie algebras generated by the comoulds. This strategy enters in the general program proposed by J. Ecalle and D. Schlomiuk in \cite{es} to study the size and splitting of some Lie ideals for the linearisability problem.

math.DS↗
arXiv2016

A Time scales Noether's theorem

We prove a time scales version of the Noether's theorem relating group of symmetries and conservation laws. Our result extends the continuous version of the Noether's theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres in \cite{BT}.

math-ph↗
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↗