Complex quadratic polynomial

From Wikipedia, the free encyclopedia

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

1 Forms
2 Dynamical system
3 Critical point
- 3.1 Critical value
- 3.2 Critical orbit
4 Planes
- 4.1 Dynamical plane
- 4.2 Parameter plane
5 Derivative
- 5.1 Derivative with respect to c
- 5.2 Derivative with respect to z
6 See also
7 References
8 External links

[edit] Forms

In the case of one variable there are main 2 forms :

general form, $f(x) = a_2 x^2 + a_1 x + a_0 \,$ .

monic and centered form

The monic and centered form has one variable $z\,$ and one parameter $c\,$

$P_c(z) = z^2 +c\,$

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)^[1],
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.^[2]
a unimodal function,
a rational function,
an entire function.

Since $P_c(z) \,$ 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.

[edit] Dynamical system

When it creates discrete nonlinear dynamical system:

$z_{n+1} = P_c(z_n) = P_c^n(z) \,$

it is named quadratic map.

Here $P_c^n(z) \,$ denotes iterated function not exponentiation.

[edit] Critical point

A critical point of $P_c\,$ is a point $z_{cr} \,$ such that

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

Since

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

implies

$z_{cr} = 0\,$

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

[edit] Critical value

A critical value of $P_c\,$ is a point $z_{cv} \,$ such that

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

Since

$z_{cr} = 0\,$

then

$z_{cv} = c \,$

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

[edit] Critical orbit

critical orbit with 3-period cycle

Julia set and critical orbit.

Orbit of critical point is called critical orbit. It is very important because it is attracted by periodic orbit.

[edit] Planes

One can use Julia-Mandelbrot 4-dimensional space for global analysis of this dynamical system^[3].

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

dynamical plane or c-plane,
parameter plane or z-plane.

There is also third plane used to analyze such dynamical systems : standard plane or w-plane.

[edit] Dynamical 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.

Mandelbrot set is on parameter plane. There are many diffrent subtypes of parameter plane^[4]