ECTS Credits: 6
Laurette Tuckerman (EPSCI), Stephan Fauve (ENS Paris)
Most problems in dynamics encountered in physics or in other fields
are governed by nonlinear differential equations. In contrast to
linear equations, they usually display multiple solutions with
different qualitative characteristics, often different symmetries. We
study the bifurcations, i.e. the transitions, between these different
solutions when a parameter of the system is varied. We show that the
dynamics in the vicinity of these bifurcations is governed by
universal equations called normal forms that mostly depend on the
broken symmetries at the transition. We emphasize the analogy with
phase transitions, but also point out differences such as limit cycles
or chaotic behaviors which do not occur at equilibrium.
The first part of the course concerns bifurcation theory for maps and ordinary
differential equations and an introduction to pattern-forming
instabilities and reaction-diffusion equations. Nonlinear waves and
solitons as well as instabilities in spatially extended systems are
considered in the second part of the lectures, mostly using the
concept of amplitude equations which is also applied to problems in
condensed-matter physics such as commensurate-incommensurate
transitions, magnetic domains and superconductivity.
Through these lectures, our aim is to show that symmetry arguments together with a
qualitative analysis of differential equations and the use of
perturbation techniques provide tools that can be used to understand
many phenomena in various fields of physics and elsewhere.
• Dynamical systems: stationary bifurcations, spectra of matrices,
Hopf bifurcations, global bifurcations.
• Convection and Lorenz model: Rayleigh-Benard convection, linear stability analysis for idealized case, derivation and behavior of
Lorenz model, real-world convection behavior
• Symmetry: reflection, rotation, groups
• Maps and period doubling: fixed points and iteration, steady bifurcations, period-doubling bifurcations
• Floquet analysis: theory, Faraday instability, cylinder wake.
• Stripes, patterns and instabilities : Swift-Hohenberg and Newell Whitehead, squares and hexagons, Eckhaus and zig-zag instabilities.
• Reaction-diffusion equations: overview, excitable systems, Turing
patterns, spatial analysis.
• Waves in a nonlinear and dispersive medium: nonlinear Schrodinger
equation, Benjamin Feir instability, solitary waves, bifurcations in conservative systems.
• Nonlinear waves: canonical nonlinear wave equations in different
fields of physics, shock waves, solitons, methods for exact solutions.
• Amplitude equations for pattern-forming instabilities: stationary or Hopf bifurcations in an extended domain, symmetry arguments,potential versus conservative dynamics.
• The amplitude equation formalism applied to some topics
from condensed-matter physics: commensurate-incommensurate
transitions, magnetic domains, superfluids, superconductivity.
• Broken symmetries and collective modes: neutral modes related
to spontaneously broken symmetries, phase dynamics of patterns, effet
of phase modes on secondary instabilities, interaction of phase modes.
• Topological defects: kinks and their dynamics, Ising versus Bloch
walls and the effect of chirality on their dynamics, vortices or dislocations, spatiotemporal chaos mediated by defect dynamics.
Bibliography:
• P. G. Drazin, Nonlinear systems, (Cambridge University Press, 1992).
• P. Manneville, Dissipative structures and weak turbulence, (Academic
Press, 1990).
• S. Strogatz, Nonlinear Dynamics and Chaos, (Westview Press, 1994).
• S. Fauve, Pattern-forming instabilities, in Hydrodynamics and nonlinear instabilities, edited by C. Godreche and P. Manneville (Cambridge University Press, 1998) https://darchive.mblwhoilibrary.org/handle/1912/802.
• R. Hoyle, Pattern Formation. An Introduction to Methods, (Cambridge
University Press, 2006).