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

Forms

In the case of one variable there are 3 main forms:

general form, $f(x) = a_2 x^2 + a_1 x + a_0 \,$ .
logistic form, $f_r(x) = r x ( 1-x ) \,$
$f_{\theta}(x) = x^2 + e^{2 \pi \theta} x \,$ which has an indifferent fixed point with multiplier $e^{2 \pi \theta} \,$ at the origin^[1]

monic and centered form, $f_c(x) = x^2 +c\,$

The monic and centered form is:

the simplest form of a nonlinear function with one coefficient ( parameter),
a univariate polynomial ( = it has one variable ),
a unicritical polynomial, i.e. it has one critical point,
centered polynomial (sum of 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]
a unimodal function,
a rational function,
an entire function.

Conjugation

Since $f_c(x) \,$ is affine conjugate to general form of 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 $c \,$ :

$c = c(\theta) = \frac {e^{2 \pi \theta}}{2} (1 - \frac {e^{2 \pi \theta}}{2})$ ^[4]

Family

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

the Douady-Hubbard family of quadratic polynomials ^[5]
quadratic family

Map

Monic and centered form is typically used with variable $z\,$ and parameter $c\,$

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

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

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

it is named quadratic map^[6]

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

Notation

Here $f^n \,$ denotes n-th iteration of function not exponentiation

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

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

Because above notation have many meanings Milnor writes $f^{\circ n}\,$ for nth iterate of function $f.\,$

Critical items

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 $z_{cr} = 0\,$ .

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 cadioid for angle 1/6

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.^[7]^[8]

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

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

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

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

$... \,$

This orbit falls into attracting periodic cycle

Critical sector

Critical sector is a sector of dynamical plane containing critical point .

Critical polynomial

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

$P_0(c)= 0 \,$

$P_1(c) = c \,$

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

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

These polynomials are used to :

finding centers of Mandelbrot set components of period n. Centers are roots of n-th critical polynomial

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

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

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

Critical curves

Diagrams of critical polynomials are called critical curves.^[9]

These curves creates skeleton of bifurcation diagram ^[10] ( the dark lines ^[11])

Planes

w-plane and c-plane

One can use Julia-Mandelbrot 4-dimensional space for global analysis of this dynamical system^[12].

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

dynamical ( dynamic ) plane, $f_c\,$ -plane or c-plane,
parameter plane or z-plane.

There is also third plane used to analyze such dynamical systems : conjugation plane^[13] , standard plane , $f_0\,$ -plane or w-plane.

Parameter plane

Phase space of quadratic map is called parameter plane. Here:

$z0 = z_{cr} \,$ is constant and $c\,$ is variable.

There is no dynamics here. It is only a set of parameter values. There are no orbits on parameter plane and one should not draw orbits on parameter plane.

Parameter plane consists of :

Mandelbrot set
- bifurcation locus = boundary of Mandelbrot set
- bounded hyperbolic components of Mandelbrot set = interior of Mandelbrot set = Connectedness locus
complement of Mandelbrot set $M^c = \mathbb{\hat{C}}\setminus M$ = unbounded hyperbolic component ^[14]

There are many different subtypes of parameter plane^[15]

Dynamical plane

On dynamical plane one can find :

Dynamical plane consists of :

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

2-D dynamical plane can be treated as a Poincare cross-section of 3-D space of continous dynamical system.^[16] ^[17]

Derivative

Derivative with respect to c

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

$\frac{d}{dc} f_c^n(z).$

This derivative can be found by iteration starting with

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

and then

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

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

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

Derivative with respect to z

at fixed point $z_0\,$

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

at periodic point z₀ of period n

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

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

References

^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
^ B Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark
^ Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
^ 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
^ Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Web Resourc
^ M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
^ 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
^ 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
^ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-50-682-3. http://power.itp.ac.cn/~hao/.
^ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint
^ Julia-Mandelbrot Space at Mu-ency by Robert Munafo
^ 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
^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
^ Alternate Parameter Planes by David E. Joyce
^ Mandelbrot set by Saratov group of theoretical nonlinear dynamics
^ Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,

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Complex quadratic polynomial

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