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Laboratory of Mathematical Physics

Mitchell Feigenbaum
Toyota Professor

Attempts at the analytical description of nature regularly encounter strong nonlinearities. The majority of existing methods treat only weak nonlinearities and consist of corrections to behaviors that are simple distortions of linear behavior. The methods in the strong case, on the other hand, are largely recent developments and pertain to behaviors qualitatively distinct from linear ones. One of the most striking such behaviors is the appearance of highly erratic spatial configurations and/or highly erratic temporal evolution, a phenomenon called "chaos."

The hallmark of chaotic motion is a lack of predictability despite the total absence of any random ingredients. Even if one should want to determine just statistical properties of these motions, the methods of statistics cannot be applied in any straightforward manner. This impediment is a consequence of the fact that the motion, rather than exploring all possibilities allowed to it by the constraints of finite energy, finite resources, etc., instead lies in a highly complicated subspace -- a so-called strange set or strange attractor. Thus an a priori calculated average over "everything" will generally produce erroneous results.

Since the equations of motion of an object usually can be cast in a form that is indifferent to where the origin of time is taken, one encounters a process that repeatedly applies the same rule to whatever happens to be there at a given moment. It also often transpires that these rules are "scale invariant," so that what the rule evolves from a given size detail is loosely proportional to that size. In consequence of those two observations, the strange sets encountered in chaotic dynamics can be amenable to a treatment that hierarchically, by rules of scaling, builds up more highly complicated details from those less so. Generically, such objects are called fractals. The method that treats them is termed "renormalization" from which it is to be inferred that after applying a scaling-down rule followed by readjustment of the overall size successively, a definite but highly complicated object continually reemerges.

It is the main effort of this laboratory to extend the scope of the problems over which these methods hold sway, to extend the methods themselves developing appropriate mathematics, and to improve the methods for extracting such information from actual experimental studies ranging from fluids to brains.