Complex quadratic polynomial

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

Forms

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

• The general form: $f(x) = a_2 x^2 + a_1 x + a_0 \qquad \,$ where $\qquad a_2 \ne 0$
• The factored form used for logistic map $f_r(x) = r x ( 1-x ) \,$
• $f_{\theta}(x) = x^2 + e^{2 \pi \theta i} x \,$ which has an indifferent fixed point with multiplier $\lambda = e^{2 \pi \theta i} \,$ at the origin^[1]
• The monic and centered form, $f_c(x) = x^2 +c\,$

The monic and centered form has the following properties:

• It is the simplest form of a nonlinear function with one coefficient (parameter),
• It is an unicritical polynomial, i.e. it has one critical point,
• It is a centered polynomial (the sum of its critical points is zero),^[2]
• 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.^[3]
• It is a unimodal function,
• It is a rational function,
• It is an entire function.

Conjugation

Between forms

Since $f_c(x) \,$ 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 $\theta\,$ to $c \,$:^[4]

$c = c(\theta) = \frac {e^{2 \pi \theta i}}{2} \left(1 - \frac {e^{2 \pi \theta i}}{2}\right).$

When one wants change from $r\,$ to $c , \,$ the parameter transformation is^[5]

$c = c(r)\,=\,\frac{1- (r-1)^2}{4}=-\frac{r}{2}\left(\frac{r-2}{2}\right)$

and the transformation between the variables in $z_{t+1}=z_t^2+c$ and $x_{t+1}=rx_t(1-x_t)$ is

$z=r\left(\frac{1}{2}-x\right).$

With doubling map

There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.

Map

The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,^[6] is typically used with variable $z\,$ and parameter $c\,$:

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

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

$z_{n+1} = f_c(z_n) \,$

it is named the quadratic map:^[7]

$f_c : z \to z^2 + c. \,$

The Mandelbrot set is the set of values of the parameter c for which the initial condition z₀ = 0 does not cause the iterates to diverge to infinity.

Notation

Here $f^n \,$ denotes the n-th iteration of the function $f \,$ (and not exponentiation of the function):

$f_c^n(z) = f_c^1(f_c^{n-1}(z)) \,$

so

$z_n = f_c^n(z_0). \,$

Because of the possible confusion with exponentiation, some authors write $f^{\circ n}\,$ for the nth iterate of the function $f.\,$

Critical items

Critical point

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

$f_c'(z_{cr}) = 0. \,$

Since

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

implies

$z_{cr} = 0\,$

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

$z_0$ is an initial point for Mandelbrot set iteration.^[8]

Critical value

A critical value $z_{cv} \$ of $f_c\,$ is the image of a critical point:

$z_{cv} = f_c(z_{cr}) \,$

Since

$z_{cr} = 0\,$

we have

$z_{cv} = c. \,$

So the parameter $c \,$ is the critical value of $f_c(z). \,$

Critical orbit

Dynamical plane with critical orbit falling into 3-period cycle

Dynamical plane with Julia set and critical orbit.

Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6

Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259

The 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]

$z_0 = z_{cr} = 0\,$

$z_1 = f_c(z_0) = c\,$

$z_2 = f_c(z_1) = c^2 +c\,$

$z_3 = f_c(z_2) = (c^2 + c)^2 + c\,$

$... \,$

This orbit falls into an attracting periodic cycle if one exists.

Critical sector

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

Critical polynomial

$P_n(c) = f_c^n(z_{cr}) = f_c^n(0) \,$

so

$P_0(c)= 0 \,$

$P_1(c) = c \,$

$P_2(c) = c^2 + c \,$

$P_3(c) = (c^2 + c)^2 + c \,$

These polynomials are used for:

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

$centers = \{ c : P_n(c) = 0 \}\,$

• finding roots of Mandelbrot set components of period n (local minimum of $P_n(c) \,$)
• Misiurewicz points

$M_{n,k} = \{ c : P_k(c) = P_{k+n}(c) \}\,$

Critical curves

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

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

Planes

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, $f_c\,$-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

Gamma parameter plane for complex logistic map $z_{n+1} = \gamma z_n \left(1 - z_n\right),$

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

$z_0 = z_{cr} \,$ is constant and $c\,$ 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:

• The Mandelbrot set
  • The bifurcation locus = boundary of Mandelbrot set
  • Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set^[18]

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

Dynamical plane

On the dynamical plane one can find:

• The Julia set
• The Filled Julia set
• The Fatou set
• Orbits

The dynamical plane consists of:

• Fatou set
• Julia set

Here, $c\,$ is a constant and $z\,$ 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 :

• $f_0$ plane for $c = 0$ ( see complex squaring map )
• $f_c$ planes ( all other planes for $c \ne 0$ )

Derivatives

Derivative with respect to c

On the parameter plane:

• $c$ is a variable
• $z_0 = 0$ is constant

The first derivative of $f_c^n(z_0)$ with respect to c is

$z_n' = \frac{d}{dc} f_c^n(z_0).$

This derivative can be found by iteration starting with

$z_0' = \frac{d}{dc} f_c^0(z_0) = 1$

and then replacing at every consecutive step

$z_{n+1}' = \frac{d}{dc} f_c^{n+1}(z_0) = 2\cdot{}f_c^n(z)\cdot\frac{d}{dc} f_c^n(z_0) + 1 = 2 \cdot z_n \cdot z_n' +1.$

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 the dynamical plane:

• $z$ is a variable;
• $c$ is a constant.

At a fixed point $z_0\, ,$

$f_c'(z_0) = \frac{d}{dz}f_c(z_0) = 2z_0 .$

At a periodic point z₀ of period p,

$(f_c^p)'(z_0) = \frac{d}{dz}f_c^p(z_0) = \prod_{i=0}^{p-1} f_c'(z_i) = 2^p \prod_{i=0}^{p-1} z_i.$

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

At a nonperiodic point, the derivative, denoted by $z'_n, \,$ can be found by iteration starting with

$z'_0 = 1, \,$

and then using

$z'_n= 2*z_{n-1}*z'_{n-1}. \,$

This derivative is used for computing the external distance to the Julia set.

Schwarzian derivative

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

$(Sf)(z) = \frac{f'''(z)}{f'(z)} - \frac{3}{2} \left ( \frac{f''(z)}{f'(z)}\right ) ^2$.

See also

• Misiurewicz point
• Periodic points of complex quadratic mappings
• Mandelbrot set
• Julia set
• Milnor–Thurston kneading theory

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 Resource
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

External links

• M. Nevins and D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers"
• Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002