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Takens' Theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of that system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. It is the most commonly used method for attractor reconstruction. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a -dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rössler Attractor Reconstructed By Taken's Theorem, Using Different Delay Lengths
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Rössler is a surname and may refer to: * Ervin Rössler (1876–1933), Croatian zoologist * Fritz Rössler (1912–1987), German Nazi politician * Günter Rössler (1926–2012), German photographer and photo-journalist * Jaroslav Rössler (1902–1990), Czech photographer * Otto Rössler (born 1940), German biochemist * Willi Rössler (1924–2007), German fencer See also * Rößler * Roessler * Roeseler * Rössler attractor The Rössler attractor () is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s... These differential equations define a continuous-time dynamical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James A
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada-language film * "James", a television Adventure Time (season 5)#ep42, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Sugihara
George Sugihara (born in Tokyo, Japan) is currently a professor of biological oceanography in the Physical Oceanography Research Division at the Scripps Institution of Oceanography, where he is the inaugural holder of the McQuown Chair in Natural Science. Sugihara is a theoretical biologist who works across a variety of fields ranging from ecology and landscape ecology, to epidemiology, to genetics, to geoscience and atmospheric science, to quantitative finance and economics. His work involves inductive theoretical approaches to understanding nature from observational data. The general approach is different from most theory and involves minimalist inductive theory – Inductive data-driven explorations of nature using minimal assumptions. The aim is to avoid inevitable assumptions of deductive first-principle models and produce an understanding that passes the validation test of out-of-sample prediction. His initial work on fisheries as complex, chaotic systems led to work on fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricardo Mañé
Ricardo Mañé Ramirez (Montevideo, 14 January 1948 – Montevideo, 9 March 1995) was a Uruguayan mathematician, known for his contributions to dynamical systems and ergodic theory. He was a doctoral student of Jacob Palis at IMPA. He was an invited speaker at the International Congresses of Mathematicians of 1983 and 1994http://www2.profmat-sbm.org.br/sitesbm/socios_honorarios.asp and is a recipient of the 1994 TWAS Prize. Selected publications *Expansive diffeomorphisms, Proceedings of the Symposium on Dynamical Systems (University of Warwick, 1974) ''Lecture Notes in Mathematics'' Vol. 468 pp. 162–174, Springer-Verlag, 1975. *Persistent manifolds are normally hyperbolic, ''Transactions of the American Mathematical Society'', Vol. 246, (Dec., 1978), pp. 261–283. *On the dimension of the compact invariant sets of certain non-linear maps, Springer, ''Lectures Notes in Mathematics'' Vol. 898 (1981) 230–242. *An ergodic closing lemma, ''Annals of Mathematics'' Secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Shaw (physicist)
Robert Stetson Shaw (born 1946) is an American physicist who was part of Eudaemonic Enterprises in Santa Cruz in the late 1970s and early 1980s. In 1988 he was awarded a MacArthur Fellowship for his work in chaos theory. Chaos theory Shaw was one of the pioneers of chaos theory and his work at University of California, Santa Cruz on the subject was among the first research into the relationship between predictable motion and chaos in a landmark PhD thesis. He was part of the Dynamical Systems Collective with J. Doyne Farmer, Norman Packard, and James Crutchfield. The collective, also known as the Santa Cruz Chaos Cabal, was best known for its work in probing chaotic systems for signs of order. A documentary is being made about Shaw's life, his art, and his science, entitled ''Strange Attractors: a movie for curious people''. Roulette While at the University of California, Santa Cruz, Shaw also worked briefly with the Eudaemons, a group of physicists attempting to create a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Doyne Farmer
J. Doyne Farmer (born June 22, 1952) is an American complex systems scientist and entrepreneur with interests in chaos theory, complexity and econophysics. He is Baillie Gifford Professor of Complex Systems Science at the Smith School of Enterprise and the Environment, Oxford University, where he is also director of the Complexity Economics programme at the Institute for New Economic Thinking at the Oxford Martin School. Additionally, he is an external professor at the Santa Fe Institute. His current research is on complexity economics, focusing on systemic risk in financial markets and technological progress. During his career he has made important contributions to complex systems, chaos, artificial life, theoretical biology, time series forecasting and econophysics. He co-founded Prediction Company, one of the first companies to do fully automated quantitative trading. While a graduate student he led a group that called itself Eudaemonic Enterprises and built the first weara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James P
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada-language film * "James", a television Adventure Time (season 5)#ep42, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Packard
Norman Harry Packard (born 1954 in Billings, Montana) is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. He is an alumnus of Reed College, with a PhD from the University of California, Santa Cruz. Packard is known for his contributions to chaos theory, complex systems, and artificial life. He coined the phrase "the edge of chaos". Biography Between 1976 and 1981, Packard formed the Dynamical Systems Collective at UC Santa Cruz with fellow physics graduate students, Robert Shaw (Physicist), Rob Shaw, J. Doyne Farmer, Doyne Farmer, and James P. Crutchfield, James Crutchfield. The collective was best known for its work in probing chaotic systems for signs of order. Around the same time, he worked with J. Doyne Farmer, Doyne Farmer and other friends in Santa Cruz, California to form the Eudaemons collective, to develop a strategy for beating the roulette wheel using a toe-operated computer. The computer could, in theory, predict in what area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Dimensionality Reduction
Nonlinear dimensionality reduction, also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. Applications of NLDR High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Embedding Theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real - dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real -space, if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the real projective spaces of dimension cannot be embedded into real -space if is a power of two (as can be seen from a characteristic class argument, also due to Whitney). *The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an immersion provided . This last result is sometimes called the Whitney immersion theorem. About the proof Weak embedding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
