James A. Yorke

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James Alan Yorke
James A Yorke.jpg
Born
James Alan Yorke

(1941-08-03) August 3, 1941 (age 79)
Nationality United States
Alma mater
Known for Kaplan–Yorke conjecture
Awards Japan Prize (2003)
Scientific career
Fields Math and Physics (theoretical)
Institutions University of Maryland, College Park
Doctoral students Tien-Yien Li and 50 more

James A. Yorke (born August 3, 1941) is a Distinguished University Research Professor of Mathematics and Physics and former chair of the Mathematics Department at the University of Maryland, College Park.

Contents

Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. Yorke is now a Distinguished University Research Professor of Mathematics and Physics with the Institute for Physical Science and Technology at the University of Maryland. In June 2013, Dr. Yorke retired as chair of the University of Maryland's Math department. He devotes his university efforts to collaborative research in chaos theory and genomics.

He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology: Yorke was selected for his work in chaotic systems. In 2003 He was elected a Fellow of the American Physical Society. [1] and in 2012 became a fellow of the American Mathematical Society. [2]

He received the Doctor Honoris Causa degree from the Universidad Rey Juan Carlos, Madrid, Spain, in January 2014. [3] In June 2014, he received the Doctor Honoris Causa degree from Le Havre University, Le Havre, France. [4] He received the Thompson Reuters Citations Laureate in Physics 2016. [5]

Contributions

Period three implies chaos

He and his co-author T.Y. Li coined the mathematical term chaos in a paper they published in 1975 entitled Period three implies chaos, [6] in which it was proved that any one-dimensional continuous map

F: RR

that has a period-3 orbit must have two properties:

(1) For each positive integer p, there is a point in R that returns to where it started after p applications of the map and not before.

This means there are infinitely many periodic points (any of which may or may not be stable): different sets of points for each period p. This turned out to be a special case of Sharkovskii's theorem. [7]

The second property requires some definitions. A pair of points x and y is called “scrambled” if as the map is applied repeatedly to the pair, they get closer together and later move apart and then get closer together and move apart, etc., so that they get arbitrarily close together without staying close together. The analogy is to an egg being scrambled forever, or to typical pairs of atoms behaving in this way. A set S is called a scrambled set if every pair of distinct points in S is scrambled. Scrambling is a kind of mixing.

(2) There is an uncountably infinite set S that is scrambled.

A map satisfying Property 2 is sometimes called "chaotic in the sense of Li and Yorke". [8] [9] Property 2 is often stated succinctly as their article's title phrase "Period three implies chaos". The uncountable set of chaotic points may, however, be of measure zero (see for example the article Logistic map), in which case the map is said to have unobservable nonperiodicity [10] :p. 18 or unobservable chaos.

O.G.Y control method

He and his colleagues (Edward Ott and Celso Grebogi) had shown with a numerical example that one can convert a chaotic motion into a periodic one by a proper time-dependent perturbations of the parameter. This article is considered as one among the classic works in the control theory of chaos and their control method is known as the O.G.Y. method.

Books

Together with Kathleen T. Alligood and Tim D. Sauer, he was the author of the book Chaos: An Introduction to Dynamical Systems.

Related Research Articles

Chaos theory Field of mathematics

Chaos theory is a branch of mathematics focusing on the study of chaos — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a hurricane in China.

Dynamical system Mathematical model which describes the time dependence of a point in a geometrical space

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

Feigenbaum constants

In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.

In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

Attractor Concept in dynamical systems

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

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King Juan Carlos University University from Spain

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Standard map

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Miguel Ángel Fernández Sanjuán

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Celso Grebogi is a Brazilian theoretical physicist who works in the area of chaos theory. He is one among the pioneers in the nonlinear and complex systems and chaos theory. Currently he works at the University of Aberdeen as the "Sixth Century Chair in Nonlinear and Complex Systems". He has done extensive research in the field of plasma physics before his work on the theory of dynamical systems. He and his colleagues have shown with a numerical example that one can convert a chaotic attractor to any one of numerous possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. This article is considered as one among the classic works in the control theory of chaos and their control method is known as the famous O.G.Y. method. He was listed in the 2016 Thomson Reuters Citation Laureates.

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References

  1. "APS Fellow Archive". APS. Retrieved 17 September 2020.
  2. List of Fellows of the American Mathematical Society , retrieved 2013-09-01
  3. Doctor Honoris Causa degree from the Universidad Rey Juan Carlos, Madrid, Spain
  4. Doctor Honoris Causa degree from Le Havre University, Le Havre, France
  5. Thompson Reuters Citations Laureate in Physics
  6. T.Y. Li, and J.A. Yorke, Period Three Implies Chaos, American Mathematical Monthly 82, 985 (1975).
  7. Sharkovskii, A. N. (1964). "Co-existence of cycles of a continuous mapping of the line into itself". Ukrainian Math. J. 16: 61–71.
  8. Blanchard, F.; Glasner, E.; Kolyada, S.; Maass, A. (2002). "On Li–Yorke pairs". Journal für die reine und angewandte Mathematik . 547: 51–68.
  9. Akin, E.; Kolyada, S. (2003). "Li–Yorke sensitivity". Nonlinearity . 16 (4): 1421–1433. Bibcode:2003Nonli..16.1421A. doi:10.1088/0951-7715/16/4/313.
  10. Collet, Pierre; Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems . Birkhäuser. ISBN   3-7643-3510-6.