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The complex Ginzburg-Landau equation is one of the most-studied equations in applied mathematics. It describes qualitatively, and often quantitatively, a vast array of phenomena including nonlinear waves, second-order phase transitions, Rayleigh-Bénard convection and superconductivity. The equation describes the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation, for which a continuous spectrum of unstable wavenumbers is taken into account. It can be viewed as a highly general normal form for a large class of bifurcations and nonlinear wave phenomena in spatially extended systems.

In this tutorial, a broad overview of the behaviour of the equation is given, with a focus on the one-dimensional case. The linear stability problem of plane wave solutions to the equation is expounded and the analysis is complemented with a graphical representation of the various observed behaviours. Two MATLAB programs are also provided, which simulate the 1D and 2D versions of the complex Ginzburg-Landau equation, allowing the reader to verify the results presented here and to conduct their own exploration of the equation.

IP1: Collapse of the Atlantic Ocean Circulation, Henk Dijkstra, Utrecht University, The Netherlands

IP2: Dynamics, Instability, and Bifurcation in the Mechanics of Biological Growth, Alain Goriely, University of Arizona

IP3: Network Topology: Sensors and Systems, Robert W. Ghrist, University of Pennsylvania

IP4: Mechanisms of Instability in Nearly Integrable Hamiltonian Systems, Tere M. Seara, Universidad Politecnica de Catalunya, Spain

IP5: Analysis of Large-Scale Interconnected Dynamical Systems, Igor Mezic, University of California, Santa Barbara

IP6: The Multiscale Dynamics of Lightning and a Moving Boundary Problem, Ute Ebert, Centrum voor Wiskunde en Informatica (CWI), Netherlands

IP7: Systems Biology: How Dynamical Systems Theory Can Help to Understand the Basis of Life, Frank Allgöwer, University of Stuttgart, Germany

IP8: Stochasticity in Deterministic Systems, Ian Melbourne, University of Surrey, United Kingdom

IP9: Living on the Edge of Noise-Driven Order, Rachel Kuske, University of British Columbia, Canada

IP10: The Fluid Trampoline: Droplets Bouncing on a Soap Film, John Bush Massachusetts Institute of Technology

In these lectures we consider, first, the continuation of solutions (to any general problem posed by the objects we are looking for) when they depend on some parameter. Then, the corresponding analysis of bifurcations is presented when the differential of the function determining the solutions has non-maximal rank.

After a quick review on fixed points and their stability and on numerical integrators, the computation of Poincaré maps and their differentials is presented. This is used for the computation of periodic orbits, their stability and continuation. Some methods to compute also quasi-periodic orbits are given.

As indicators of the behaviour of general orbits we stress on the computation of Lyapunov exponents, warning about the correct interpretation of what is really computed.

Concerning invariant manifolds, it is useful to have good local analytic approximations. To this end some symbolic manipulation can be required. This is simple close to fixed points. Near periodic orbits or invariant tori it can pose more difficulties, but the general principle is always the same: to ask for invariance. Having a local approximation at hand we can globalize the manifolds numerically. Finally, knowing how to compute invariant manifolds, the computation of homoclinic and heteroclinic points, their tangencies, and the variation with respect to parameters is shown to be a relatively simple problem. The formulations are presented in general, and several examples illustrate a sample of topics.

Reprinted from "Les Méthodes Modernes de la Mecénique Céleste" (Course given at Goutelas, France, 1989), D. Benest and C. Froeschlé (eds.), pp. 285--329, Editions Frontières, Paris, 1990.

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