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Douady Rabbit
The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map. It is named after French mathematician Adrien Douady. Formula The rabbit is generated by iterating the Mandelbrot set map z=z^2+c on the complex plane with c fixed to lie in the period three bulb off the main cardiod and z ranging over the plane. The pixels in the image are then colored to show whether for a particular value of z the iteration converged or diverged Variants Twisted rabbit or rabbits with twisted ears = is the composition of the “rabbit” polynomial with n-th powers of the Dehn twists about its ears. Corabbit is symmetrical image of rabbit. Here parameter c \approx -0.1226 -0.7449i It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit. 3D The julia set has no direct analog in 3D 4D Quaternion julia set with parameters c = −0,123 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filled Julia Set
The filled-in Julia set K(f) of a polynomial f is a Julia set and its interior, non-escaping set Formal definition The filled-in Julia set K(f) of a polynomial f is defined as the set of all points z of the dynamical plane that have bounded orbit with respect to f K(f) \overset \left \ where: * \mathbb is the set of complex numbers * f^ (z) is the k -fold composition of f with itself = iteration of function f Relation to the Fatou set The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. K(f) = \mathbb \setminus A_(\infty) The attractive basin of infinity is one of the components of the Fatou set. A_(\infty) = F_\infty In other words, the filled-in Julia set is the complement of the unbounded Fatou component: K(f) = F_\infty^C. Relation between Julia, filled-in Julia set and attractive basin of infinity The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity J(f) = \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Disc
Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation. Description Given a holomorphic endomorphism f:S\to S on a Riemann surface S we consider the dynamical system generated by the iterates of f denoted by f^n=f\circ\stackrel\circ f. We then call the orbit \mathcal^+(z_0) of z_0 as the set of forward iterates of z_0. We are interested in the asymptotic behavior of the orbits in S (which will usually be \mathbb, the complex plane or \mathbb=\mathbb\cup\, the Riemann sphere), and we call S the phase plane or ''dynamical plane''. One possible asymptotic behavior for a point z_0 is to be a fixed point, or in general a ''periodic point''. In this last case f^p(z_0)=z_0 where p is the period and p=1 means z_0 is a fixed point. We can then define the ''multiplier'' of the orbit as \rho=(f^p)'(z_0) and this enables us to classify periodic orbits as ''attracting'' if , \rho, 1 and indifferent if \rho=1. Indifferent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herman Ring
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou componentJohn Milnor''Dynamics in one complex variable'' Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006. where the rational function is conformally conjugate to an irrational rotation of the standard annulus. Formal definition Namely if ''ƒ'' possesses a Herman ring ''U'' with period ''p'', then there exists a conformal mapping :\phi:U\rightarrow\ and an irrational number \theta, such that :\phi\circ f^\circ\phi^(\zeta)=e^\zeta. So the dynamics on the Herman ring is simple. Name It was introduced by, and later named after, Michael Herman (1979) who first found and constructed this type of Fatou component. Function * Polynomials do not have Herman rings. * Rational functions can have Herman rings. According to the result of Shishikura, if a rational function ''ƒ'' possesses a Herman ring, then the degree of ''ƒ'' is at least ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dragon Curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently. Heighway dragon The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column ''Mathematical Games'' in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel ''Jurassic Park''. Construction The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
External Rays
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External may refer to: * External (mathematics), a concept in abstract algebra * Externality, in economics, the cost or benefit that affects a party who did not choose to incur that cost or benefit * Externals, a fictional group of X-Men antagonists See also * *Internal (other) Internal may refer to: *Internality as a concept in behavioural economics *Neijia, internal styles of Chinese martial arts *Neigong or "internal skills", a type of exercise in meditation associated with Daoism *''Internal (album)'' by Safia, 2016 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filled Julia Set
The filled-in Julia set K(f) of a polynomial f is a Julia set and its interior, non-escaping set Formal definition The filled-in Julia set K(f) of a polynomial f is defined as the set of all points z of the dynamical plane that have bounded orbit with respect to f K(f) \overset \left \ where: * \mathbb is the set of complex numbers * f^ (z) is the k -fold composition of f with itself = iteration of function f Relation to the Fatou set The filled-in Julia set is the (absolute) complement of the attractive basin of infinity. K(f) = \mathbb \setminus A_(\infty) The attractive basin of infinity is one of the components of the Fatou set. A_(\infty) = F_\infty In other words, the filled-in Julia set is the complement of the unbounded Fatou component: K(f) = F_\infty^C. Relation between Julia, filled-in Julia set and attractive basin of infinity The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity J(f) = \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volodymyr Nekrashevych
Volodymyr ( uk, Володи́мир, Volodýmyr, , orv, Володимѣръ) is a Ukrainian given name of Old East Slavic origin. The related Ancient Slavic, such as Czech, Russian, Serbian, Croatian, etc. form of the name is Володимѣръ ''Volodiměr'', which in other Slavic languages became Vladimir (from cu, Владимѣръ, Vladiměr). Diminutives include Volodyk, Volodia, Lodgo and Vlodko People known as Volodymyr * Volodymyr the Great (aka St. Volodymyr, Volodymyr I of Kyiv), Grand Prince of Kyiv * Volodymyr II Monomakh, Grand Prince of Kyiv * Volodymyr Atamanyuk (born 1955), Soviet footballer * Volodymyr Bahaziy (1902–1942), Ukrainian nationalist * Volodymyr Barilko (born 1994), Ukrainian football striker * Volodymyr Bezsonov (born 1958), Ukrainian football manager and player * Volodymyr Chesnakov (born 1988), Ukrainian footballer * Volodymyr Demchenko (born 1981), Ukrainian sprinter who competed in the 2004 Summer Olympics * Volodymyr Dyudya (born 1983), U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Bartholdi
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Laurent may refer to: *Laurent (name), a French masculine given name and a surname **Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent **Pierre Alphonse Laurent, mathematician **Joseph Jean Pierre Laurent, amateur astronomer, discoverer of minor planet (51) Nemausa *Laurent, South Dakota, a proposed town for the Deaf to be named for Laurent Clerc See also *Laurent series, in mathematics, representation of a complex function ''f(z)'' as a power series which includes terms of negative degree, named for Pierre Alphonse Laurent *Saint-Laurent (other) *Laurence (name), feminine form of "Laurent" *Lawrence (other) Lawrence may refer to: Education Colleges and universities * Lawrence Technological University, a university in Southfield, Michigan, United States * Lawrence University, a liberal arts university in Appleton, Wisconsin, United States Preparator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
