{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} 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 monic and centered form has the following properties:

## 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.

$c=c(\theta )={\frac {e^{2\pi \theta i}}{2}}\left(1-{\frac {e^{2\pi \theta i}}{2}}\right)$ .
$c=c(r)\,=\,{\frac {1-(r-1)^{2}}{4}}$ .

### With doubling map

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

## Map

$f_{c}(z)=z^{2}+c.\,$ When it is used as an evolution function of the discrete nonlinear dynamical system:

$z_{n+1}=f_{c}(z_{n})\,$ $f_{c}:z\to z^{2}+c.\,$ This iteration leads to the Mandelbrot set.

## Notation

$f_{c}^{n}(z)=f_{c}^{1}(f_{c}^{n-1}(z))\,$ so

$z_{n}=f_{c}^{n}(z_{0}).\,$ Because of the possible confusion it is customary to write $f^{\circ n}\,$ for the nth iterate of the function $f.\,$ ## Critical items

### Critical point

$f_{c}'(z_{cr})=0.\,$ Since

$f_{c}'(z)={\frac {d}{dz}}f_{c}(z)=2z$ implies

$z_{cr}=0\,$ ### Critical value

$z_{cv}=f_{c}(z_{cr})\,$ Since

$z_{cr}=0\,$ we have

$z_{cv}=c.\,$ ### Critical orbit Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6

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.

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

so

These polynomials are used for:

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

### Critical curves

Diagrams of critical polynomials are called critical curves.

These curves create skeleton of bifurcation diagram. (the dark lines)

## Planes

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

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

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

• the conjugation plane
• model plane

### Parameter plane

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

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.

### Dynamical plane

On the dynamical plane one can find:

The dynamical plane consists of:

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

Dynamical z-planes can be divided in two groups :

## Derivatives

### Derivative with respect to c

On parameter plane:

$z_{n}'={\frac {d}{dc}}f_{c}^{n}(z_{0}).$ 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 dynamical plane:

$f_{c}'(z_{0})={\frac {d}{dz}}f_{c}(z_{0})=2z_{0}$ at a periodic point z0 of period p

$(f_{c}^{p})'(z_{0})={\frac {d}{dz}}f_{c}^{p}(z_{0})=\prod _{i=0}^{p-1}f_{c}'(z_{i})=2^{p}\prod _{i=0}^{p-1}z_{i}.$ It is used to check the stability of periodic (also fixed) points.

at nonperiodic point:

$z'_{n}\,$ This derivative can be found by iteration starting with

$z'_{0}=1\,$ and then :

$z'_{n}=2*z_{n-1}*z'_{n-1}\,$ This dervative is used for computing external distance to Julia set.

### Schwarzian derivative

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

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