Strange Beauty: The Lorenz Attractor

The Lorenz oscillator gives one of the most famous images in mathematics – the Lorenz Attractor in dynamic systems

This must be one of the most beautiful images in mathematics. Meterologist, Edward Lorenz discovered it by chance in 1961 while running computer simulations to study atmospheric convection and weather patterns. When he reran an earlier numerical solution with what he thought was identical initial data, he got wildly different results. Lorenz eventually traced the divergence to tiny differences in the starting values of some of the system’s parameters. Two years later, he published his findings in a journal where he explicitly showed how a simple linear model could produce highly complex nonlinear chaotic behavior. Chaos theory is now one of the big success stories of 20th century mathematics and science.

I first encountered the Lorenz Attractor while enrolled in an applied math course in 1980s and was enthralled by it for weeks. The trademark of a chaotic system is extreme sensitivity to initial conditions, famously expressed by the metaphor of a butterfly which flaps its wings in Brazil and ended up stirring a tornado in Texas. There are now dozens of programs on the Internet which let users tweak initial values and watch the attractor’s “butterfly wings” flap and twist wildly on the screen, like a choreographed sequence of dance moves. See for example,

The word “attractor” has a special meaning in mathematics. Briefly, a set A is attracting if it has a neighborhood U such that
{\displaystyle \bigcap \limits_{i=0}} f_t (U)
where f_t (x) is the solution of the dynamic system (defined by a set of ordinary differential equations or ODE) at time t with initial condition x. That is, U flows towards the attracting set A. The Lorenz Attractor is a so-called Strange Attractor because one never knows exactly where on the attractor the system will be. Two points on the attractor can be near each other at one time, yet be arbitrarily far apart at later times. The only restriction is that the state of system remain on the attractor. Strange attractors never close on themselves which means they never settle to a steady state. The system is in perpetual motion and never repeats, i.e., is non-periodic [1].

Weather is a good example of a chaotic system (Lorenz was a meteorologist after all). The fickleness of weather has taught meteorologists not to rely on deterministic equations to model weather patterns. Financial markets too, have some semblance of chaotic systems in their tendency to overreact to news and sentiment, which makes them difficult to predict. The universal message of Chaos Theory echoes that of the Uncertainty Principle in Quantum Mechanics: when one lacks precision on the initial conditions of a system, any uncertainty will be amplified, predictions will go awry and all bets should be off.

From a recreation angle, you may wonder whether the “dance moves” of the Lorenz Attractor can be used to choreograph real dance moves. This idea was explored by computer scientists, Elizabeth Bradley and Joshua Stuart from the University of Colorado at Boulder in a recent study [2]. They designed a technique that maps the dynamics of a chaotic attractor to symbols representing the body positions of a dance piece. The results (below, middle and bottom panel) are what you might call a sequence of chaotic dance moves. Their similarity to real dance moves (top panel) is striking. Mathematical chaos, it seems, can be artistically sublime.

A ballet jump. The top panel : the original sequence, middle panel: a chaotic variation and the bottom panel: an interpolated version of that variation.

Technical Notes
[1] In 1998, Fields Medalist Stephen Smale in his list of 18 “Mathematical Problems for the Next Century made his 14th problem, a rigorous proof that the dynamics of the Lorenz oscillator as defined as a set of deterministic differential equations is a strange attractor with various additional properties that make it a “geometric Lorenz attractor”. The problem was solved by Australian mathematician, Warwick Tucker in 2002 by computer using an ingenious application of the classic Euler method for solving ODEs. For that work, Tucker was awarded the Moore Prize in 2002 and the EMS Prize in 2004.

[2] Bradley and Stuart used a combination of computational geometry techniques (in particular, the Voriono diagram) and rigid body dynamics in mechanics in their work. For full details, see their paper, “Using Chaos to Generate Choreographic Variations”,  Proceedings of the Experimental Chaos Conference, Boca Raton FL; August 1997.

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