Modeling of stochastic equations, Applications in physics and biology
Modeling of stochastic equations, Applications in physics and biology
Fichiers
Date
2021-11-30
Auteurs
KHELOUFI Yasmina
Nom de la revue
ISSN de la revue
Titre du volume
Éditeur
Université oran1 Ahmed Ben Bella
Résumé
In this thesis, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties are stability/instability of equilibrium points and simplictic /Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. 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. Finally, we select the stochastic models to Hodgkin-Huxley Reaction-Diffusion preserving viability. Then, we project these results on a Networks represented by a graph of N neurons, adding an excitatory nonlinear coupling between neurons.
Description
Mots-clés
Stochastic differential equations, model validation, Landau-Lifshitz Equation, Itô equations, Stratonovich equations, equilibrium points, ferromagnetism, Poisson Hamiltonian, Hodgkin-Huxley, N neurons