HOUSE_OVERSIGHT_013527.jpg

down” the expansive flow back onto the road, the entire constrictive field called the stable manifold. This influence herds points into shadowing the main road of the dynamics, like the hair of the dog that stay close to the real body of the animal in motion. It is in this way that just a few often slightly off the mark points nonetheless shadow the real (called fiduciary) orbits of the attractor, outlining its global geometry with just a little information.

The intuitive reason shadowing works is built into these natural countervailing tendencies of hyperbolic dynamics, which on one hand tends to spread out nearby initial points and brings disparate others together. The latter inclination is the one that smoothes down the escaping points onto surfaces of actions that mathematicians call manifolds. However, the details of the orbital paths don’t look that orderly due to the mixing of the sequence of points in hyperbolic motion. The mixing process on manifolds has been analogized to that of the bundled pink loops of the stretching (expanding) and folding (contracting) taffy puller at the carnival candy stand. The process gets sequences of small particles of candy out of sequential order while maintaining the taffy’s overall geometrically ovoid shape. Disorder is local with the entropy being generated by the repeatedly shuffling of the line up of the original orbital sequence. This results in the impossibility of any point-to-point prediction for more than a few points even though the over all shape is maintained. Exactly what minute a habitually late sleeper awakes can’t be predicted. On the other hand, the skeletal manifold of the global structure is entirely in evidence from almost the beginning. Late risers remain late risers even without a precise, minute-to-minute, predictable schedule.

It is also interesting that a uniformly hyperbolic dynamical system, unlike the fixed-point attractors of stylistic fixation, resist perturbation-induced changes in global dynamical form. In an apparent paradox worthy of metaphysical allusion, the dynamically hyperbolic kind of formlessness has structural stability. The global geometric predictability of this point-to-point, completely unpredictable system can be both the subject and object of Zen frustration and thoughtful meditation.

During weekly professorial rounds at Los Angeles’s Neuropsychiatric Institute, I assigned a standard exercise for psychiatric residents on clinical rounds,

27

HOUSE_OVERSIGHT_013527