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Wikipedia / 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:

  • It is a unicritical polynomial, i.e. it has one finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: the basin of infinity and basin of the finite critical point (if the finite critical point does not escape)

  • It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic.

  • It is a unimodal function,

  • It is a rational function,

  • It is an entire function.

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:

is:

  • "The first family of dynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"

Conjugation

Between forms

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.

:

, the parameter transformation is

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

denotes the n-th iterate:

so

for the n.

Parameter

can be marked by:

of the ray that lands:

  • At c in Mandelbrot set on the parameter plane

  • On the critical value:z = c in Julia set on the dynamic plane

so :

Examples:

Map

The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, and parameter:

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 on the dynamical plane such that the derivative vanishes:

Since

implies that

.

is an initial point for Mandelbrot set iteration.

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, a complex quadratic polynomial has 2d-2 critical points. In this model, zero and infinity are critical points.

Critical value

A critical value is the image of a critical point:

Since

we have

.

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:

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

    finding roots of Mandelbrot set components of period n (local minimum)

  • Misiurewicz points

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:

  • -plane or c-plane

  • the parameter plane or z-plane

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

  • the conjugation plane

  • model plane

2D Parameter plane

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

is variable.

There are neither dynamics nor orbits on the parameter plane. However, there is a set of parameter values.

The parameter plane consists of:

The Mandelbrot set

  • The bifurcation locus = boundary of Mandelbrot set with root points

  • Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set with internal rays

Exterior of Mandelbrot set with

Besides the aforementioned set, there are many different sub-types of the parameter plane.

See also :

  • Boettcher map which maps exterior of Mandelbrot set to the exterior of unit disc

  • multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc

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:

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:

  • is a variable

  • is constant

The first derivative 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:

  • is a variable;

  • is a constant.

At a fixed point,

At a periodic point z0 of period p the first derivative of a function

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, 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:

See also

References