The description and prediction of natural events that exhibit erratic behavior, such as whorls in fluids, remain challenges in physics and mathematics. Mitchell Feigenbaum’s laboratory helped establish the field of chaotic dynamics, which seeks the understanding of just such phenomena. Its overall goal is to enlarge the applicability of mathematics to science.
Feigenbaum is a pioneer in the science of “chaos,” the mathematics of erratic dynamical systems—objects with unpredictable behavior and an attendant fractal geometry.
Chaotic motion shows a lack of predictability despite the total absence of any random ingredients. While the various constraints on a system, such as bounded resources, can allow it to move regularly on a smooth space, its chaotic motion lies in a highly complicated subspace—a so-called strange attractor.
Using computers and inventing mathematics, Feigenbaum developed a full, precise mathematical description of systems during their transition from order to disorder—for example, a dripping faucet changing from a steady drip into an erratic one. The mathematics that underlies this changeover holds true for all systems undergoing this “period doubling” onset of chaos, with all scaling details identically independent of a system’s precise nature, including fluctuating animal populations, electrical signals in circuits, lasers, and various chemical reactions. Feigenbaum has shown that all these phenomena prominently exhibit numbers determined by his theory, for example 4.6692016…, a constant of nature called Feigenbaum’s constant that determines the rate of onset.
A fractal is a complex object built hierarchically of finer and finer details, all similar apart from their successively reduced scales. These intricate formations in space, reminiscent of objects such as mountains and snowflakes, as well as complex formations in time, can be described by mathematical rules called scaling functions, which Feigenbaum discovered. Scaling functions describe the evolution of an object, whatever its current form or size, and so, unchanged, can be repeatedly reapplied.
In this circumstance, the object produced is scale invariant: as it evolves from a given size, its details remain loosely proportional to that size, and so, are fractal. Looking at systems from this perspective, Feigenbaum has made important contributions to numerous fields, including cartography: as a consultant to the Hammond Corporation, he developed techniques that allow computers to draw, with unprecedented accuracy, maps of archival quality using a dataset of just one high, fixed resolution. This atlas, published in 1992, contains Feigenbaum’s optimal conformal projections for arbitrarily chosen regions on Earth, and these projections are at least three times more accurate than all previous ones. The computer-automated labeling of maps was also accomplished by a new directed-annealing algorithm for dynamical systems.
At Rockefeller, Feigenbaum has taken part in numerous collaborations. Among them are efforts to study the electrical fluctuations of single neurons to quantitatively determine their chaotic properties; measure the way fibroblasts travel to the site of injury, observing that as their path appears not to be random, they are moving chaotically; and analyzing the outcome of optical imaging of cortical activity.