World Library  
Flag as Inappropriate
Email this Article

Periodic points of complex quadratic mappings

Article Id: WHEBN0000869255
Reproduction Date:

Title: Periodic points of complex quadratic mappings  
Author: World Heritage Encyclopedia
Language: English
Subject: Filled Julia set, Mandelbrot set, Multiplier, Orbit portrait, Complex dynamics
Collection: Complex Dynamics, Fractals, Limit Sets
Publisher: World Heritage Encyclopedia

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 1
  • Stability of periodic points (orbit) - multiplier 2
  • Period-1 points (fixed points) 3
    • Finite fixed points 3.1
      • Complex dynamics 3.1.1
      • Special cases 3.1.2
      • Only one fixed point 3.1.3
    • Infinite fixed point 3.2
  • Period-2 cycles 4
    • First method of factorization 4.1
    • Second method of factorization 4.2
      • Special cases 4.2.1
  • Cycles for period greater than 2 5
  • References 6
  • Further reading 7
  • External links 8




be the complex quadric mapping, where z and c are complex-valued.

Notationally, \ f^{(k)} _c (z) is the \ k -fold composition of f _c\, with itself—that is, the value after the k-th iteration of function f _c.\, Thus

\ f^{(k)} _c (z) = f_c(f^{(k-1)} _c (z)).

Periodic points of a complex quadratic mapping of period \ p are points \ z of the dynamical plane such that

f^{(p)} _c (z) = z,

where \ p is the smallest positive integer for which the equation holds at that z.

We can introduce a new function:

\ F_p(z,f) = f^{(p)} _c (z) - z,

so periodic points are zeros of function \ F_p(z,f) : points z satisfying

F_p(z,f) = 0,

which is a polynomial of degree 2^p.

Stability of periodic points (orbit) - multiplier

Stability index of periodic points along horizontal axis
boundaries of regions of parameter plane with attracting orbit of periods 1-6
Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with abs(multiplier)=0.99993612384259

The multiplier ( or eigenvalue, derivative ) m(f^p,z_0)=\lambda \, of a rational map f\, iterated p times, at cyclic point z_0\, is defined as:

m(f^p,z_0)=\lambda = \begin{cases} f^{p \prime}(z_0), &\mbox{if }z_0\ne \infty \\ \frac{1}{f^{p \prime} (z_0)}, & \mbox{if }z_0 = \infty \end{cases}

where f^{p\prime} (z_0) is the first derivative of \ f^p with respect to z\, at z_0. \,

Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit.

The multiplier is:

  • a complex number;
  • invariant under conjugation of any rational map at its fixed point;[1]
  • used to check stability of periodic (also fixed) points with stability index abs(\lambda). \,

A periodic point is[2]

  • attracting when abs(\lambda) < 1; \,
    • super-attracting when abs(\lambda) = 0; \,
    • attracting but not super-attracting when 0 < abs(\lambda) < 1; \,
  • indifferent when abs(\lambda) = 1; \,
  • repelling when abs(\lambda) > 1. \,

Periodic points

  • that are attracting are always in the Fatou set;
  • that are repelling are in the Julia set;
  • that are indifferent fixed points may be in one or the other.[3] A parabolic periodic point is in the Julia set.

Period-1 points (fixed points)

Finite fixed points

Let us begin by finding all finite points left unchanged by one application of f. These are the points that satisfy \ f_c(z)=z. That is, we wish to solve


which can be rewritten as

\ z^2-z+c=0.

Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:

\alpha_1 = \frac{1-\sqrt{1-4c}}{2} and \alpha_2 = \frac{1+\sqrt{1-4c}}{2}.

So for c \in C \setminus [1/4,+\inf ] we have two finite fixed points \alpha_1 \, and \alpha_2\, .


\alpha_1 = \frac{1}{2}-m and \alpha_2 = \frac{1}{2}+ m where m = \frac{\sqrt{1-4c}}{2}

then \alpha_1 + \alpha_2 = 1 .\,.

Thus fixed points are symmetrical around z = 1/2.\,

This image shows fixed points (both repelling)

Complex dynamics

Fixed points for c along horizontal axis
Fatou set for F(z)=z*z with marked fixed point

Here different notation is commonly used:[4]

\alpha_c = \frac{1-\sqrt{1-4c}}{2} with multiplier \lambda_{\alpha_c} = 1-\sqrt{1-4c}\,


\beta_c = \frac{1+\sqrt{1-4c}}{2} with multiplier \lambda_{\beta_c} = 1+\sqrt{1-4c}.\,

Using Viète's formulas one can show that:

\alpha_c + \beta_c = 1 .

Since the derivative with respect to z is

P_c'(z) = \frac{d}{dz}P_c(z) = 2z ,


P_c'(\alpha_c) + P_c'(\beta_c)= 2 \alpha_c + 2 \beta_c = 2 (\alpha_c + \beta_c) = 2 . \,

This implies that P_c \, can have at most one attractive fixed point.

These points are distinguished by the facts that:

  • \beta_c \, is :
    • the landing point of external ray for angle=0 for c \in M \setminus \left \{ \frac{1}{4} \right \}
    • the most repelling fixed point, belongs to Julia set,
    • 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).[5]
  • \alpha_c \, is:
    • the landing point of several rays
    • 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 )
    • parabolic at the root point of the limb of Mandelbrot set
    • repelling for other c values

Special cases

An important case of the quadratic mapping is c=0. In this case, we get \alpha_1 = 0 and \alpha_2=1. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

Only one fixed point

We have \alpha_1=\alpha_2 exactly when 1-4c=0. This equation has one solution, c=1/4, in which case \alpha_1=\alpha_2=1/2. In fact c=1/4 is the largest positive, purely real value for which a finite attractor exists.

Infinite fixed point

We can extend the complex plane \mathbb{C} to the Riemann sphere (extended complex plane) \mathbb{\hat{C}} by adding infinity :

\mathbb{\hat{C}} = \mathbb{C} \cup \{ \infty \}

and extending polynomial f_c\, such that f_c(\infty)=\infty. \,

Then infinity is :

  • superattracting
  • a fixed point of polynomial f_c: \,[6]

Period-2 cycles

Bifurcation from period 1 to 2 for complex quadratic map

Period-2 cycles are two distinct points \beta_1 and \beta_2 such that f_c(\beta_1) = \beta_2 and f_c(\beta_2) = \beta_1.

We write f_c(f_c(\beta_n)) = \beta_n:

f_c(f_c(z)) = (z^2+c)^2+c = z^4 + 2cz^2 + c^2 + c.\,

Equating this to z, we obtain

z^4 + 2cz^2 - z + c^2 + c = 0.

This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are \alpha_1 and \alpha_2, computed above, since if these points are left unchanged by one application of f, then clearly they will be unchanged by 2 applications (or more).

Our 4th-order polynomial can therefore be factored in 2 ways:

First method of factorization

(z-\alpha_1)(z-\alpha_2)(z-\beta_1)(z-\beta_2) = 0.\,

This expands directly as x^4 - Ax^3 + Bx^2 - Cx + D = 0 (note the alternating signs), where

D = \alpha_1 \alpha_2 \beta_1 \beta_2, \,
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, \,
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, \,
A = \alpha_1 + \alpha_2 + \beta_1 + \beta_2.\,

We already have two solutions, and only need the other two. Hence the problem is equivalent to solving a quadratic polynomial. In particular, note that

\alpha_1 + \alpha_2 = \frac{1-\sqrt{1-4c}}{2} + \frac{1+\sqrt{1-4c}}{2} = \frac{1+1}{2} = 1


\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.

Adding these to the above, we get D = c \beta_1 \beta_2 and A = 1 + \beta_1 + \beta_2. Matching these against the coefficients from expanding f, we get

D = c \beta_1 \beta_2 = c^2 + c and A = 1 + \beta_1 + \beta_2 = 0.

From this, we easily get

\beta_1 \beta_2 = c + 1 and \beta_1 + \beta_2 = -1.

From here, we construct a quadratic equation with A' = 1, B = 1, C = c+1 and apply the standard solution formula to get

\beta_1 = \frac{-1 - \sqrt{-3 -4c}}{2} and \beta_2 = \frac{-1 + \sqrt{-3 -4c}}{2}.

Closer examination shows that :

f_c(\beta_1) = \beta_2 and f_c(\beta_2) = \beta_1,

meaning these two points are the two points on a single period-2 cycle.

Second method of factorization

We can factor the quartic by using polynomial long division to divide out the factors (z-\alpha_1) and (z-\alpha_2), which account for the two fixed points \alpha_1 and \alpha_2 (whose values were given earlier and which still remain at the fixed point after two iterations):

(z^2+c)^2 + c -z = (z^2 + c - z)(z^2 + z + c +1 ). \,

The roots of the first factor are the two fixed points. They are repelling outside the main cardioid.

The second factor has the two roots

-\frac{1}{2} \pm (-\frac{3}{4} - c)^\frac{1}{2}. \,

These two roots, which are the same as those found by the first method, form the period-2 orbit.[7]

Special cases

Again, let us look at c=0. Then

\beta_1 = \frac{-1 - i\sqrt{3}}{2} and \beta_2 = \frac{-1 + i\sqrt{3}}{2},

both of which are complex numbers. We have | \beta_1 | = | \beta_2 | = 1. Thus, both these points are "hiding" in the Julia set. Another special case is c=-1, which gives \beta_1 = 0 and \beta_2 = -1. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

Cycles for period greater than 2

The degree of the equation f^{(n)}(z)=z is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation.

There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals.[8]

In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case z_{n+1}=z_n^2-2 is equivalent to the logistic map case r = 4: x_{n+1}=4x_n(1-x_n). Here the equivalence is given by z=2-4x. One of the k-cycles of the logistic variable x (all of which cycles are repelling) is

\sin^2\left(\frac{2\pi}{2^k-1}\right), \, \sin^2\left(2\cdot\frac{2\pi}{2^k-1}\right), \, \sin^2\left(2^2\cdot\frac{2\pi}{2^k-1}\right), \, \sin^2\left(2^3\cdot\frac{2\pi}{2^k-1}\right), \dots , \sin^2\left(2^{k-1}\frac{2\pi}{2^k-1}\right).


  1. ^ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
  2. ^ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, page 99
  3. ^ Some Julia sets by Michael Becker
  4. ^ On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178.
  5. ^ Periodic attractor by Evgeny Demidov
  6. ^ 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
  7. ^ Period 2 orbit by Evgeny Demidov
  8. ^ Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram

Further reading

  • Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2
  • 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
  • Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
  • The permutations of periodic points in quadratic polynominials by J Leahy

External links

  • by Donald D. CrossAlgebraic solution of Mandelbrot orbital boundaries
  • by Robert P. MunafoBrown Method
  • arXiv:hep-th/0501235v2 V.Dolotin, A.Morozov: Algebraic Geometry of Discrete Dynamics. The case of one variable.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.