Mitchell J. Feigenbaum, a mathematician and physicist, died on June 30 at a hospital in Manhattan at the age of 74. Among his staggering achievements, he was the first to discover that many different physical systems follow a common “period doubling” path to chaos, thus pioneering the study of chaos theory.^{[1]}

Mitchell Feigenbaum was born in 1994 in Philadelphia to Abraham Joseph Feigenbaum and Mildred Sugar, who were emigrants from Poland and Ukraine, respectively. Mitchell was the middle child of three, with an older brother named Edward and a younger sister, Glenda. When he was twelve, Feigenbaum started high school in Brooklyn, New York. At school, Feigenbaum learnt more by studying by himself than through formal lessons. He taught himself how to play the piano at about twelve years old, and during high school, he self-studied calculus.^{[2]}

In 1960, Feigenbaum attended the City College of New York. Then, in 1964, he began his graduate studies at the Massachusetts Institute of Technology. Though he enrolled for graduate study in electrical engineering, he eventually switched to physics, completing his doctorate in 1970. After short positions at Cornell University and the Virginia Polytechnic Institute and State University, Feigenbaum was offered a long-term post at the Los Alamos National Laboratory.^{[3]} He writes, “When I arrived at Los Alamos, the theory division head, P Carruthers, felt that the time was right, and I was the appropriate person, to see if Wilson’s renormalisation group ideas could solve the century and a half old problem of turbulence. In a nutshell, it couldn’t — or so far hasn’t — but led me off in wonderful directions.”^{[4]} He began studying turbulence in fluids, which then led him to study chaotic maps. Around this time, Feigenbaum received his own programmable calculator for the first time, and “invented new ODE solvers, minimisation routines, interpolation methods, etc.”^{[4]}

Feigenbaum considered the logistic map that lead to chaotic dynamics, defined by \( x_{n+1} \mapsto r x_n (1-x_n) \). Let \( x_{n+1} \mapsto r x_n (1-x_n) \), where \( x_0 \in [0, 1] \) and \( r \in (0, 4] \). This ensures that \( x_0 \in [0, 1] \). Why? Consider the limit of the sequence as \( n \rightarrow \infty \). For \( r \leq 1 \), the sequence approaches \( 0 \). For \( 1 < r < 3 \), the sequence converges to the stable fixed point \( x_{*} = (r-1)/r \). If \( r \geq 3 \), however, this fixed point would be unstable. To illustrate this, suppose that \( x_n = (r-1)/r + \epsilon \). Then, \( x_{n+1} = (r-1)/r – (r-2+\epsilon)\epsilon \), so for \( r \geq 3 \), the error between \( x \) and \( (r-1)/r \) increases. Thus, in this case, the sequence does not converge to \( (r-1)/r \).

It turns out that when \( 3 < r < a_1 \), for which \( a_1 \approx 3.449 \), the sequence oscillates between two values. Similarly, for \( a_1 < r < a_2 \), where \( a_2 \approx 3.544 \), the sequence oscillates between four values. This pattern continues, the period doubling each time \( r \) becomes greater than a critical value \( a \), in a process known as bifurcation.^{[4]}

What Feigenbaum noticed is that \( \lim\limits_{n \to \infty} (a_{n+1}-a_n)/(a_{n+2}-a_{n+1}) = \delta \approx 4.669 \). Of course, it is a rather interesting result that this limit exists, that the gaps between period doubling approaches a geometric sequence. But what makes it more significant that this \( \delta = 4.669 \) is true for many other maps as well. For example, take the seemingly different map \( x_{n+1} \mapsto a \sin (\pi x_{n}) \). Feigenbaum found that the \( \delta \) value of this map is also \( 4.669 \). Eventually, he was able to show that this holds for all one-dimensional maps with a single quadratic maximum. This number \( 4.6692016910 \ldots \), now known as the first Feigenbaum number, describes the bifurcation rate for every chaotic system following this description.

In 1976, Feigenbaum wrote a report announcing his results, with mixed reception. Physicists wondered how it related to physics. Mathematicians wondered about its status, since it came from experimental mathematics, without proof. It did not help that Feigenbaum’s explanations were hard to follow.

But in the summer of 1979, Albert Libchaber in Paris reported results of the transition to turbulence in convection in liquid helium, where period doubling is observed as well, precisely at the exponent \( \delta \) Feigenbaum had calculated. Apparently, this \( \delta \) wasn’t just about mathematics; it appeared in physical systems, too. Soon, Feigenbaum was famous. In 1986, he was awarded the Wolf Prize in Physics “for his pioneering theoretical studies demonstrating the universal character of non-linear systems, which has made possible the systematic study of chaos.” In the same year, he moved to the Rockefeller University in New York. Feigenbaum continued to make discoveries, implementing fractal geometry in cartography and creating a physics-based algorithm to place labels on maps. In addition, he founded a company called Numerix to commercialize the use of physics-like methods for financial instruments.^{[4]} Feigenbaum remained a professor at Rockefeller until his death in June.^{[2]}

Sources

- Fenz, K. (2019, July 2). Celebrating and remembering Mitchell Feigenbaum, physicist who pioneered chaos theory. Retrieved from https://www.rockefeller.edu/news/26289-mitchell-feigenbaum-physicist-pioneered-chaos-theory-died/.
- Mitchell Jay Feigenbaum. (n.d.). Retrieved October 19, 2019, from https://www-history.mcs.st-andrews.ac.uk/Biographies/Feigenbaum.html.
- M J Feigenbaum, Computer generated physics, in
*Twentieth Century Physics*(New York, 1995), 1823-1853. - Mitchell Feigenbaum (1944‑2019). (2019, July 23). Retrieved October 19, 2019, from https://writings.stephenwolfram.com/2019/07/mitchell-feigenbaum-1944-2019-4-66920160910299067185320382/.
- Feigenbaum, M. J. (n.d.).
*Universal Behavior in Nonlinear Systems*. Retrieved from https://fas.org/sgp/othergov/doe/lanl/pubs/00818090.pdf.