James A. Yorke (born August 3, 1941) is a Distinguished University Research Professor of
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and former chair of the Mathematics Department at the
University of Maryland, College Park
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public land-grant research university in College Park, Maryland. Founded in 1856, UMD is the flagship institution of the University System of Mary ...
.
Born in
Plainfield, New Jersey
Plainfield is a city in Union County, in the U.S. state of New Jersey, known by its nickname as "The Queen City." ,
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
, Yorke attended
The Pingry School
The Pingry School is a coeducational, independent school, independent, University-preparatory school, college preparatory country day school in New Jersey, with a Lower School (K–5) campus in the Short Hills, New Jersey, Short Hills neighbo ...
, 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
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
were the recipients of the 2003
Japan Prize
is awarded to people from all parts of the world whose "original and outstanding achievements in science and technology are recognized as having advanced the frontiers of knowledge and served the cause of peace and prosperity for mankind." The P ...
in Science and Technology: Yorke was selected for his work in
chaotic system
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
s. In 2003 He was elected a
Fellow of the American Physical Society
The American Physical Society honors members with the designation ''Fellow'' for having made significant accomplishments to the field of physics.
The following lists are divided chronologically by the year of designation.
* List of American Physic ...
. and in 2012 became a fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.
He received the Doctor Honoris Causa degree from the Universidad Rey Juan Carlos, Madrid, Spain, in January 2014. In June 2014, he received the Doctor Honoris Causa degree from Le Havre University, Le Havre, France. He received the Thompson Reuters Citations Laureate in Physics 2016.
Contributions
Period three implies chaos
He and his co-author
T.Y. Li coined the mathematical term ''
chaos
Chaos or CHAOS may refer to:
Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
'' in a paper they published in 1975 entitled ''Period three implies chaos'', in which it was proved that any one-dimensional continuous map
:''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
Sharkovskii's theorem
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 r ...
.
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
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
''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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
zero (see for example the article
Logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
), 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
Edward Ott (born 22 December 1941) is an American physicist most noted for his contributions to the development of chaos theory.
Ott was born and grew up in New York City. He attended Stuyvesant High School, received his bachelor's degree in El ...
and
Celso Grebogi
Celso Grebogi (born 1947) 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 "Si ...
) had shown with a numerical example that one can convert a chaotic motion into a periodic one by a proper time-dependent perturbation of the parameter. This article is considered a classic among the 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 boo
Chaos: An Introduction to Dynamical Systems
References
External links
Website at the University of Maryland*
{{DEFAULTSORT:Yorke, James A.
1941 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Chaos theorists
Columbia University alumni
Fellows of the American Physical Society
Fellows of the American Mathematical Society
Theoretical physicists
University of Maryland, College Park alumni
University of Maryland, College Park faculty
Fellows of the Society for Industrial and Applied Mathematics
People from Plainfield, New Jersey
Mathematicians from New Jersey