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Mitchell Feigenbaum
Mitchell Feigenbaum in 2006
Born	Mitchell Jay Feigenbaum (1944-12-19)December 19, 1944 Philadelphia, Pennsylvania, US
Died	June 30, 2019(2019-06-30) (aged 74) New York City, New York, US
Nationality	American
Alma mater	City College of New York (BS) Massachusetts Institute of Technology (PhD)
Known for	Feigenbaum constants Feigenbaum function Feigenbaum universality
Awards	MacArthur Fellow (1984) Wolf Prize (1986) Heineman Prize (2008)
Scientific career
Fields	Mathematical physics
Institutions	Rockefeller University
Doctoral advisor	Francis E. Low

Mitchell Jay Feigenbaum /ˈfaɪɡənˌbaʊm/ (December 19, 1944 – June 30, 2019) was an American mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.

Early life

Feigenbaum was born in Philadelphia, Pennsylvania, to Jewish emigrants from Poland and Ukraine. He attended Samuel J. Tilden High School, in Brooklyn, New York, and the City College of New York. In 1964, he began his graduate studies at the Massachusetts Institute of Technology (MIT). Enrolling for graduate study in electrical engineering, he changed his area of study to physics. He completed his doctorate in 1970 for a thesis on dispersion relations, under the supervision of Professor Francis E. Low.

Career

After short positions at Cornell University (1970–1972) and the Virginia Polytechnic Institute and State University (1972–1974), he was offered a longer-term post at the Los Alamos National Laboratory in New Mexico to study turbulence in fluids. He was at Cornell from 1982 to 1986 and then joined Rockefeller University as Toyota Professor in 1987. Although a complete theory of turbulent fluids remains elusive, Feigenbaum's research paved the way for chaos theory, providing groundbreaking insight into the many dynamical systems in which scientists and mathematicians find chaotic maps.

In 1983, he was awarded a MacArthur Fellowship, and in 1986, alongside Rockefeller University colleague Albert Libchaber, 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". He was a member of the Board of Scientific Governors at the Scripps Research Institute. He remained at Rockefeller University as Toyota Professor from 1987 until his death.

Work

Some mathematical mappings involving a single linear parameter exhibit the apparently random behavior known as chaos when the parameter lies within certain ranges. As the parameter is increased towards this region, the mapping undergoes bifurcations at precise values of the parameter. At first, one stable point occurs, then bifurcates to an oscillation between two values, then bifurcating again to oscillate between four values, and so on. Feigenbaum discovered in 1975, using an HP-65 calculator, that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692... He was able to provide a mathematical argument of that fact, and he then showed that the same behavior, with the same mathematical constant, would occur within a wide class of mathematical functions, prior to the onset of chaos. This universal result enabled mathematicians to take their first steps to unraveling the apparently intractable "random" behavior of chaotic systems. The "ratio of convergence" measured in this study is now known as the first Feigenbaum constant.

The logistic map is a prominent example of the mappings that Feigenbaum studied in his noted 1978 article: "Quantitative Universality for a Class of Nonlinear Transformations".

Feigenbaum's other contributions include the development of important new fractal methods in cartography, starting when he was hired by Hammond to develop techniques to allow computers to assist in drawing maps. The introduction to the Hammond Atlas (1992) states:

Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable of reconfiguring coastlines, borders, and mountain ranges to fit a multitude of map scales and projections. Dr. Feigenbaum also created a new computerized type-placement program which places thousands of map labels in minutes, a task that previously required days of tedious labor.

In another practical application of his work, he founded Numerix with Michael Goodkin in 1996. The company's initial product was a software algorithm that dramatically reduced the time required for Monte Carlo pricing of exotic financial derivatives and structured products.

The press release made on the occasion of his receiving the Wolf Prize summed up his works:

The impact of Feigenbaum's discoveries has been phenomenal. It has spanned new fields of theoretical and experimental mathematics ... It is hard to think of any other development in recent theoretical science that has had so broad an impact over so wide a range of fields, spanning both the very pure and the very applied.

Works

• Feigenbaum, Mitchell J. (May 1983). "Universal behavior in nonlinear systems" (PDF). Physica D: Nonlinear Phenomena. 7 (1–3): 16–39. Bibcode:1983PhyD....7...16F. doi:10.1016/0167-2789(83)90112-4. Archived from the original (PDF) on 2010-01-07. A semipopular account of the universal scaling theory for the period doubling route to chaos is presented.
• "Feigenbaum, Mitchell J." Publications. Astrophysics Data System.
• Feigenbaum, Mitchell J. (1 July 1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. Bibcode:1978JSP....19...25F. doi:10.1007/BF01020332. S2CID 124498882.
• Feigenbaum, Mitchell J. (March 1987). "Some characterizations of strange sets". Journal of Statistical Physics. 46 (5–6): 919–924. Bibcode:1987JSP....46..919F. doi:10.1007/BF01011148. S2CID 121418123.

See also

• Theory of Colours (book)

References

1. "Mitchell Feigenbaum, physicist who pioneered chaos theory, has died". Rockefeller University. July 2, 2019. Archived from the original on January 21, 2020. Retrieved July 3, 2019.
2. ^ "Mitchell Jay Feigenbaum". University of St Andrews.
3. A New Kind of Science
4. Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976
5. Feigenbaum, M. J. (1978). "Quantitative Universality for a Class of Non-Linear Transformations". J. Stat. Phys. 19 (1): 25–52. Bibcode:1978JSP....19...25F. CiteSeerX 10.1.1.418.9339. doi:10.1007/BF01020332. S2CID 124498882.
6. Hammond World Atlas. Hammond Inc. 1992. ISBN 978-0-8437-1604-7.

External links

• O'Connor, John J.; Robertson, Edmund F., "Mitchell Feigenbaum", MacTutor History of Mathematics Archive, University of St Andrews
• Miller, Antony .D. (2023). "Origins of Chaos Theory in Science and Society: Exploring this Concept in a Troubled Society." Feigenbaum pp. 27-35. Hardcover & Paperback – August 3 & 7, 2023. Otgontenger University, Mongolia. ISBN 979-8-3977-8000-1 (Hardcopy), ISBN 979-8-8563-3203-1 (Paperback)
• Feigenbaum, Mitchell J. (6 June 2008). "The Theory of Relativity - Galileo's Child". arXiv:0806.1234 .
• Cvitanović, P. "A very brief history of universality in period doubling"
• Cvitanović, P. "A not so short history of Universal Function"

v t e Chaos theory
Concepts	Core Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy Theorems Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem
Theoretical branches	Bifurcation theory Control of chaos Dynamical system Ergodic theory Quantum chaos Stability theory Synchronization of chaos
Chaotic maps (list)	Discrete Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map Continuous Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator
Physical systems	Chua's circuit Convection Double pendulum Elastic pendulum FPUT problem Hénon–Heiles system Kicked rotator Multiscroll attractor Population dynamics Swinging Atwood's machine Tilt-A-Whirl Weather
Chaos theorists	Michael Berry Rufus Bowen Mary Cartwright Chen Guanrong Leon O. Chua Mitchell Feigenbaum Peter Grassberger Celso Grebogi Martin Gutzwiller Brosl Hasslacher Michel Hénon Svetlana Jitomirskaya Bryna Kra Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Hee Oh Edward Ott Henri Poincaré Itamar Procaccia Mary Rees Otto Rössler David Ruelle Caroline Series Yakov Sinai Oleksandr Mykolayovych Sharkovsky Nina Snaith Floris Takens Audrey Terras Mary Tsingou Marcelo Viana Amie Wilkinson James A. Yorke Lai-Sang Young
Related articles	Butterfly effect Complexity Edge of chaos Predictability Santa Fe Institute

• 1944 births
• 2019 deaths
• Scientists from Philadelphia
• 20th-century American mathematicians
• 21st-century American mathematicians
• American people of Polish-Jewish descent
• American people of Ukrainian-Jewish descent
• 21st-century American physicists
• Chaos theorists
• City College of New York alumni
• Jewish American physicists
• MacArthur Fellows
• Members of the United States National Academy of Sciences
• Scripps Research
• Wolf Prize in Physics laureates
• American mathematical physicists
• Los Alamos National Laboratory personnel
• Samuel J. Tilden High School alumni
• Mathematicians from New York (state)
• Cornell University faculty
• MIT Center for Theoretical Physics people