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 general form: $f(x) = a_2 x^2 + a_1 x + a_0$ where $a_2 \ne 0$
* The factored form used for the logistic map: $f_r(x) = r x (1-x)$
* $f_(x) = x^2 +\lambda x$ which has an indifferent fixed point with multiplier $\lambda = e^$ at the origin
* The monic and centered form, $f_c(x) = x^2 +c$

The monic and centered form has been studied extensively, and has the following properties:

* It is the simplest form of a nonlinear function with one coefficient (parameter),
* It is a centered polynomial (the sum of its critical points is zero).
* it is a binomial

The lambda form $f_(z) = z^2 +\lambda z$ is:
* the simplest non-trivial perturbation of unperturbated system $z \mapsto \lambda z$
* "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

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 $c$:

:$c = c(\theta) = \frac \left(1 - \frac \right).$

When one wants change from $r$ to $c$, the parameter transformation is
:$c = c(r) = \frac = -\frac \left(\frac\right)$

and the transformation between the variables in $z_=z_t^2+c$ and $x_=rx_t(1-x_t)$ is

:$z=r\left(\frac-x\right).$

With doubling map

There is semi-conjugacy between the dyadic transformation of a quadratic map is called its parameter plane. Here:

$z_0 = z_$ 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:
* 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
** external rays
** equipotential lines

There are many different subtypes 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 Kauko

On the dynamical plane one can find:
* The Julia set
* The Filled Julia set
* The Fatou set
* Orbits

The dynamical plane consists of:
* Fatou set
* Julia set

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:
* $f_0$ plane for $c = 0$ (see complex squaring map)
* $f_c$ planes (all other planes for $c \ne 0$)

Riemann sphere

The extended complex plane plus a point at infinity
* the Riemann sphere

Derivatives

First derivative with respect to ''c''

On the parameter plane:
* $c$ is a variable
* $z_0 = 0$ is constant

The first derivative of $f_c^n(z_0)$ with respect to ''c'' is

: $z_n' = \frac f_c^n(z_0).$

This derivative can be found by iteration starting with

: $z_0' = \frac f_c^0(z_0) = 1$

and then replacing at every consecutive step

: $z_' = \frac f_c^(z_0) = 2\cdotf_c^n(z)\cdot\frac f_c^n(z_0) + 1 = 2 \cdot z_n \cdot z_n' +1.$

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.

First derivative with respect to ''z''

On the dynamical plane:
* $z$ is a variable;
* $c$ is a constant.

At a fixed point $z_0$,

: $f_c'(z_0) = \fracf_c(z_0) = 2z_0 .$

At a periodic point ''z''₀ of period ''p'' the first derivative of a function

: $(f_c^p)'(z_0) = \fracf_c^p(z_0) = \prod_^ f_c'(z_i) = 2^p \prod_^ z_i = \lambda$

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

: $z'_0 = 1,$

and then using

:$z'_n= 2*z_*z'_.$

This derivative is used for computing the external distance to the Julia set.

Schwarzian derivative

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

:$(Sf)(z) = \frac - \frac \left ( \frac\right ) ^2 .$

See also

* Misiurewicz point
*Periodic points of complex quadratic mappings
*Mandelbrot set
*Julia set
*Milnor–Thurston kneading theory
*Tent map
*Logistic map

References

External links

* Monica Nevins and Thomas D. Rogers,
Quadratic maps as dynamical systems on the p-adic numbers

Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
*More about Quadratic Maps

{{DEFAULTSORT:Complex Quadratic Polynomial
Complex dynamics
Fractals
Polynomials