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| This article describes [[periodic point]]s of some [[Complex quadratic polynomial|complex quadratic map]]s. 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 powers one and two; and a '''complex''' map is one in which the variable is a [[complex number]]. A [[periodic point]] of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.
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| This theory is applied in relation with the theories of [[Fatou set|Fatou]] and [[Julia set]]s.
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| ==Definitions==
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| Let
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| :<math>f_c(z)=z^2+c\,</math>
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| where <math>z</math> and <math>c</math> are [[Complex number|complex-valued]]. (This <math>\ f</math> is the ''[[Complex quadratic polynomial|complex quadratic mapping]]'' mentioned in the title.) This article explores the ''[[periodic point]]s'' of this [[Map (mathematics)|mapping]] - that is, the points that form a periodic cycle when <math>\ f</math> is repeatedly applied to them.
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| <math> \ f^{(k)} _c (z) </math> is the <math>\ k</math> -fold [[Function composition|compositions]] of <math>f _c\,</math> with itself = [[Iterated function|iteration of function]] <math>f _c\,</math> or,
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| <math> \ f^{(k)} _c (z) = f_c(f^{(k-1)} _c (z))</math>
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| Periodic points of a [[Complex quadratic polynomial|complex quadratic mapping]] of [[Frequency|period]] <math>\ p</math> are points <math> \ z</math> of the [[phase space|dynamical plane]] such that :
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| <math> \ z : f^{(p)} _c (z) = z</math>
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| where <math>\ p</math> is the smallest positive integer.
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| We can introduce a new function:
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| <math> \ F_p(z,f) = f^{(p)} _c (z) - z</math>
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| so periodic points are zeros of function <math> \ F_p(z,f) </math> :
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| <math> \ z : F_p(z,f) = 0</math>
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| which is a polynomial of degree <math> \ = 2^p</math>
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| ==Stability of periodic points (orbit) - multiplier==
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| [[File:Multiplier4 f.png|right|thumb|Stability index of periodic points along horizontal axis]]
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| [[File:Mandelbrot set Components.jpg|right|thumb|boundaries of regions of parameter plane with attracting orbit of periods 1-6]]
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| [[File:Critical orbit 3d.png|right|thumb|Critical orbit of discrete dynamical system based on [[complex quadratic polynomial]]. It tends to weakly [[Attractor|attracting]] [[Fixed point (mathematics)|fixed point]] with abs(multiplier)=0.99993612384259]]
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| The '''[[Multiplication|multiplier]]''' ( or eigenvalue, derivative ) <math>m(f,z_0)=\lambda \,</math> of rational map <math>f\,</math> at fixed point <math>z_0\,</math> is defined as :
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| <math>
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| m(f,z_0)=\lambda =
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| \begin{cases}
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| f_c'(z_0), &\mbox{if }z_0\ne \infty \\
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| \frac{1}{f_c'(z_0)}, & \mbox{if }z_0 = \infty
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| \end{cases}
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| </math>
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| where <math>f_c'(z_0)\,</math> is [[Complex quadratic polynomial|first derivative]] of <math> \ f_c</math> with respect to <math>z\,</math> at <math>z_0\,</math>.
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| Because the multiplier is the same at all periodic points, it can be called a multiplier of periodic [[orbit (dynamics)|orbit]].
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| [[Multiplication|Multiplier]] is:
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| *[[complex number]],
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| *invariant under conjugation of any rational map at its fixed point<ref>Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41</ref>
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| *used to check stability of periodic (also fixed) points with '''stability index''' : <math>abs(\lambda) \,</math>
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| Periodic point is :<ref>Alan F. Beardon, ''Iteration of Rational Functions'', Springer 1991, ISBN 0-387-95151-2, page 99</ref>
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| * attracting when <math>abs(\lambda) < 1 \,</math>
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| ** super-attracting when <math>abs(\lambda) = 0 \,</math>
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| ** attracting but not super-attracting when <math>0 < abs(\lambda) < 1 \,</math>
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| * indifferent when <math>abs(\lambda) = 1 \,</math>
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| ** rationally indifferent or parabolic if <math>abs(\lambda) \,</math> is a [[root of unity]]
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| ** [[Siegel disc|irrationally indifferent]] if <math>abs(\lambda)=1 \,</math> but multiplier is not a root of unity
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| * repelling when <math>abs(\lambda) > 1 \,</math>
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| Where do periodic points belong?
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| * attracting is always in [[Classification of Fatou components|Fatou set]]
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| * repelling is in the Julia set
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| * Indifferent fixed points may be in the one or in the other.<ref>[http://www.ijon.de/mathe/julia/some_julia_sets_1_en.html Some Julia sets by Michael Becker]</ref> Parabolic periodic point is in Julia set.
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| ==Period-1 points (fixed points)==
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| ===Finite fixed points===
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| Let us begin by finding all [[Wikt:finite|finite]] points left unchanged by 1 application of <math>f</math>. These are the points that satisfy <math>\ f_c(z)=z</math>. That is, we wish to solve
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| : <math>z^2+c=z\,</math>
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| which can be rewritten
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| : <math>\ z^2-z+c=0.</math>
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| Since this is an ordinary quadratic equation in 1 unknown, we can apply [[Quadratic equation|the standard quadratic solution formula]]. Look in any standard mathematics textbook, and you will find that there are two solutions of <math>\ Ax^2+Bx+C=0</math> are given by
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| : <math>x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}</math>
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| In our case, we have <math>A=1, B=-1, C=c</math>, so we will write
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| : <math>\alpha_1 = \frac{1-\sqrt{1-4c}}{2}</math> and <math>\alpha_2 = \frac{1+\sqrt{1-4c}}{2}.</math>
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| So for <math>c \in C \setminus [1/4,+\inf ]</math> we have two [[Wikt:finite|finite]] fixed points <math>\alpha_1 \,</math> and <math>\alpha_2\, </math>. | |
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| Since
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| : <math>\alpha_1 = \frac{1}{2}-m</math> and <math>\alpha_2 = \frac{1}{2}+ m</math> where <math>m = \frac{\sqrt{1-4c}}{2}</math>
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| then <math>\alpha_1 + \alpha_2 = 1 \,</math>.
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| It means that fixed points are symmetrical around <math>z = 1/2\,</math>.
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| [[File:JuliaRay3.png|thumb|right|This image shows fixed points (both repelling)]]
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| ====Complex dynamics====
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| [[File:Fixed.png|right|thumb|Fixed points for c along horizontal axis]]
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| [[File:Julia0bb.jpg|thumb| [[Fatou set]] for F(z)=z*z with marked fixed point]]
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| Here different notation is commonly used:<ref>[http://www.math.nagoya-u.ac.jp/~kawahira/works/cauliflower.pdf On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178. ]</ref>
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| : <math>\alpha_c = \frac{1-\sqrt{1-4c}}{2}</math> with multiplier <math>\lambda_{\alpha_c} = 1-\sqrt{1-4c}\,</math>
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| and | |
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| :<math>\beta_c = \frac{1+\sqrt{1-4c}}{2}</math> with multiplier <math>\lambda_{\beta_c} = 1+\sqrt{1-4c}\,</math>
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| Using [[Viète's formulas]] one can show that:
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| :<math> \alpha_c + \beta_c = -\frac{B}{A} = 1 </math>
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| Since [[Complex quadratic polynomial#Derivative with respect to z|derivative with respect to z]] is :
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| :<math>P_c'(z) = \frac{d}{dz}P_c(z) = 2z </math>
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| then | |
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| :<math>P_c'(\alpha_c) + P_c'(\beta_c)= 2 \alpha_c + 2 \beta_c = 2 (\alpha_c + \beta_c) = 2 \,</math>
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| It implies that <math>P_c \,</math> can have at most one attractive fixed point.
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| This points are distinguished by the facts that:
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| * <math>\beta_c \,</math> is :
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| **the landing point of [[external ray]] for angle=0 for <math>c \in M \setminus \left \{ \frac{1}{4} \right \}</math>
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| **the most repelling fixed point, belongs to Julia set,
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| ** the one on the right ( whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower).<ref>[http://www.ibiblio.org/e-notes/MSet/Attractor.htm Periodic attractor by Evgeny Demidov]</ref>
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| * <math>\alpha_c \,</math> is:
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| ** landing point of several rays
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| ** is :
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| ***attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, it means belongs to Fatou set ( strictly to basin of attraction of finite fixed point )
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| ***parabolic at the root point of the limb of Mandelbrot set
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| ***repelling for other c values
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| ====Special cases====
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| An important case of the quadratic mapping is <math>c=0</math>. In this case, we get <math>\alpha_1 = 0</math> and <math>\alpha_2=1</math>. In this case, 0 is a superattractive [[Fixed point (mathematics)|fixed point]], and 1 belongs to the [[Julia set]].
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| ====Only one fixed point====
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| We might wonder what value <math>c</math> should have to cause <math>\alpha_1=\alpha_2</math>. The answer is that this will happen exactly when <math>1-4c=0</math>. This equation has 1 solution: <math>c=1/4</math> (in which case, <math>\alpha_1=\alpha_2=1/2</math>). This is interesting, since <math>c=1/4</math> is the largest positive, purely real value for which a finite attractor exists.
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| ===Infinite fixed point===
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| We can extend [[complex plane]] <math>\mathbb{C}</math> to [[Riemann sphere|Riemann sphere (extended complex plane)]] <math>\mathbb{\hat{C}}</math> by
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| adding [[Point at infinity|infinity]]
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| <math>\mathbb{\hat{C}} = \mathbb{C} \cup \{ \infty \}</math>
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| and extend [[Complex quadratic polynomial|polynomial]] <math>f_c\,</math> such that <math>f_c(\infty)=\infty\,</math>
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| Then [[Point at infinity|infinity]] is :
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| *superattracting
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| *fixed point of [[Complex quadratic polynomial|polynomial]] <math>f_c\,</math><ref>R L Devaney, L Keen (Editor): Chaos and Fractals: The Mathematics Behind the Computer Graphics. Publisher: Amer Mathematical Society July 1989, ISBN 0-8218-0137-6 , ISBN 978-0-8218-0137-6</ref>
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| <math>f_c(\infty)=\infty=f^{-1}_c(\infty)\,</math>
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| ==Period-2 cycles==
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| [[File:Bifurcation1-2.png|right|thumb|300px|Bifurcation from period 1 to 2 for [[complex quadratic polynomial|complex quadratic map]]]]
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| Suppose next that we wish to look at ''period-2 cycles''. That is, we want to find two points <math>\beta_1</math> and <math>\beta_2</math> such that <math>f_c(\beta_1) = \beta_2</math>, and <math>f_c(\beta_2) = \beta_1</math>.
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| Let us start by writing <math>f_c(f_c(\beta_n)) = \beta_n</math>, and see where trying to solve this leads.
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| : <math>f_c(f_c(z)) = (z^2+c)^2+c = z^4 + 2z^2c + c^2 + c.\,</math>
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| Thus, the equation we wish to solve is actually <math>z^4 + 2cz^2 - z + c^2 + c = 0</math>.
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| This equation is a polynomial of degree 4, and so has 4 (possibly non-distinct) solutions. ''However'', actually, we already know 2 of the solutions. They are <math>\alpha_1</math> and <math>\alpha_2</math>, computed above. It is simple to see why this is; if these points are left unchanged by 1 application of <math>f</math>, then clearly they will be unchanged by 2 applications (or more).
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| Our 4th-order polynomial can therefore be factored in 2 ways :
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| ===First method===
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| : <math>(z-\alpha_1)(z-\alpha_2)(z-\beta_1)(z-\beta_2) = 0.\,</math>
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| This expands directly as <math>x^4 - Ax^3 + Bx^2 - Cx + D = 0</math> (note the alternating signs), where
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| : <math>D = \alpha_1 \alpha_2 \beta_1 \beta_2\,</math>
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| : <math>C = \alpha_1 \alpha_2 \beta_1 + \alpha_1 \alpha_2 \beta_2 + \alpha_1 \beta_1 \beta_2 + \alpha_2 \beta_1 \beta_2\,</math>
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| : <math>B = \alpha_1 \alpha_2 + \alpha_1 \beta_1 + \alpha_1 \beta_2 + \alpha_2 \beta_1 + \alpha_2 \beta_2 + \beta_1 \beta_2\,</math>
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| : <math>A = \alpha_1 + \alpha_2 + \beta_1 + \beta_2.\,</math>
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| We already have 2 solutions, and only need the other 2. This is as difficult as solving a quadratic polynomial. In particular, note that
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| : <math>\alpha_1 + \alpha_2 = \frac{1-\sqrt{1-4c}}{2} + \frac{1+\sqrt{1-4c}}{2} = \frac{1+1}{2} = 1</math>
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| and
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| : <math>\alpha_1 \alpha_2 = \frac{(1-\sqrt{1-4c})(1+\sqrt{1-4c})}{4} = \frac{1^2 - (\sqrt{1-4c})^2}{4}= \frac{1 - 1 + 4c}{4} = \frac{4c}{4} = c.</math>
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| Adding these to the above, we get <math>D = c \beta_1 \beta_2</math> and <math>A = 1 + \beta_1 + \beta_2</math>. Matching these against the coefficients from expanding <math>f</math>, we get
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| : <math>D = c \beta_1 \beta_2 = c^2 + c</math> and <math>A = 1 + \beta_1 + \beta_2 = 0.</math>
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| From this, we easily get :
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| <math>\beta_1 \beta_2 = c + 1</math> and <math>\beta_1 + \beta_2 = -1</math>. | |
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| From here, we construct a quadratic equation with <math>A' = 1, B = 1, C = c+1</math> and apply the standard solution formula to get
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| : <math>\beta_1 = \frac{-1 - \sqrt{-3 -4c}}{2}</math> and <math>\beta_2 = \frac{-1 + \sqrt{-3 -4c}}{2}.</math>
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| Closer examination shows (the formulas are a tad messy) that :
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| <math>f_c(\beta_1) = \beta_2</math> and <math>f_c(\beta_2) = \beta_1</math>
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| meaning these two points are the two halves of a single period-2 cycle.
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| ===Second method of factorization===
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| <math>(z^2+c)^2 + c -z = (z^2 + c - z)(z^2 + z + c +1 ) \,</math>
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| The roots of the first factor are the two fixed points <math>z_{1,2}\,</math> . They are repelling outside the main cardioid.
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| The second factor has two roots
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| <math>z_{3,4} = -\frac{1}{2} \pm (-\frac{3}{4} - c)^\frac{1}{2}. \,</math>
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| These two roots form period-2 orbit.<ref>[http://www.ibiblio.org/e-notes/MSet/Attractor.htm Period 2 orbit by Evgeny Demidov]</ref>
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| ====Special cases====
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| Again, let us look at <math>c=0</math>. Then
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| : <math>\beta_1 = \frac{-1 - i\sqrt{3}}{2}</math> and <math>\beta_2 = \frac{-1 + i\sqrt{3}}{2}</math>
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| both of which are complex numbers. By doing a little algebra, we find <math>| \beta_1 | = | \beta_2 | = 1</math>. Thus, both these points are "hiding" in the Julia set.
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| Another special case is <math>c=-1</math>, which gives <math>\beta_1 = 0</math> and <math>\beta_2 = -1</math>. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.
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| ==Cycles for period>2==
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| [[Abel–Ruffini theorem|There is no general solution]] in [[Nth root|radicals]] to polynomial equations of degree five or higher, so it must be computed using [[Root-finding algorithm|numerical methods]].
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| ==References==
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| {{Reflist}}
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| == Further reading ==
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| *Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2
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| *Michael F. Barnsley (Author), Stephen G. Demko (Editor), Chaotic Dynamics and Fractals (Notes and Reports in Mathematics in Science and Engineering Series) Academic Pr (April 1986), ISBN 0-12-079060-2
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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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| *[http://hdl.handle.net/10090/3895 The permutations of periodic points in quadratic polynominials by J Leahy]
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| ==External links==
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| {{Wikibooks|Fractals }}
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| *[http://cosinekitty.com/mandel_orbits_analysis.html ''Algebraic solution of Mandelbrot orbital boundaries'' by Donald D. Cross ]
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| *[http://www.mrob.com/pub/muency/brownmethod.html ''Brown Method'' by Robert P. Munafo]
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| *[http://arxiv.org/abs/hep-th/0501235 arXiv:hep-th/0501235v2] V.Dolotin, A.Morozov: ''Algebraic Geometry of Discrete Dynamics''. The case of one variable.
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| *[http://arxiv.org/abs/0802.2565 Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram]
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| {{DEFAULTSORT:Periodic Points Of Complex Quadratic Mappings}}
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| [[Category:Complex dynamics]]
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| [[Category:Fractals]]
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| [[Category:Limit sets]]
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