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Bifurcation Locus
In complex dynamics, the bifurcation locus of a parameterized family of one-variable holomorphic functions informally is a locus of those parameterized points for which the dynamical behavior changes drastically under a small perturbation of the parameter. Thus the bifurcation locus can be thought of as an analog of the Julia set in parameter space. Without doubt, the most famous example of a bifurcation locus is the boundary of the Mandelbrot set. Parameters in the complement of the bifurcation locus are called J-stable. References * Alexandre E. Eremenko and Mikhail Yu. Lyubich, ''Dynamical properties of some classes of entire functions'', Annales de l'Institut Fourier 42 (1992), no. 4, 989–1020, http://www.numdam.org/item?id=AIF_1992__42_4_989_0. * Mikhail Yu. Lyubich, ''Some typical properties of the dynamics of rational mappings (Russian)'', Uspekhi Mat. Nauk 38 (1983), no. 5(233), 197–198. * Ricardo Mañé, Paulo Sad and Dennis Sullivan, ''On the dynamics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Dynamics
Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem ** Poincaré metric **Schwarz lemma **Riemann mapping theorem ** Carathéodory's theorem (conformal mapping) ** Böttcher's equation * Combinatorial ** Hubbard trees ** Spider algorithm ** Tuning ** Laminations ** Devil's Staircase algorithm (Cantor function) **Orbit portraits ** Yoccoz puzzles Parts * Holomorphic dynamics (dynamics of holomorphic functions) ** in one complex variable ** in several complex variables * Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable. See also * Arithmetic dynamics *Chaos theory *Complex analysis * Complex quadratic polynomial * Fatou set *Infinite compositions of analytic functions * Julia set *Mandelbrot set * Symbolic dynamics Notes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifurcation Theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them . Bifurcation types It is useful to divide bifurcations into two principal classes: * Local bifurcations, which can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locus (mathematics)
In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.. In other words, the set of the points that satisfy some property is often called the ''locus of a point'' satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be ''located'' or may move. History and philosophy Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the ''locus'' of a point that is at a given d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant holomorphic function from the Riemann sphere o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbrot Set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Lyubich
Mikhail Lyubich (born 25 February 1959 in Kharkiv, Ukraine) is a mathematician who made important contributions to the fields of holomorphic dynamics and chaos theory. Lyubich graduated from Kharkiv University with a master's degree in 1980, and obtained his PhD from Tashkent University in 1984. Currently, he is a Professor of Mathematics at Stony Brook University and the Director of the Institute of Mathematical Sciences at Stony Brook. From 2002-2008, he also held a position of Canada Research Chair at the University of Toronto. He is credited with several important contributions to the study of dynamical systems. In his 1984 Ph.D. thesis, he proved fundamental results on ergodic theory and the structural stability of rational mapping. Due to this work, the measure of maximal entropy of a rational map (the Mané-Lyubich measure) bears his name. In 1999, he published the first non-numerical proof of the universality of the Feigenbaum constants in chaos theory. He receive ... [...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'' S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University. Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022. Early life and education Sullivan was born in Port Huron, Michigan, on February 12, 1941.. His family moved to Houston soon afterwards. He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words: He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, ''Triangu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curtis McMullen
Curtis Tracy McMullen (born May 21, 1958) is an American mathematician who is the Cabot Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory. Biography McMullen graduated as valedictorian in 1980 from Williams College and obtained his PhD in 1985 from Harvard University, supervised by Dennis Sullivan. He held post-doctoral positions at the Massachusetts Institute of Technology, the Mathematical Sciences Research Institute, and the Institute for Advanced Study, after which he was on the faculty at Princeton University (1987–1990) and the University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ... (1990–1997), before joining Harvard in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connectedness Locus
In one-dimensional complex dynamics, the connectedness locus of a parameterized family of one-variable holomorphic functions is a subset of the parameter space which consists of those parameters for which the corresponding Julia set is connected. Examples Without doubt, the most famous connectedness locus is the Mandelbrot set, which arises from the family of complex quadratic polynomials : :f_c(z) = z^2+c\, The connectedness loci of the higher-degree unicritical families, : z\mapsto z^d+c\, (where d\geq 3\,) are often called 'Multibrot sets'. For these families, the bifurcation locus In complex dynamics, the bifurcation locus of a parameterized family of one-variable holomorphic functions informally is a locus of those parameterized points for which the dynamical behavior changes drastically under a small perturbation of the pa ... is the boundary of the connectedness locus. This is no longer true in settings, such as the full parameter space of cubic polynomials, where ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
