Ondes localisées dans des systèmes mécaniques discrets excitables .

Keywords :

• ondes solitaires
• fronts
• équations différentielles sur réseau
• systèmes mécaniques excitables
• modèle de Burridge-Knopoff
• frottement non linéaire
• équations différentielles avec avance-retard
• systèmes dynamiques linéaires par morceaux
• extinction de propagation

This thesis analyses localized travelling waves for some classes of non-linear lattice differential equations describing excitable mechanical systems. These systems correspond to an infinite chain of blocks connected by springs and sliding on a surface in the presence of a nonlinear velocity-dependent friction force. We investigate both the Burridge-Knopoff model (with blocks attached to springs pulled at constant velocity) and a chain of free blocks sliding on an inclined plane under the effect of gravity. For a class of non-monotonic friction functions, both systems display a large response to perturbations above a threshold, one of the main properties of excitable systems. This response induces the propagation of either solitary waves or fronts, depending on the model and parameter regime. We study these localized waves numerically and theoretically for a broad range of friction laws  and parameter regimes, which leads to the analysis of nonlinear advance-delay differential equations. Phenomena of propagation failure and oscillations of the travelling wave profile are also investigated. The introduction of a piecewise linear friction function allows one to construct localized waves explicitly in the form of oscillatory integrals and to analyse some of their properties such as shape and wave speed. An existence proof for solitary waves is obtained for the excitable Burridge-Knopoff model in the weak coupling regime.

Directors:

• Mr Arnaud Tonnelier (Chargé de recherche - INRIA )
• Mr Guillaume James (Professeur - Grenoble INP )

Raporteurs:

• Mr Jean-Pierre Françoise (Professeur - Université Pierre Marie Curie )
• Mme Anna Vainchtein (Professeure - Université de Pittsburgh )

Examinators:

• Mr Bruno Lombard (Chargé de recherche - CNRS, LMA, Marseille )
• Mr Michael Herrmann (Professeur - Université de Münster )
• Mr Hidde de Jong (Directeur de recherche - INRIA )