The Importance of Mathematics by Timothy Gowers

The Importance of Mathematics by Timothy Gowers

This is the general audience talk on "The Importance of Mathematics" by Timothy Gowers presented at The Millennium Meeting (2000) A celebration of the universality of Mathematical thought in Paris.

The complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation

Second Prize, DSWeb Tutorials Contest

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.


Slides of Invited Presentations, SIAM Conference on Applications of Dynamical Systems, Snowbird Utah, 2009

Slides of Invited Presentations, SIAM Conference on Applications of Dynamical Systems, Snowbird Utah, 2009

Each talk is in the supplementary files below. For abstracts, see the conference program.

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


On the Analytical and Numerical Approximation of Invariant Manifolds

On the Analytical and Numerical Approximation of Invariant Manifolds

The study of Dynamical Systems and, in particular, Celestial Mechanics, requires a combination of analytical and numerical methods. Most of the relevant objects in the phase space can be found as solutions of equations, either in the phase space itself or in a suitable functional space (which is approximated by a finite-dimensional truncation in numerical computations).

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.


Mathematics in the Wind

Mathematics in the Wind

In any sport or human endeavor, coaches regularly state "play to your strengths." One might not guess that a land-locked, mountainous country like Switzerland would have strengths that would give them a chance at winning the oldest, most competitive sailing competition in the world, the America's Cup. But it does: Switzerland has mathematics! The Swiss yacht Alinghi, two-time winner of the America's Cup (2003 and 2007).

In 2003, in the Harukai Gulf of New Zealand, a Swiss yacht called Alinghi was the surprise winner of the America's Cup. And in 2007, in the Mediterranean sea near Valencia, Alinghi confirmed its dominance by defeating the New Zealand Team 5 to 2 in a breathtaking final match race.

Mathematics and computational science were involved in Alinghi's first successes in 2003, when it won the Louis Vuitton Cup and then the America’s Cup by defeating the defending Black Magic Team of New Zealand. Mathematicians helped create the design of the boat which brought the defender Alinghi its second triumph in Valencia in 2007. Mathematical models were used in the design phase and again during the competition.

Of course mathematical models alone are not enough to guarantee success in a race as competitive as the America's Cup: a great team and much luck are needed. We know that people and explorers have been sailing literally since the stone age. It may seem astonishing that, with all the technological progress that has been achieved, sailing is still such a challenge. Alinghi has shown that part of the progress that will likely be made in the future rests on mathematical models and their numerical solution.

This tutorial appears on Why Do Math.


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