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Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.

Forms

When the quadratic polynomial has only one variable (univariate), one can distinguish its 4 main forms:

  • 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>
  • The factored form used for logistic map <math>f_r(x) = r x ( 1-x ) \,</math>
  • <math> f_{\theta}(x) = x^2 + e^{2 \pi \theta i} x \,</math> which has an indifferent fixed point with multiplier <math>\lambda = e^{2 \pi \theta i} \,</math> at the origin[1]
  • The monic and centered form, <math>f_c(x) = x^2 +c\,</math>

The monic and centered form has the following properties:

Conjugation

Between forms

Since <math>f_c(x) \,</math> is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia and Fatou sets.

When one wants change from <math>\theta\,</math> to <math>c \,</math>:[4]

<math>c = c(\theta) = \frac {e^{2 \pi \theta i}}{2} \left(1 - \frac {e^{2 \pi \theta i}}{2}\right) </math>.

When one wants change from <math>r\,</math> to <math>c \,</math>:[5]

<math>

c = c(r)\,=\,\frac{1- (r-1)^2}{4} </math>.

With doubling map

There is semi-conjugacy between the dyadic transformation (here named doubling map) and the quadratic polynomial.

Family

The family of quadratic polynomials <math>f_c : z \to z^2 +c\,</math> parametrised by <math> c \in \mathbb{C} \,</math> is called:

  • the Douady-Hubbard family of quadratic polynomials[6]
  • quadratic family

Map

The monic and centered form is typically used with variable <math>z\,</math> and parameter <math>c\,</math>:

<math>f_c(z) = z^2 +c.\,</math>

When it is used as an evolution function of the discrete nonlinear dynamical system:

<math>z_{n+1} = f_c(z_n) \,</math>

it is named quadratic map:[7]

<math>f_c : z \to z^2 + c. \,</math>

This iteration leads to the Mandelbrot set.

Notation

Here <math> f^n \,</math> denotes the n-th iteration of the function <math> f \,</math> not exponentiation

<math>f_c^n(z) = f_c^1(f_c^{n-1}(z)) \,</math>

so

<math>z_n = f_c^n(z_0). \,</math>

Because of the possible confusion it is customary to write <math>f^{\circ n}\,</math> for the nth iterate of the function <math>f.\,</math>

Critical items

Critical point

A critical point of <math>f_c\,</math> is a point <math> z_{cr} \,</math> in the dynamical plane such that the derivative vanishes:

<math>f_c'(z_{cr}) = 0. \,</math>

Since

<math>f_c'(z) = \frac{d}{dz}f_c(z) = 2z </math>

implies

<math> z_{cr} = 0\,</math>

we see that the only (finite) critical point of <math>f_c \,</math> is the point <math> z_{cr} = 0\,</math>.

<math>z_0</math> is an initial point for Mandelbrot set iteration.[8]

Critical value

A critical value <math>z_{cv} \ </math> of <math>f_c\,</math> is the image of a critical point:

<math>z_{cv} = f_c(z_{cr}) \,</math>

Since

<math> z_{cr} = 0\,</math>

we have

<math>z_{cv} = c. \,</math>

So the parameter <math> c \,</math> is the critical value of <math>f_c(z). \,</math>

Critical orbit

File:Cr orbit 3.png
Dynamical plane with critical orbit falling into 3-period cycle
File:Miimcr.png
Dynamical plane with Julia set and critical orbit.
File:6furcation.gif
Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6
File:Critical orbit 3d.png
Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259

Forward orbit of a critical point is called a critical orbit. Critical orbits are very important because every attracting periodic orbit attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[9][10][11]

<math>z_0 = z_{cr} = 0\,</math>

<math>z_1 = f_c(z_0) = c\,</math>

<math>z_2 = f_c(z_1) = c^2 +c\,</math>

<math>z_3 = f_c(z_2) = (c^2 + c)^2 + c\,</math>

<math>... \,</math>

This orbit falls into an attracting periodic cycle.

Critical sector

The critical sector is a sector of the dynamical plane containing the critical point.

Critical polynomial

<math>P_n(c) = f_c^n(z_{cr}) = f_c^n(0) \,</math>

so

<math>P_0(c)= 0 \,</math>

<math>P_1(c) = c \,</math>

<math>P_2(c) = c^2 + c \,</math>

<math>P_3(c) = (c^2 + c)^2 + c \,</math>

These polynomials are used for:

  • finding centers of these Mandelbrot set components of period n. Centers are roots of n-th critical polynomials

<math>centers = \{ c : P_n(c) = 0 \}\,</math>

<math>M_{n,k} = \{ c : P_k(c) = P_{k+n}(c) \}\,</math>

Critical curves

Diagrams of critical polynomials are called critical curves.[12]

These curves create skeleton of bifurcation diagram.[13] (the dark lines[14])

Planes

File:Iray.png
w-plane and c-plane

One can use the Julia-Mandelbrot 4-dimensional space for a global analysis of this dynamical system.[15]

In this space there are 2 basic types of 2-D planes:

  • the dynamical (dynamic) plane, <math>f_c\,</math>-plane or c-plane
  • the parameter plane or z-plane

There is also another plane used to analyze such dynamical systems w-plane:

  • the conjugation plane[16]
  • model plane[17]

Parameter plane

File:MandelbrotLambda.jpg
Gamma parameter plane for complex logistic map <math>z_{n+1} = \gamma z_n \left(1 - z_n\right),</math>

The phase space of a quadratic map is called its parameter plane. Here:

<math>z0 = z_{cr} \,</math> is constant and <math>c\,</math> is variable.

There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.

The parameter plane consists of:

There are many different subtypes of the parameter plane.[19][20]

Dynamical plane

On the dynamical plane one can find:

The dynamical plane consists of:

Here, <math>c\,</math> is a constant and <math>z\,</math> is a variable.

The two-dimensional dynamical plane can be treated as a Poincaré cross-section of three-dimensional space of continuous dynamical system.[21][22]

Dynamical z-planes can be divided in two groups :

  • <math>f_0</math> plane for <math>c = 0</math>
  • <math>f_c</math> planes ( all other planes for <math>c \ne 0</math> )

Derivatives

Derivative with respect to c

On parameter plane:

  • <math>c</math> is a variable
  • <math>z_0 = 0 </math> is constant

The first derivative of <math>f_c^n(z_0)</math> with respect to c is

<math>z_n' = \frac{d}{dc} f_c^n(z_0).</math>

This derivative can be found by iteration starting with

<math>z_0' = \frac{d}{dc} f_c^0(z_0) = 1</math>

and then replacing at every consecutive step

<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>

This can easily be verified by using the chain rule for the derivative.

This derivative is used in the distance estimation method for drawing a Mandelbrot set.

Derivative with respect to z

On dynamical plane:

  • <math>z</math> is a variable
  • <math>c </math> is a constant

at a fixed point <math>z_0\,</math>

<math>f_c'(z_0) = \frac{d}{dz}f_c(z_0) = 2z_0 </math>

at a periodic point z0 of period p

<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>

It is used to check the stability of periodic (also fixed) points.

at nonperiodic point:

<math>z'_n\,</math>

This derivative can be found by iteration starting with

<math>z'_0 = 1 \,</math>

and then :

<math>z'_n= 2*z_{n-1}*z'_{n-1}\,</math>

This dervative is used for computing external distance to Julia set.

Schwarzian derivative

The Schwarzian derivative (SD for short) of f is:[23]

<math> (Sf)(z) = \frac{f'(z)}{f'(z)} - \frac{3}{2} \left ( \frac{f(z)}{f'(z)}\right ) ^2 </math>.

See also

References

  1. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  2. ^ Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark
  3. ^ Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
  4. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  5. ^ stackexchange questions : Show that the familiar logistic map ...
  6. ^ 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
  7. ^ Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Web Resourc
  8. ^ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  9. ^ M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
  10. ^ 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
  11. ^ Khan Academy : Mandelbrot Spirals 2
  12. ^ The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653
  13. ^ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. 
  14. ^ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint
  15. ^ Julia-Mandelbrot Space at Mu-ency by Robert Munafo
  16. ^ 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
  17. ^ Holomorphic motions and puzzels by P Roesch
  18. ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  19. ^ Alternate Parameter Planes by David E. Joyce
  20. ^ exponentialmap by Robert Munafo
  21. ^ Mandelbrot set by Saratov group of theoretical nonlinear dynamics
  22. ^ Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,
  23. ^ The Schwarzian Derivative & the Critical Orbit by Wes McKinney 18.091 20 April 2005

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