AdiAbAtic dynAmicAl SyStemS And HyStereSiS

This work is dedicated to the study of Dynamical Systems depending on a slowly vary ing parameter It contains in particular a detailed analysis of memory e ects such as hysteresis which frequently appear in systems involving several time scales In a rst part of this dissertation we develop a mathematical framework to deal with adiabatic di erential equations We do this whenever possible by favouring the geomet rical approach to the theory which allows to derive qualitative properties of the dynamics such as existence of hysteresis cycles and scaling laws with a minimum of analytic calcu lations We begin by analysing one dimensional adiabatic systems of the form x f x We rst show existence of adiabatic solutions which remain close to equilibrium branches of the system and admit asymptotic series in the adiabatic parameter We then provide a method to analyse solutions near bifurcation points and show that they scale in a nontrivial way with with an exponent that can be easily computed The analysis is concluded by examining global properties of the ow in particular existence of hysteresis cycles These results are then extended to the n dimensional case The discussion of adi abatic solutions carries over in a natural way The dynamics of neighboring solutions is however more di cult to analyse We rst provide a method to diagonalize linear equations dynamically and show that eigenvalue crossings lead to similar behaviours than bifurcations We then introduce some methods to deal with nonlinear terms in particular adiabatic manifolds and dynamic normal forms In a second part of this work we apply the previously developed methods to some selected examples We rst discuss the dynamics of some low dimensional nonlinear oscil lators In particular we present the example of a damped pendulum on a table rotating with a slowly oscillating angular frequency This system displays chaotic motion even for arbitrarily small adiabatic parameter This phenomenon is explained by computing an asymptotic expression of the Poincar e map As a second application we analyse a few models of ferromagnetism Starting from a lattice model with stochastic spin ip dynamics we show how to derive a deterministic equation of motion of Ginzburg Landau type in the case of in nite range interactions and in the thermodynamic limit We analyse the in uence of dimensionality and interaction anisotropy on shape and scaling properties of hysteresis cycles A few simple approxima tions to the dynamics of an Ising model are also discussed We conclude this work by extending some properties of adiabatic di erential equations to iterated maps We give some results on existence of adiabatic invariants for near integrable slow fast maps and apply them to billiards