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 is:

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: When one wants change from r to c, the parameter transformation is and the transformation between the variables in and 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 f^n denotes the n-th iterate of the function f: so Because of the possible confusion with exponentiation, some authors write f^{\circ n} for the nth iterate of f.

Parameter

The monic and centered form can be marked by: so : Examples:

Map

The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable z and parameter c: 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 z0 = 0 does not cause the iterates to diverge to infinity.

Critical items

Critical points

complex plane

A critical point of f_c is a point z_{cr} on the dynamical plane such that the derivative vanishes: Since implies 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. For the quadratic family the critical point z = 0 is the center of symmetry of the Julia set Jc, so it is a convex combination of two points in Jc.

Extended complex plane

In the Riemann sphere polynomial has 2d-2 critical points. Here zero and infinity are critical points.

Critical value

A critical value z_{cv} of f_c is the image of a critical point: Since we have So the parameter c is the critical value of f_c(z).

Critical level curves

A critical level curve the level curve which contain critical point. It acts as a sort of skeleton of dynamical plane Example : level curves cross at saddle point, which is a special type of critical point.

Critical limit set

Critical limit set is the set of forward orbit of all critical points

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 set

Critical set is a set of critical points

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 two basic types of 2D 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: 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: 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 Kauko"On the dynamical plane one can find: The dynamical plane consists of: 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. Dynamical z-planes can be divided into 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 f_c^n(z_0) with respect to c is This derivative can be found by 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 distance estimation method for drawing a Mandelbrot set.

First derivative with respect to z

On the dynamical plane: At a fixed point z_0, At a** periodic point** z0 of period p the first derivative of a function is often represented by \lambda and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. Absolute value of multiplier 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 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: