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| | Hi there, I am Alyson Boon even though it is not the title on my birth certificate. For many years he's been living in Alaska and he doesn't plan on altering it. She is really fond of caving but she doesn't have the time recently. I am presently a travel agent.<br><br>My webpage - clairvoyants ([http://chungmuroresidence.com/xe/reservation_branch2/152663 http://chungmuroresidence.com/xe/reservation_branch2/152663]) |
| A '''complex quadratic polynomial''' is a [[quadratic polynomial]] whose [[coefficient]]s are [[complex number]]s.
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| ==Forms==
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| When the quadratic polynomial has only one variable ([[univariate]]), one can distinguish its 4 main forms:
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| * The general form: <math> f(x) = a_2 x^2 + a_1 x + a_0 \qquad \, </math> where <math> \qquad a_2 \ne 0</math>
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| * The factored form used for [[logistic map]] <math>f_r(x) = r x ( 1-x ) \,</math>
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| * <math> f_{\theta}(x) = x^2 + e^{2 \pi \theta i} x \,</math> which has an indifferent [[Fixed point (mathematics)|fixed point]] with [[Periodic_points_of_complex_quadratic_mappings#Stability_of_periodic_points_.28orbit.29_-_multiplier|multiplier]] <math>\lambda = e^{2 \pi \theta i} \,</math> at the [[Origin (mathematics)|origin]]<ref>[http://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials. ]</ref>
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| * The monic and centered form, <math>f_c(x) = x^2 +c\,</math>
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| The '''[[Monic polynomial|monic]] and centered form''' has the following properties:
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| * It is the simplest form of a [[Nonlinearity (disambiguation)|nonlinear]] [[Function (mathematics)|function]] with one [[coefficient]] ([[parameter]]),
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| * It is an unicritical polynomial, i.e. it has one [[Critical point (mathematics)|critical point]],
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| * It is a centered polynomial (the sum of its critical points is zero),<ref>[http://www.mat.dtu.dk/English/Medarbejdere/MAT_VIP.aspx?lg=showcommon&id=260&type=person B Branner]: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark</ref>
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| * It can be '''postcritically finite''', i.e. If the orbit of the critical point is finite. It is when critical point is periodic or preperiodic.<ref>[http://arxiv.org/abs/math/9305207 Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits ]</ref>
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| * It is a [[unimodal]] [[Function (mathematics)|function]],
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| * It is a [[rational function]],
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| * It is an [[entire function]].
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| ==Conjugation==
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| ===Between forms===
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| Since <math>f_c(x) \,</math> is [[Affine transformation|affine]] [[Topological conjugation|conjugate]] to the general form of the quadratic polynomial it is often used to study [[complex dynamics]] and to create images of [[Mandelbrot set|Mandelbrot]], [[Julia set|Julia]] and [[Fatou set]]s.
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| When one wants change from <math>\theta\,</math> to <math>c \,</math>:<ref>[http://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials. ]</ref>
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| :<math>c = c(\theta) = \frac {e^{2 \pi \theta i}}{2} \left(1 - \frac {e^{2 \pi \theta i}}{2}\right) </math>.
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| When one wants change from <math>r\,</math> to <math>c \,</math>:
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| :<math>
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| c = c(r)\,=\,\frac{1- (r-1)^2}{4}
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| </math>.
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| ===With doubling map===
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| There is semi-conjugacy between the [[dyadic transformation]] (here named doubling map) and the quadratic polynomial.
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| ==Family==
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| The family of quadratic polynomials <math>f_c : z \to z^2 +c\,</math> parametrised by <math> c \in \mathbb{C} \,</math> is called:
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| * '''the Douady-Hubbard family of quadratic polynomials'''<ref>[http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/2004MandLocConn.pdf Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points ] Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264</ref>
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| *'''quadratic family'''
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| ==Map==
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| The monic and centered form is typically used with [[Variable (mathematics)|variable]] <math>z\,</math> and [[parameter]] <math>c\,</math>:
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| :<math>f_c(z) = z^2 +c.\,</math>
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| When it is used as an [[Dynamical system (definition)|evolution function]] of [[Dynamical system|the discrete nonlinear dynamical system]]:
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| : <math>z_{n+1} = f_c(z_n) \,</math>
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| it is named '''[[Quadratic polynomial|quadratic]] [[Map (mathematics)|map]]''':<ref>[http://mathworld.wolfram.com/QuadraticMap.html Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Web Resourc]</ref>
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| :<math>f_c : z \to z^2 + c. \,</math>
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| ==Notation==
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| Here <math> f^n \,</math> denotes the ''n''-th [[iterated function|iteration]] of the function <math> f \,</math> not [[exponentiation]]
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| :<math>f_c^n(z) = f_c^1(f_c^{n-1}(z)) \,</math>
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| so
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| :<math>z_n = f_c^n(z_0). \,</math>
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| Because of the possible confusion it is customary to write <math>f^{\circ n}\,</math> for the ''n''th iterate of the function <math>f.\,</math>
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| ==Critical items==
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| ===Critical point===
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| A '''[[Critical point (mathematics)|critical point]]''' of <math>f_c\,</math> is a point <math> z_{cr} \,</math> in the dynamical plane such that the [[derivative]] vanishes:
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| : <math>f_c'(z_{cr}) = 0. \,</math>
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| Since
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| : <math>f_c'(z) = \frac{d}{dz}f_c(z) = 2z </math>
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| implies
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| : <math> z_{cr} = 0\,</math>
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| we see that the only (finite) critical point of <math>f_c \,</math> is the point <math> z_{cr} = 0\,</math>.
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| <math>z_0</math> is an initial point for [[Mandelbrot set]] iteration.<ref>[http://mathesim.degruyter.de/jws_en/show_simulation.php?id=1052&type=RoessMa&lang=en Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations]</ref>
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| ===Critical value===
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| A '''[[critical value]]''' <math>z_{cv} \ </math> of <math>f_c\,</math> is the image of a critical point:
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| : <math>z_{cv} = f_c(z_{cr}) \,</math>
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| Since
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| : <math> z_{cr} = 0\,</math>
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| we have
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| : <math>z_{cv} = c. \,</math>
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| So the parameter <math> c \,</math> is the critical value of <math>f_c(z). \,</math>
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| ===Critical orbit===
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| [[File:Cr orbit 3.png|thumb|right|Dynamical plane with critical orbit falling into 3-period cycle]]
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| [[File:Miimcr.png|thumb|right|Dynamical plane with Julia set and critical orbit.]]
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| [[File:6furcation.gif|thumb|right|Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6]]
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| [[File:Critical orbit 3d.png|right|thumb|Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259]]
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| [[orbit (dynamics)|Forward orbit]] of a critical point is called a '''critical orbit'''. Critical orbits are very important because every attracting [[Periodic points of complex quadratic mappings|periodic orbit]] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the [[Fatou set]].<ref>[http://www.iec.csic.es/~miguel/miguel.html M. Romera], [http://www.iec.csic.es/~gerardo/Gerardo.html G. Pastor], and F. Montoya : [http://www.iec.csic.es/~miguel/Preprint4.ps Multifurcations in nonhyperbolic fixed points of the Mandelbrot map.] [http://users.utcluj.ro/~mdanca/fractalia/ Fractalia] 6, No. 21, 10-12 (1997)</ref><ref>[http://myweb.cwpost.liu.edu/aburns/index.html Burns A M] : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116</ref>
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| <math>z_0 = z_{cr} = 0\,</math>
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| <math>z_1 = f_c(z_0) = c\,</math>
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| <math>z_2 = f_c(z_1) = c^2 +c\,</math>
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| <math>z_3 = f_c(z_2) = (c^2 + c)^2 + c\,</math>
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| <math>... \,</math>
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| This orbit falls into an [[Periodic points of complex quadratic mappings|attracting periodic cycle]].
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| ===Critical sector===
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| The [[Orbit_portrait#Sectors|critical sector]] is a sector of the dynamical plane containing the critical point.
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| ===Critical polynomial===
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| <math>P_n(c) = f_c^n(z_{cr}) = f_c^n(0) \,</math>
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| so
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| <math>P_0(c)= 0 \,</math>
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| <math>P_1(c) = c \,</math>
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| <math>P_2(c) = c^2 + c \,</math>
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| <math>P_3(c) = (c^2 + c)^2 + c \,</math>
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| These polynomials are used for:
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| * finding centers of these Mandelbrot set components of period n. Centers are roots of n-th critical polynomials
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| <math>centers = \{ c : P_n(c) = 0 \}\,</math>
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| * finding roots of Mandelbrot set components of period n ([[local minimum]] of <math>P_n(c) \,</math>)
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| * [[Misiurewicz point]]s
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| <math>M_{n,k} = \{ c : P_k(c) = P_{k+n}(c) \}\,</math> | |
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| ===Critical curves===
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| Diagrams of critical polynomials are called '''critical curves'''.<ref>The Road to Chaos is Filled with Polynomial Curves
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| by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653</ref>
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| These curves create skeleton of [[bifurcation diagram]].<ref>{{Cite book
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| | last = Hao
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| | first = Bailin
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| | authorlink = Bailin Hao
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| | coauthors =
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| | title = Elementary Symbolic Dynamics and Chaos in Dissipative Systems
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| | publisher = [[World Scientific]]
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| | year = 1989
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| | location =
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| | pages =
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| | url = http://power.itp.ac.cn/~hao/
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| | doi =
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| | id =
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| | isbn =9971-5-0682-3 }}</ref> (the dark lines<ref>[http://www.iec.csic.es/~gerardo/publica/Romera96b.pdf M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint]</ref>)
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| ==Planes==
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| [[File:Iray.png|right|thumb|w-plane and c-plane]]
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| One can use the Julia-Mandelbrot 4-dimensional space for a global analysis of this dynamical system.<ref>[http://www.mrob.com/pub/muency/juliamandelbrotspace.html Julia-Mandelbrot Space at Mu-ency by Robert Munafo ]</ref>
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| In this space there are 2 basic types of 2-D planes:
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| * the dynamical (dynamic) plane, <math>f_c\,</math>-plane or '''c-plane'''
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| * the parameter plane or '''z-plane'''
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| There is also another plane used to analyze such dynamical systems '''w-plane''':
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| * the conjugation plane<ref>Carleson, Lennart, Gamelin, Theodore W.: Complex Dynamics Series: Universitext, Subseries: Universitext: Tracts in Mathematics, 1st ed. 1993. Corr. 2nd printing, 1996, IX, 192 p. 28 illus., ISBN 978-0-387-97942-7</ref>
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| * model plane<ref>Holomorphic motions and puzzels by P Roesch</ref>
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| ===Parameter plane===
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| [[File:MandelbrotGammaDouadyRabbit.jpg|right|thumb|Gamma parameter plane for complex logistic map <math>z_{n+1} = \gamma z_n \left(1 - z_n\right),</math>]]
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| The [[phase space]] of a quadratic map is called its '''parameter plane'''. Here:
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| <math>z0 = z_{cr} \,</math> is [[Coefficient|constant]] and <math>c\,</math> is variable.
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| There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
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| The parameter plane consists of:
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| * The [[Mandelbrot set]]
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| ** The [[bifurcation locus]] = boundary of [[Mandelbrot set]]
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| ** Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set<ref>[http://front.math.ucdavis.edu/0805.1658 [[Lasse Rempe]], Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity]</ref>
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| There are many different subtypes of the parameter plane.<ref>[http://aleph0.clarku.edu/~djoyce/julia/altplane.html Alternate Parameter Planes by David E. Joyce ]</ref><ref>[http://mrob.com/pub/muency/exponentialmap.html exponentialmap by Robert Munafo]</ref>
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| ===Dynamical plane===
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| On the dynamical plane one can find:
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| * The [[Julia set]]
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| * The [[Filled Julia set]]
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| * The [[Fatou set]]
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| * Orbits
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| The dynamical plane consists of:
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| * [[Fatou set]]
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| * [[Julia set]]
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| Here, <math>c\,</math> is a [[Coefficient|constant]] and <math>z\,</math> is a variable.
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| The two-dimensional dynamical plane can be treated as a [[Poincaré map|Poincaré cross-section]] of three-dimensional space of continuous dynamical system.<ref>[http://www.sgtnd.narod.ru/science/complex/eng/main.htm Mandelbrot set by Saratov group of theoretical nonlinear dynamics]</ref><ref>[http://www.scholarpedia.org/article/Periodic_orbit#Stability_of_a_Periodic_Orbit|Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,]</ref>
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| ==Derivatives==
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| ===Derivative with respect to ''c''===
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| On parameter plane:
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| * <math>c</math> is a variable
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| * <math>z_0 = 0 </math> is constant
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| The first [[derivative]] of <math>f_c^n(z_0)</math> with respect to ''c'' is
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| : <math>z_n' = \frac{d}{dc} f_c^n(z_0).</math>
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| This [[derivative]] can be found by [[Iterated function|iteration]] starting with
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| : <math>z_0' = \frac{d}{dc} f_c^0(z_0) = 1</math>
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| and then replacing at every consecutive step
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| : <math>z_{n+1}' = \frac{d}{dc} f_c^{n+1}(z_0) = 2\cdot{}f_c^n(z)\cdot\frac{d}{dc} f_c^n(z_0) + 1 = 2 \cdot z_n \cdot z_n' +1.</math>
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| This can easily be verified by using the [[chain rule]] for the derivative.
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| This derivative is used in the [[Mandelbrot_set#Distance_estimates|distance estimation method for drawing a Mandelbrot set]].
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| ===Derivative with respect to ''z''===
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| On dynamical plane:
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| * <math>z</math> is a variable
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| * <math>c </math> is a constant
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| at a '''fixed point''' <math>z_0\,</math>
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| : <math>f_c'(z_0) = \frac{d}{dz}f_c(z_0) = 2z_0 </math>
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| at a''' periodic point''' ''z''<sub>0</sub> of period ''p''
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| : <math>(f_c^p)'(z_0) = \frac{d}{dz}f_c^p(z_0) = \prod_{i=0}^{p-1} f_c'(z_i) = 2^p \prod_{i=0}^{p-1} z_i. </math>
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| It is used to check the [[Stability theory|stability]] of [[Periodic points of complex quadratic mappings|periodic (also fixed) points]].
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| at '''nonperiodic point''':
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| :<math>z'_n\,</math>
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| This [[derivative]] can be found by [[Iterated function|iteration]] starting with
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| : <math>z'_0 = 1 \,</math>
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| and then :
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| :<math>z'_n= 2*z_{n-1}*z'_{n-1}\,</math>
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| This dervative is used for computing external distance to Julia set.
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| ===Schwarzian derivative===
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| The [[Schwarzian derivative]] (SD for short) of f is:<ref>[http://ocw.mit.edu/courses/mathematics/18-091-mathematical-exposition-spring-2005/lecture-notes/lecture09.pdf The Schwarzian Derivative & the Critical Orbit by Wes McKinney 18.091 20 April 2005]</ref>
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| :<math> (Sf)(z) = \frac{f'''(z)}{f'(z)} - \frac{3}{2} \left ( \frac{f''(z)}{f'(z)}\right ) ^2 </math>.
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| ==See also==
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| *[[Misiurewicz point]]
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| *[[Periodic points of complex quadratic mappings]]
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| *[[Mandelbrot set]]
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| *[[Julia set]]
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| *[[Milnor–Thurston kneading theory]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{sisterlinks|Complex quadratic polynomial}}
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| *[http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/nevins.pdf M. Nevins and D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers" ]
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| *[http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002]
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| {{Chaos theory}}
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| {{DEFAULTSORT:Complex Quadratic Polynomial}}
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| [[Category:Complex dynamics]]
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| [[Category:Fractals]]
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| [[Category:Polynomials]]
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