Periodic points of complex quadratic mappings

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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 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.

This theory is applied in relation with the theories of Fatou and Julia sets.

Definitions[edit]

Let

f_c(z)=z^2+c\,

where z and c are complex-valued. (This \ f is the complex quadratic mapping mentioned in the title.) This article explores the periodic points of this mapping - that is, the points that form a periodic cycle when \ f is repeatedly applied to them.

 \ f^{(k)} _c (z) is the \ k -fold compositions of f _c\, with itself = iteration of function f _c\, or,

 \ 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 :

 \ z : f^{(p)} _c (z) =   z

where \ p is the smallest positive integer.

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)  :

 \ z : F_p(z,f) = 0

which is a polynomial of degree  \ = 2^p

Stability of periodic points (orbit) - multiplier[edit]

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,z_0)=\lambda \, of rational map f\, at fixed point z_0\, is defined as :


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

where f_c'(z_0)\, is first derivative of  \ f_c with respect to z\, at z_0\,.

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

Multiplier is:

  • 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) \,

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 \,

Where do periodic points belong?

  • attracting is always in Fatou set
  • repelling is in the Julia set
  • Indifferent fixed points may be in the one or in the other.[3] Parabolic periodic point is in Julia set.

Period-1 points (fixed points)[edit]

Finite fixed points[edit]

Let us begin by finding all finite points left unchanged by 1 application of f. These are the points that satisfy \ f_c(z)=z. That is, we wish to solve

z^2+c=z\,

which can be rewritten

\ z^2-z+c=0.

Since this is an ordinary quadratic equation in 1 unknown, we can apply the standard quadratic solution formula. Look in any standard mathematics textbook, and you will find that there are two solutions of \ Ax^2+Bx+C=0 are given by

x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}

In our case, we have A=1, B=-1, C=c, so we will write

\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\, .

Since

\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 \,.

It means that fixed points are symmetrical around z = 1/2\,.

This image shows fixed points (both repelling)

Complex dynamics[edit]

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}\,

and

\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 = -\frac{B}{A} = 1

Since derivative with respect to z is :

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

then

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

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

This 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:
    • landing point of several rays
    • is :
      • 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[edit]

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[edit]

We might wonder what value c should have to cause \alpha_1=\alpha_2. The answer is that this will happen exactly when 1-4c=0. This equation has 1 solution: c=1/4 (in which case, \alpha_1=\alpha_2=1/2). This is interesting, since c=1/4 is the largest positive, purely real value for which a finite attractor exists.

Infinite fixed point[edit]

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

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

and extend polynomial f_c\, such that f_c(\infty)=\infty\,

Then infinity is :

f_c(\infty)=\infty=f^{-1}_c(\infty)\,

Period-2 cycles[edit]

Bifurcation from period 1 to 2 for complex quadratic map

Suppose next that we wish to look at period-2 cycles. That is, we want to find two points \beta_1 and \beta_2 such that f_c(\beta_1) = \beta_2, and f_c(\beta_2) = \beta_1.

Let us start by writing f_c(f_c(\beta_n)) = \beta_n, and see where trying to solve this leads.

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

Thus, the equation we wish to solve is actually z^4 + 2cz^2 - z + c^2 + c = 0.

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 \alpha_1 and \alpha_2, computed above. It is simple to see why this is; if these points are left unchanged by 1 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[edit]

(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 2 solutions, and only need the other 2. This is as difficult as 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

and

\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 (the formulas are a tad messy) that :

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

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

Second method of factorization[edit]

(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 z_{1,2}\, . They are repelling outside the main cardioid.

The second factor has two roots

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

These two roots form period-2 orbit.[7]

Special cases[edit]

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. By doing a little algebra, we find | \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>2[edit]

There is no general solution in radicals to polynomial equations of degree five or higher, so it must be computed using numerical methods.

References[edit]

  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

Further reading[edit]

External links[edit]