James A. Yorke

This article is about the 20th century mathematician. For the 18th century clergyman, see James Yorke (bishop).

James Alan Yorke
File:James A Yorke.jpg
Born	James Alan Yorke (1941-08-03) August 3, 1941 Plainfield, New Jersey
Nationality	United States
Fields	Math and Physics (theoretical)
Institutions	University of Maryland, College Park
Alma mater	A.B., Columbia University, 1963 Ph.D., University of Maryland, College Park, 1966
Doctoral students	Tien-Yien Li
Notable awards	Japan Prize (2003)

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

Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. In June 2013, Dr. Yorke retired and stepped down as chair of the University of Maryland's Math department. Yorke is now a Distinguished University Professor Emeritus with the Institute for Physical Science and Technology department at the University of Maryland.

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 2012 he became a fellow of the American Mathematical Society.

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, in which it was proved that any continuous function

F: R →R

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 Sharkovsky's theorem.

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". 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 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, Tim D. Sauer and Edward Ott he was the author of the book Chaos in Dynamical Systems an introduction.

External links

• Website at the University of Maryland
• James A. Yorke at the Mathematics Genealogy Project