Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.
Properties.
Quadratic polynomials have the following properties, regardless of the form:
Forms.
When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms:
The monic and centered form has been studied extensively, and has the following properties:
The lambda form formula_7 is:
Conjugation.
Between forms.
Since formula_9 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 formula_10 to formula_11:
When one wants change from formula_13 to formula_11, the parameter transformation is
and the transformation between the variables in formula_16 and formula_17 is
With doubling map.
There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of "c" = –2.
Notation.
Iteration.
Here formula_19 denotes the "n"-th iteration of the function formula_20 (and not exponentiation of the function):
so
Because of the possible confusion with exponentiation, some authors write formula_23 for the "n"th iteration of the function formula_24.
Parameter.
The monic and centered form formula_6 can be marked by:
so :
formula_28
formula_29
Map.
The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable formula_30 and parameter formula_11:
When it is used as an evolution function of the discrete nonlinear dynamical system
it is named the quadratic map:
The Mandelbrot set is the set of values of the parameter "c" for which the initial condition "z"0 = 0 does not cause the iterates to diverge to infinity.
Critical items.
Critical point.
A critical point of formula_35 is a point formula_36 in the dynamical plane such that the derivative vanishes:
Since
implies
we see that the only (finite) critical point of formula_35 is the point formula_39.
formula_42 is an initial point for Mandelbrot set iteration.
Critical value.
A critical value formula_43 of formula_35 is the image of a critical point:
Since
we have
So the parameter formula_11 is the critical value of formula_49.
Critical orbit.
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.
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.
so
These polynomials are used for:
Critical curves.
Diagrams of critical polynomials are called critical curves.
These curves create the skeleton (the dark lines) of a bifurcation diagram.
Spaces, planes.
4D space.
One can use the Julia-Mandelbrot 4-dimensional (4D) 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: 
2D Parameter plane.
The phase space of a quadratic map is called its parameter plane. Here:
formula_64 is constant and formula_11 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.
See also :
2D Dynamical plane.
"The polynomial Pc maps each dynamical ray to another ray doubling the angle (which we measure in full turns, i.e. 0 = 1 = 2π rad = 360◦), and the dynamical rays of any polynomial “look like straight rays” near infinity. This allows us to study the Mandelbrot and Julia sets combinatorially, replacing the dynamical plane by the unit circle, rays by angles, and the quadratic polynomial by the doubling modulo one map." Virpi KaukoOn the dynamical plane one can find:
The dynamical plane consists of:
Here, formula_11 is a [[Coefficient|constant]] and formula_30 is a variable.
The two-dimensional dynamical plane can be treated as a [[Poincaré map|Poincaré cross-section]] of three-dimensional space of continuous dynamical system.
Dynamical z-planes can be divided in two groups :
Riemann sphere.
The extended complex plane plus a [[point at infinity]]
Derivatives.
First derivative with respect to "c".
On the parameter plane:
The first [[derivative]] of formula_74 with respect to "c" is
This [[derivative]] can be found by [[Iterated function|iteration]] starting with
and then replacing at every consecutive step
This can easily be verified by using the [[chain rule]] for the derivative.
This derivative is used in the [[Mandelbrot set#Distance estimates|distance estimation method for drawing a Mandelbrot set]].
First derivative with respect to "z".
On the dynamical plane:
At a fixed point formula_42,
At a [[Periodic points of complex quadratic mappings|periodic point]] "z"0 of period "p" the first derivative of a function
is often represented by formula_83 and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. It used to check the [[Stability theory|stability]] of [[Periodic points of complex quadratic mappings|periodic (also fixed) points]].
At a nonperiodic point, the derivative, denoted by formula_84, can be found by [[Iterated function|iteration]] starting with
and then using
This derivative is used for computing the external distance to the Julia set.
Schwarzian derivative.
The [[Schwarzian derivative]] (SD for short) of f is:
External links.
[[Category:Complex dynamics]]
[[Category:Fractals]]
[[Category:Polynomials]]