Contributions
Period three implies chaos
He and his co-author T.Y. Li coined the mathematical term ''chaos (mathematics), chaos'' 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. 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 (mathematics), mixing. (2) There is an uncountable set, 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 (mathematics), 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 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 Control of chaos#OGY method, O.G.Y. method.Books
Together with Kathleen T. Alligood and Tim D. Sauer, he was the author of the booReferences
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