James Gleick

Chaos: Making a New Science


Chapter 5. Strange Attractors

Turbulence: eddies within eddies at all scales. Unstable, highly dissipative. Arises in smooth flow in smooth pipe.

Laminar (pre-turbulent) flow already handled in 19th c. Little progress beyond that despite attention of major figures.

Kolmogorov's theory of turbulence: eddies fill the fluid. Poincare already noticed areas of calm in areas of turbulence. Closer to fractal, but that also isn't right.(?)

Lev D. Landau's theory of onset of turbulence:author of standard fluid dynamics text. Higher energy input adds dissonant overtones to system. Infinitely many degrees of freedom, hence mathematically useless. Fit the facts (compare to degrees of freedom in Ptolemy's theory). No prediction of how much energy produces the new frequency or why. Theory accepted without test. A way of giving up.

Harry Swinney: experiments on phase transitions. Thermal conductivity of CO2 increases a thousandfold at phase transition.

Phase transition similar to chaos: major qualitative changes at slight changes in energy. Also studied liquid-vapor transition.

Guenter Ahlers, Bell Lab: Superfluidity of hydrogen (almost no viscosity at phase transition).

By 1970s, looking for new problems to work on. Natural analogy between phase transition and onset of turbulence. Both involve qualitative bifurcation.

1975 Jerry Gollub: Physics Dept., Haverford College. Sabbattical with Swinney. Reproduced Couette-Taylor flow experiment: flow between counter-rotating cylinders (1/8" between) to constrain flow to 2 dimensions.

First bifurcation: stack of doughnuts, withi rotation out and around each doughnut.

They had in mind a normal science confirmation of Landau's theory of turbulence as collection of dissonant frequencies.

Gollub and Swinney applied nonstandard apparatus from phase transition experiments to the Couette-Taylor flow experiment: Laser doppler interferometry and computer processing of data.

Sociological boundary between fluid dynamics and physics: Fluid dynamicists didn't believe the results. Not recommended for NSF funding.

Successive phase transitions predicted by Landau did not appear. The experiment jumped immediately to the fully confused state without successive wavelengths being added.

David Ruelle: Hautes Etudes Scientifiques. Heard Smale describe horseshoe map. Knew about Landau's theory. Suspected that the two ideas don't match. No background in fluid mechanics. "Always nonspecialists find the new things".

Floris Takens, Dutch mathematician visiting the Institute. Co-authored paper (1971). Mathematical propositions.

Instead of infinite degrees of freedom piling up new frequencies, they claimed to obtain turbulence with just three degrees of freedom (compare to Ptolemy and Copernicus). Permanent contributions of paper debatable.

Takens and Ruelle coin new word: strange attractor.

System states: a specification of all the physically relevant properties of a system at a time.

Phase space: the set of all possible states of a system.

Degrees of freedom: dimensions of phase space (independent variables).

Trajectory: the successive states adopted by the system through time.

Attractor: a set of points that the system eventually remains arbitrarily close to.

Basin of attraction: the initial conditions that generate trajectories that converge to a given attractor.

Dogma: all attractors are either cycles or points. Damped, driven systems continue to move, so can't have point attractors. Therefore, must have cycles.

Strange attractor: Alternative possibility: bounded, low-dimensional attractor that never repeats itself and that generates continuous frequency spectrum. Would have to be an infinitely long, non-overlapping line in a finite area (fractal, but term not yet invented). Ruelle and Takens argue that such a thing mathematically exists.

Poincare section of attractor: cross-sections reveal internal structure obscured by crossing lines in two-dimensional projection.

Physicists unhappy with idea: continuous frequency spectrum requires infinite degrees of freedom.

Lorenz had already published a picture of his strange 3-d attractor for convection in 1963. Two wings of Lorenz attractor correspond to two different directions of revolution. Stable, low-dimensional, non-periodic.

Ruelle learned of Lorenz's work in 1973.

New activities.

  1. Study math of strange attractors.
  2. Try to find them in experiments.

Yshisuke Ueda: attractors of oscillating circuits.

Otto Roessler: twisted band and convoluted sausage.

Michel Henon, Nice Observatory. Simplest strange attractor comes from astronomy.

Problem of globular clusters

Moved to orbits around the galactic center: can't use Newton's point mass approximation because disk is not spherical. Studied using Poincare sections.

1976, heard of Ruelle and Takens. Heard a talk on Lorenz attractor. Henon returned to difference equations.

Ruelle visits lab of Gollub and Swinney in 1974. Situation: sketchy math, Landau's theory doesn't stand up to Couette-Taylor flow experiment.

Computer simulations of strange attractors were very suggestive to scientists who started looking for them everywhere.

Problems:

  1. Nobody saw a strange attractor in a real experiment.
  2. Not clear how to find one.
  3. Nobody knew how to attach numbers to sensitivity to initial conditions or mixing (new concepts required).
  4. Nobody knew how to measure the fractal dimension of attractors.
  5. Knowledge of one system didn't transfer to others. Each case unique.

Beauty of pictures and availability of computers fueled further interest.