Adiabatic dynamical systems and hysteresis
This work is dedicated to the study of Dynamical Systems, depending on a slowly varying parameter. It contains, in particular, a detailed analysis of memory effects, such as hysteresis, which frequently appear in systems involving several time scales. In a first part of this dissertation, we develop a mathematical framework to deal with adiabatic differential equations. We do this, whenever possible, by favouring the geometrical 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 calculations. We begin by analysing one-dimensional adiabatic systems of the form εẋ = f(x,τ). We first show existence of adiabatic solutions, which remain close to equilibrium branches of the system, and admit asymptotic series in the adiabatic parameter ε. We then provide a method to analyse solutions near bifurcation points, and show that they scale in a nontrivial way with ε, with an exponent that can be easily computed. The analysis is concluded by examining global properties of the flow, in particular existence of hysteresis cycles. These results are then extended to the n-dimensional case. The discussion of adiabatic solutions carries over in a natural way. The dynamics of neighboring solutions is, however, more difficult to analyse. We first 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 first discuss the dynamics of some low-dimensional nonlinear oscillators. 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 Poincaré map. As a second application, we analyse a few models of ferromagnetism. Starting from a lattice model with stochastic spin flip dynamics, we show how to derive a deterministic equation of motion of Ginzburg-Landau type, in the case of infinite range interactions and in the thermodynamic limit. We analyse the influence of dimensionality and interaction anisotropy on shape and scaling properties of hysteresis cycles. A few simple approximations to the dynamics of an Ising model are also discussed. We conclude this work by extending some properties of adiabatic differential equations to iterated maps. We give some results on existence of adiabatic invariants for near-integrable slow-fast maps, and apply them to billiards.