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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] |
Julia Set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under iterated function, repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small Perturbation theory, 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 "chaos theory, 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. Form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Quadratic Polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable (mathematics), variable are complex numbers. Properties Quadratic polynomials have the following properties, regardless of the form: *It is a unicritical polynomial, i.e. it has one #Critical_points, finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: the basin of infinity and basin of the finite critical point (if the finite critical point does not escape) *It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic. * It is a Unimodality#Unimodal function, unimodal function, * It is a rational function, * It is an entire function. Forms When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms: * The general form: f(x) = a_2 x^2 + a_1 x + a_0 where a_2 \ne 0 * The factored form used for the logistic map: f_r(x) = r x (1-x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractals
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bodil Branner
Bodil Branner (born 5 February 1943, in Aarhus) is a retired Danish mathematician, one of the founders of European Women in Mathematics and a former chair of the Danish Mathematical Society. Her research concerned holomorphic dynamics and the history of mathematics. Education and career Branner studied mathematics and physics at Aarhus University, where mathematician Svend Bundgaard was one of her mentors, and in 1967 earned a master's degree (the highest degree then available) under the supervision of Leif Kristensen. She had intended to travel to the U.S. for a doctorate, but her husband, a chemist, took an industry job in Copenhagen. Branner could not get a job as a high school teacher because she did not have a teaching qualification, but Bundgaard found her a position as a faculty assistant for Frederik Fabricius-Bjerre at the Technical University of Denmark. Despite this not beginning as an actual faculty position, she eventually earned tenure there in the 1970s. Europe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Disc
A Siegel disc or Siegel disk 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 , \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douady Rabbit
A Douady rabbit is a fractal derived from the filled Julia set, Julia set of the function f_c(z) = z^2+c, when Parameter (computer programming), parameter c is near the center of one of the Mandelbrot set#Main cardioid and period bulbs, period three bulbs of the Mandelbrot set for a complex quadratic map. It is named after France, French mathematician Adrien Douady. Background The Douady rabbit is generated by iterating the Mandelbrot set, Mandelbrot set map z_=z_n^2+c on the complex plane, where parameter c is fixed to lie in one of the two period three bulb off the main cardioid and z ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value z_0 and calculating the amount of iteration, iterations required before the value of z_n escapes a bounded region, after which it will diverge toward infinity. It can also be described using the ''Complex quadratic polynomial#Forms, logistic map form'' of the complex quadratic map, specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karen Brucks
Karen Marie Brucks (February 1, 1957 – July 8, 2017) was an American mathematician known for her research in topological dynamics, and for her advocacy of women in mathematics. She worked for many years as a faculty member and administrator at the University of Wisconsin–Milwaukee. Life Brucks was born on February 1, 1957, in Chicago. She majored in mathematics at the University of Arizona, graduating in 1980. Next, she earned a master's degree in 1982 at the University of North Texas, and continued there for doctoral study, completing her Ph.D. in 1988. Her doctoral dissertation, ''Dynamics of One Dimensional Maps'', was supervised by R. Daniel Mauldin. After postdoctoral positions at Michigan State University and Stony Brook University, she became a faculty member at the University of Wisconsin–Milwaukee in 1991, eventually serving for 24 years there. She was a Fulbright Scholar in 1997–1998, on a research visit to Hungary. At the University of Wisconsin–Milwaukee, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
External Ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly in complex dynamics and geometric function theory. History External rays were introduced in Douady and Hubbard's study of the Mandelbrot set Types Criteria for classification : * plane : parameter or dynamic * map * bifurcation of dynamic rays * Stretching * landing plane External rays of (connected) Julia sets on dynamical plane are often called dynamic rays. External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays. bifurcation Dynamic ray can be: * bifurcated = branched = broken * smooth = unbranched = unbroken When the filled Julia set is connected, there are no branching external rays. When the Julia set is not connected then some external ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Quadratic Polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable (mathematics), variable are complex numbers. Properties Quadratic polynomials have the following properties, regardless of the form: *It is a unicritical polynomial, i.e. it has one #Critical_points, finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: the basin of infinity and basin of the finite critical point (if the finite critical point does not escape) *It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic. * It is a Unimodality#Unimodal function, unimodal function, * It is a rational function, * It is an entire function. Forms When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms: * The general form: f(x) = a_2 x^2 + a_1 x + a_0 where a_2 \ne 0 * The factored form used for the logistic map: f_r(x) = r x (1-x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Points Of Complex Quadratic Mappings
This article describes periodic points of some Complex quadratic polynomial, complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a Quadratic equation, quadratic map is one that involves the previous value raised to the power (mathematics), powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length. These periodic points play a role in the theories of Fatou set, Fatou and Julia sets. Definitions Let :f_c(z) = z^2+c\, be the complex quadratic polynomial, complex quadratic mapping, where z and c are complex numbers. Notationally, f^ _c (z) is the k-fold Function composition, composition of f_c with itself (not to be confused with the kth derivative of f_c)—that is, the value after the ''k''-th Iterated function, iteration of the function f _c. Thus :f^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
