James A. Yorke
|James Alan Yorke|
|Born||James Alan Yorke
August 3, 1941
Plainfield, New Jersey
|Fields||Math and Physics (theoretical)|
|Institutions||University of Maryland, College Park|
|Doctoral students||Tien-Yien Li|
|Notable awards||Japan Prize (2003)|
Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. In June of 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.
Period three implies chaos
- 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: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.
- List of Fellows of the American Mathematical Society, retrieved 2013-09-01.
- T.Y. Li, and J.A. Yorke, Period Three Implies Chaos, American Mathematical Monthly 82, 985 (1975).
- A.N. Sharkovskii, Co-existence of cycles of a continuous mapping of the line into itself, Ukrainian Math. J., 16:61-71 (1964).
- Collet, Pierre, and Eckmann, Jean-Pierre, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, 1980.