About analytic non integrability

We prove several general results on non existence of analytic first integrals for analytic diffeomorphisms possessing a hyperbolic fixed point.Comment: 14 page

Mould Calculus for Hamiltonian Vector Fields

We present the general framework of \'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \'Ecalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then prove a Kolmogorov theorem on Hamiltonian vector fields near a diophantine torus in action-angle coordinates using moulds techniques.Comment: 30 page

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 Schr\"odinger equation} via the \emph{scale Newton's equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schr\"odinger equation}

Hyperboliity versus partial-hyperbolicity and the transversality-torsion phenomenon

In this paper, we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrres of freedom Hamiltonian systems: the transversality-torsion phenomenon.Comment: 10 page

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

Generalized Euler-Lagrange equations for variational problems with scale derivatives

We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.Comment: Submitted on 03-Aug-2009; accepted for publication 16-March-2010; in "Letters in Mathematical Physics"

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, It\^o's and Stratonovich's interpretation of stochastic differential equations, improving a previous result obtained by A. Milian [A.Milian, Coll. Math. 1995] in the It\^o 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