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

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

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

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

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.

Lagrangian Coherent Structures: Analysis of time-dependent dynamical systems using finite-time Lyapunov exponents

Third Prize, DSWeb Tutorials Contest

This tutorial explains the application of finite-time Lyapunov exponents (FTLE) for studying time-dependent dynamical systems. The emphasis here is on dynamical systems with arbitrary time dependence, since there is already a nice repertory of tools to tackle time-independent and time-periodic systems. A leading source for time-dependent dynamical systems are fluid flow problems. So while the ideas stated in this tutorial are expressed in terms of a general dynamical system, we often assume that the system represents a fluid flow. The evolution of such systems is often governed by partial differential equations, yet it is often acceptable to represent such systems by ordinary differential equations when interest is on large scale transport. This is typically accomplished by either numerically solving an approximation of the Navier-Stokes equation, or taking direct measurements of the fluid. In either case, one typically ends up with with a discrete set of velocity data which represents the vector field of the dynamical system. Therefore, we also emphasize that the given dynamical system might only be known over a finite time-interval.

This tutorial is intended for readers that have had some exposure to dynamical systems concepts, however the presentation is somewhat informal and hopefully easily accessible. Those that have taken a course, or are currently taking a course, in dynamical systems are well-suited to read this tutorial.

In the tutorial, we show that finite-time Lyapunov exponents can be used to find separatrices in time-dependent systems, which are often analogous to stable and unstable manifolds of time-independent systems. These separatrices are called Lagrangian Coherent Structures (LCS). These structures divide dynamically distinct regions in the flow and reveal geometry which is often hidden when viewing the vector field or even trajectories of the system. Therefore these structures often provide a nice tool in analyzing systems with general time-dependence, especially for understanding transport.

Sections 2 through 6 are mostly dedicated to the theoretical development of the FTLE and LCS. In Section 7, we demonstrate these concepts on a few examples. Section 8 overviews the algorithmic computation of FTLE fields, and Section 9 provides access to software that can be used to perform these computations. In particular, we specify the steps needed to reproduce results shown in Section 7.3. References are located throughout the tutorial where needed, however Section 10 contains a more coherent discussion of works related to the ideas presented in this tutorial.

Excitable Media (Java Applets)

Examples of Excitable media in 0D, 1D and 2D with emphasis to cardiac dynamics. The page contains more than 40 Java Applets dedicated to teach the origen of excitability, as well as the dynamics and stability of waves in 1D and spiral waves in 2D.

Crisis-induced Intermittency in Coupled Chaotic Maps

Honorable Mention, DSWeb Tutorials Contest

Intermittent transitions between multiple dynamical states are characteristic nonlinear phenomena in dynamical systems. It is important to understand a mechanism of an onset of intermittency in mathematical models, because it often replicates an observable phenomenon in physical world. Among several types of intermittency, we focus on crisis-induced intermittency in this tutorial.

In coupled chaotic maps which are widely used in modeling networks of dynamical elements, coexistence of multiple attractors is not uncommon. As a system parameter is varied, such a system often exhibits a sudden crisis inducing intermittent behaviors. We demonstrate that a crisis can be understood by a contact between attractors and a fractal basin boundary as well as an emergence of a snap-back repeller in two coupled chaotic maps. The scenario for the crisis is also illustrated with qualitative changes in basin structure and quantitative changes of fractal dimension of the basin boundary.

The approaches to the crisis is successfully applied to an analysis of a global bifurcation inducing itinerant memory dynamics in a chaotic neural network. In addition to the viewpoints for crisis-induced intermittency, this tutorial provides several basic concepts such as invertible and non-invertible maps, smooth and fractal basin boundaries, fractal dimension, and basin bifurcations in discrete-time dynamical systems.

Community of Ordinary Differential Equations Educators

The Community of Ordinary Differential Equations Educators (CODEE) seeks to improve the teaching and learning of ordinary differential equations. One way to do this is to increase student engagement and active learning is through the use of projects involving modeling and computer experiments. So CODEE, with the support of the National Science Foundation (1992 – 97 and 2008 – 2013), has constructed a digital library which supports this purpose (www.codee.org). The digital library contains the CODEE Journal which contains articles about the teaching and learning of ODEs, as well as classroom-ready projects, and reviews of other teaching materials. The CODEE web site also describes some software that supports the teaching and learning of ODEs, as well as archives of the newsletters and other materials produced during the first NSF grant. The ultimate goal of the digital library is to create a community for instructors to find, share, and discuss resources for teaching ordinary differential equations. To learn more about submitting a manuscript to the CODEE digital library, visit the Editorial Policy page of the CODEE web site.