Dynamics of Nonlinear and Disordered Systems

The following sections are included:

• PREFACE

• CONTENTS

We introduce the qualitative analysis of dynamical systems (hyperbolic dynamics), centering in the order characterizing Morse-Smale dynamical systems and a simple model possessing chaos. Both class of systems are structurally stable.

A general survey of introductory elements in Fluid Dynamics is presented; this, with the purpose of providing some background for some of the lecture notes in this volume. After a brief discussion of the basic hypotheses, and a derivation of the conservation equations, the constitutive relations are introduced. Two alternative formulations for the problem of an incompressible newtonian fluid are given, using the Lagrangian description and an integral version of the equations. A guideline on the microscopic foundations of the hydrodynamic scheme is provided, as seen from the Kinetic and the Linear Response Theory points of view. Dimensional analysis is also considered, and illustrated using examples from Thermodynamics, Geometry, Quantum Mechanics, and Fluid Dynamics, in order to emphasize its wide possibilities. Exact solutions for viscous fluids are discussed through the optics of the Jeffery-Hamel flow, ending with some comments on the two central problems in the field: stability and turbulence. The lecture notes include a short review of some current research on an odd class of fluids, if they are fluids at all, for which a theory is still lacking: granular materials.

We study parametric instabilities as a canonical examples of phenomena which occur in dissipative systems driven far from equilibrium. We show that the response to parametric forcing is governed by evolution equations for the amplitudes of the weakly unstable modes in the vicinity of the bifurcation points. We next consider parametric forcing of waves on an array of coupled pendula, that displays most of the qualitative phenomena which arise in pattern forming instabilities. In particular, we show that various secondary instabilities trace back to the broken symmetries at the primary instability onset. We consider the parametric forcing of surface waves to adress the problem of pattern selection, and we show that it is possible to generate quasicrystalline order with a pattern forming instability in fluid mechanics.

In this set of lectures we describe a tool for analysing the dynamics of spatiotemporal systems referred to us as bi-orthogonal decompositions.

The starting point is to know how to extract the information about the dynamics which is available in the statistical data. To do that we build a symmetric scheme, in space and time. On one hand, the time history of the system is described as an orbit of vectors in, some Hilbert space. On the other hand, the local time series are viewed as space dependent vectors in another Hilbert space. The whole, signal is then considered as an operator U from the first to the second Hilbert space, and can be viewed as an exact dispersion relation between two characteristic spaces inside these Hilbert spaces.

In this context, space-time symmetries are pairs of unitary operators acting on such spaces which are intertwined by U. The full structure of U, as it is described by its spectral analysis is fixed by the symmetries.

On the basis of this analysis, some global quantities are defined and then used for detecting bifurcations in the dynamical regime of the problem.

At the end we present, as an interesting example, the analysis of the dynamics of coupled map lattices using a bi-orthogonal decomposition. Close to a Hopf bifurcation the method allows a description of the complexity of the dynamics on the basis of the relative positions of the characteristic spaces as well as the dynamics inside these spaces, when a parameter is changed.

In this paper the kinetics of aggregation phenomena is reviewed. The simplifications brought to the various problems by the use of scaling methods is particularly stressed. We also point out that the usual rate equation treatment of aggregation phenomena is an approximation, the validity of which is ill understood. We review various exactly solved cases in which the rate equations have been found to fail. Although these examples are usually quite removed from physical reality, the fact that they are tractable allows one to reach an understanding of the phenomena which may cause the breakdown of the mean-field appproach. It is found in a complex interplay between the build-up of correlations due to the presence of reactions and the transport mechanism which smoothes them out, but also carries their influence at ever increasing distances.

Some new theoretical approaches to deal with several aspects of topological disorder in solids are discussed. A method based on the agglomeration of units defined by the intermediate range order is explained and applied to treat the structure of borate glasses and to study the Penrose tiles. Finally, the dynamics of growth of solids is modeled by means of a non-linear mapping.

We present the elementary dynamics involved in the circular accelerators design.

In these lectures, I consider stochastic fluctuations in microscopic and mesoscopic systems. In microscopic systems (atoms, molecules, atomic nuclei) such fluctuations arise as a consequence of chaotic dynamics. In mesoscopic systems, where inelastic mean free path for electrons is larger than the system size, such fluctuations originate in disorder and/or chaotic scattering. In both cases, the complex Hamiltonian dynamics can be simulated in terms of random-matrix models having the correct symmetry properties. From such models, mean values and variances of observables can be calculated with the help of a technique that uses a generating function written as an intergral over commuting and anticommuting variables. The following examples are discussed. (i) Statistical nuclear cross-section; (ii) Chaotic quantum scattering; (iii) Conductance fluctuations in mesoscopic systems.