James Gleick

Chaos: Making a New Science

2. Revolution

Review of Kuhn's account of scientific revolutions.

Card suit experiment.

Normal science = puzzle solving under shared assumptions

Revolution:

• Interdisciplinary
• Driven by loneley outsiders.
• Bars to publication.
• Romantic

This pattern is realized by the chaos revolution:

• Graduae students warned that their careers would be ruined.
• Can't discuss it with colleagues.
• Midlife crisis, gambling on untried approach.
• Intellectual excitement.
• Feeling of new way of thinking.
• Ferocious reaction from establishment.
• Mathematics unfamiliar and difficult.
• Resented by traitional fluid mechanics.
• Eventually chaos theory diffused into the universitiy hierarchy.
• Chaos is a method as well as a theory: graphic images on small computers.
• Chaos theorists appeal to Kuhn explicitly.
• Evangelical, foundational, methodological publications
• New hopes, new styles, new way of seeing.
• Replacement of old paradigm.

Reprise of Kuhn on pendulum:

• Kuhn on Aristotelian vs. Galilean pendulum
• Chaos gave new perspective on pendulum again.
• Galileo saw too much: period independent of ampitude of swing.
• Changing angle yields nonlinearity negligible at small angles.
• Friction and air resistance are also nonlinear.
• Galilean science: ideal scientific world where regularities are separated from disorder of experience.
• Eventually dissipative systems with friction were handled.
• Novelty was chaos out of nonlinearity.
• Key to solution of turbulence was new understanding of pendulum.
• Damped & driven pendulum: swing pushed at the wrong times generates chaos.
• Weather: driven by sun, damped by friction.
• Damped, driven systems are unpredictable and generate natural complexity of the sort encountered in ordinary life.
• Triple magnets under pendulum. Initial positions leading to capture by a given magnet has a complex, infinitely varied geometry at arbitrarily small scales.
• Writing down equations alone is only the beginning of the story.
• Computer simulation: due to initial condition sensitivity, simulate runs amok after a few iterations! (What then is the point of the simulation?)

Stephen Smale: a Berkeley mathematician revisits the pendulum (a few years before Lorenz's discovery).

• Name in multidimensional topology: solved Poincare conjecture.
• Fields medal in mathematics.
• Anti-vietnam activist.
• Ability to lead others into new areas.
• Chaos and topology both founded by Henri Poincare.
• Poincare proposed to look at structure of solutions to equations over the entire phase space.
• Changing system parameters amounts to continuous changes in geometry of solutions, as in topology.
• False conjecture: chaotic behavior cannot be stable: slight perturbation should suck system into orderly, stable behavior.
• Colleague found counterexample: a damped, driven electronic circuit in a vacuum tube. In 1920's van der Pol noticed chaoitic transitions between frequencies and disregarded them.
• Disspation corresponds to continuous contraction of phase space (e.g., damped pendulum).
• Van der Pol circuit required a "taffy pulling" transformation called the "horseshoe" map.
• Similar idea: repeated looping of rubber band around your fingers. Two spots on opposite sides of the rubber band will get arbitrarily close and arbitrarily far apart through time.
• Great visual analogue of sensitivity to initial conditions.
• After 1930s, physicists ignored modern mathematics. Physicists started teaching students mathematics on their own and warned students away from mathematicians. Chaos revolution brought modern topology back into physics.

20 yrs later: Great red spot of Jupiter

• Highly visible anomaly prior to chaos revolution. Sequence of hypotheses:

• Lava flow.
• New moon emerging from surface.
• Solid body floating in atmosphere (1943!)
• Spot never drifted far: column of gas rising from a crater.
• Voyager photographs: 1978. Swirling eye of humongous hurricane. But not like hurricane.
• Rotates in wrong direction.
• Too persistent to be random.
• Evanescent features inside, but outside constant.
• Astronomer Philip Marcus: spot is stable chaotic system. Like Darwin after conversion to evolutionism, he bacame a new kind of scientist specializing in chaos, not mathematics, fluid dynamics, or astronomy.