Fonction périodique

Résumé

En mathématiques, une fonction périodique est une fonction qui lorsqu'elle est appliquée à une variable, reprend la même valeur si on ajoute à cette variable une certaine quantité fixe appelée période. Des exemples de telles fonctions peuvent être obtenus à partir de phénomènes périodiques, comme l'heure indiquée par la petite aiguille d'une horloge, les phases de la lune, etc. Définition thumb|La fonction sinus est périodique de période 2π. Lorsqu'une fonction est périodique, son graphe reproduit de façon répétitive n’importe quelle portion particulière de longueur une période : c'est une propriété d'invariance par translation. Par exemple, la fonction partie fractionnaire f qui associe à un nombre réel sa partie fractionnaire définie par :\forall x\in\R,\ f(x)=x-\lfloor x\rfloor Ici, \lfloor x\rfloor désigne la partie entière de x. La fonction f est périodique et de période 1. Ainsi nous avons :

Source officielle

https://fr.wikipedia.org/wiki/Fonction_p%C3%A9riodique

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Control and stabilization of the periodic fifth order Korteweg-de Vries equation

Derek Lee Smith

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in H-s(& x1d54b;), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.

2019

Chargement

In the past decade, the engineering community has conceived, manufactured and tested micro-swimmers, i.e. microscopic devices which can be steered in their intended environment. Foreseen applications range from microsurgery and targeted drug delivery to environmental decontamination. This thesis presents a mathematical analysis of the dynamics of rigid magnetic swimmers in a Stokes flow driven by an external uniform magnetic field that rotates steadily about its axis of rotation. The swimmer is assumed to be made of a permanent magnetic material and to be placed in a fluid that fills an infinite enveloping space. A specific swimmer is prescribed by its magnetic moment and mobility matrix. For a given swimmer, its dynamics depend on two parameters that can be changed during an experiment: the Mason number, related to the magnitude and angular speed of the magnetic field, and the conical angle between the magnetic field and its axis of rotation. As these two parameter vary, strikingly different regimes of responses occur. The swimmer's trajectory is entirely governed by its rotational dynamics: once its orientation dynamics are known, its position trajectory can be recovered. For neutrally buoyant swimmers, this work provides a complete classification of the steady states of the rotational dynamics, along with a study of non-steady solutions in the asymptotic limits of small and large Mason number, and small conical angle. Predicted out-of-equilibrium solutions are in good agreement with numerical simulations. Swimmer's trajectories corresponding to steady states and periodic solutions of the rotational dynamics are then recovered. Finally, the effect of buoyancy is taken into account, and the relative equilibria of swimmers with a different density than that of the fluid are determined when the axis of rotation of the magnetic field is aligned with gravity.

EPFL2019

Chargement

Blow Up Construction and Stability of Stationary Maps

Sohrab Mirshams Shahshahani

This thesis is a study of wave maps from a curved background to the standard two dimensional sphere S2. The target is always assumed to be embedded in R3 in the standard way. The do- main manifold (the "curved background") will be diffeomorphic to S2 × R, but the Lorentzian metric will not necessarily be the standard one. The current work is based on the following two results by Shatah, Tahvildar-Zadeh [47] and Krieger, Schlag, Tataru [25]. In [47] Shatah and Tahvildar-Zadeh established the existence and stability in the energy norm of a compact family of stationary (satisfying a certain periodicity condition in time) and equiv- ariant (satisfying a certain spatial symmetry) wave maps from S2 × R to S2 equipped with the standard metrics. Stability was proved under perturbations in the same spatial equivariance class. In [25], the authors construct a one parameter family of blow up solutions to the co-rotational wave maps equation from R2+1 to S2 with the standard metrics. Co-rotational means that the wave maps are assumed to have rotation number equal to one. These solutions are obtained as perturbations of the rescaled standard rotational harmonic map 2arctanr from R2 to S2. Here, we prove the conclusions of these papers for different metrics on the domains of wave maps. More specifically, our results can be grouped into two parts: stationary maps and blow ups. In the first part following the method of [25], we construct a one parameter family of co-rotational blow up wave maps from S2 ×R to S2. These solutions can be parameterized by ν ∈ ( 1 , 1]. The metric on the domain sphere is taken to be the standard one. 2 In the second part, we consider wave maps from S2 × R to S2 where the metric on the do- main S2 is now taken to be a general SO(1)−symmetric metric of the form dr2 + f 2(r)dθ2. The special case f (r ) = sin r corresponds to the standard metric considered by Shatah and Tahvildar-Zadeh. We prove the same stability result as in [47] in this general setting.

EPFL2012

Chargement

Blow Up Construction and Stability of Stationary Maps

Sohrab Mirshams Shahshahani

EPFL2012