|
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 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 on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Quadratic Polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and 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 finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity and basin of finite critical point ( if finite critical point do not escapes) *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 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) * f_(x) = x^2 +\lambda x which has an indifferent fixed point with multiplier \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractals
In mathematics, a fractal is a 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). Likewise, ... [...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 Bent 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. She was the fir ... [...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] |
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] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 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, and also goes by several 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 construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Quadratic Polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and 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 finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity and basin of finite critical point ( if finite critical point do not escapes) *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 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) * f_(x) = x^2 +\lambda x which has an indifferent fixed point with multiplier \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Points Of Complex Quadratic Mappings
This article describes periodic points of some 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 map is one that involves the previous value raised to the 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 and Julia sets. Definitions Let :f_c(z) = z^2+c\, be the complex quadric mapping, where z and c are complex numbers. Notationally, f^ _c (z) is the k-fold 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 iteration of the function f _c. Thus :f^ _c (z) = f_c(f^ _c (z)). Periodic points of a complex quadratic mapping of period p are points z of the dynamical plane such that :f^ _c (z) = z, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
