Jacky Cresson

This work focuses on the global and partial identification problem for fractional differential equations. We provide a general numerical procedure based on global and local optimization algorithms with two refinements for biological systems that ensure solution positivity and homogeneous parameter units. The method is applied to a new fractional model of Dengue outbreak called the Fractional Homogeneous Nishiura (FHN) model, calibrated using data of newly infected people in Cape Verde. We show that our identification method yields a better fit between data and model solutions than previous approaches and that our FHN model captures the dynamics of Dengue more closely than existing systems.

We give a self contained presentation of the notion of variance of a vector field introduced by Jean Ecalle and Bruno Vallet in [8] following a previous work of Jean Ecalle and Dana Schlomiuk in [7]. 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.

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.

In this note, we give a self-contained presentation of the POAV2 theory developed by Haddon to study the alignment of π-orbitals along a non-planar molecule. The classical POAV theory is purely geometric instead of the POAV2 which includes more physics in the computations. We compare the results obtained from the POAV2 and POAV theory. We prove that the difference between the two quantities is in most of the cases negligible. As a consequence, the POAV theory is sufficient for most of the purposes concerning the description of the local π-system of a molecule.

Alain H{é}naut's results on implicit web symmetry groups are revisited in the usual framework of Lie group actions on differential equations. A link is also given between the Lie algebra of fabric symmetries and the existence of Darboux polynomials.

Alain Hénaut's results on implicit web symmetry groups are revisited in the usual framework of Lie group actions on differential equations. A link is also given between the Lie algebra of fabric symmetries and the existence of Darboux polynomials.

The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica.

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [26,27], based in previous results [11,35]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [14], and formulate the restricted calculus of variations using the discrete embedding approach [12,18]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.