— This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right… (More)
— We developp a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non differentiable analogue of the classical derivative. We then define the notion… (More)
We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general… (More)
This paper is twofold. In a first part, we extend the classical differential calculus to continuous non differentiable functions by developping the notion of scale calculus. The scale calculus is based on a new approach of continuous non differentiable functions by constructing a one parameter family of differentiable functions f (t, ǫ) such that f (t, ǫ) →… (More)
— Most physical systems are modelled by an ordinary or a partial differential equation, like the n-body problem in celestial mechanics. In some cases, for example when studying the long term behaviour of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modelled or even known. One… (More)
Let N be a smooth manifold and f : N → N be a C ℓ , ℓ ≥ 2 diffeomorphism. Let M be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the λ-lemma in this case.
— Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian space R n which can be viewed as the basic bricks to construct irregular objets. They are obtain by taking… (More)