American mathematician

Mitchell Jay Feigenbaum (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.

Contents

• 1 Early life
• 2 Career
• 3 Personal life
• 4 Work
• 5 Works
• 6 See also
• 7 References
• 8 External links

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 M*achusetts Ins*ute of Technology (MIT). Enrolling for graduate study in electrical engineering, he changed his area 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 Ins*ute 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. Although a complete theory of turbulent fluids remains to be established, his research led him to study 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 Ins*ute. He was Toyota Professor at Rockefeller University from 1986 until his death.

Bifurcation diagram of the logistic map. Feigenbaum noticed in 1975 that the quotient of successive distances between bifurcation events tends to 4.6692...

Personal life

He died on June 30, 2019 at the age of 74.

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 there is one stable point, then bifurcating to an oscillation between two values, then bifurcating again to oscillate between four values and so on. In 1975, Dr. Feigenbaum, using the small HP-65 calculator he had been issued, discovered 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 cl* 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: Quan*ative Universality for a Cl* 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 *ist 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 mul*ude 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.

Mitchell Feigenbaum (right) and Joel Lebowitz (left), 1998

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. Numerix remains one of the leading software providers to financial market participants.

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. doi:10.1016/0167-2789(83)90112-4. 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). "Quan*ative universality for a cl* of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. doi:10.1007/BF01020332.
• Feigenbaum, Mitchell J. (March 1987). "Some characterizations of strange sets". Journal of Statistical Physics. 46 (5–6): 919–924. doi:10.1007/BF01011148.

See also

• Theory of Colours (book)

References

External links

• O'Connor, John J.; Robertson, Edmund F., "Mitchell Feigenbaum", MacTutor History of Mathematics archive, University of St Andrews
• Feigenbaum's webpage at Rockefeller
• 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"

Theoretical
branches

• Bifurcation theory
• Chaos theory
• Control of chaos
• Dynamical system
• Ergodic theory
• Quantum chaos
• Stability theory
• Synchronization of chaos

Chaotic
maps (list)Physical
systems

• Chua's circuit
• Convection
• Double pendulum
• Elastic pendulum
• FPUT problem
• 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 Gr*berger
• Celso Grebogi
• Martin Gutzwiller
• Brosl H*lacher
• Michel Hénon
• Svetlana Jitomirskaya
• Bryna Kra
• Edward Norton Lorenz
• Aleksandr Lyapunov
• Benoît Mandelbrot
• Hee Oh
• Edward Ott
• Henri Poincaré
• 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 Ins*ute