Complex quadratic polynomial

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: basin of infinity and basin of finite critical point (if finite critical point do not escapes)
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: where
The factored form used for the logistic map:
which has an indifferent fixed point with multiplier at the origin
The monic and centered form,
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 is:
the simplest non-trivial perturbation of unperturbated system
"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 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 to :
When one wants change from to , 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 denotes the n-th iterate of the function :
so
Because of the possible confusion with exponentiation, some authors write for the nth iterate of .
Parameter
The monic and centered form can be marked by:
the parameter
the external angle 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:
c is the landing point of the 1/6 external ray of the Mandelbrot set, and is (where i^2=-1)
c is the landing point the 5/14 external ray and is with
Map
The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable 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 of is a point on the dynamical plane such that the derivative vanishes:
Since
implies
we see that the only (finite) critical point of is the point .
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 of is the image of a critical point:
Since
we have
So the parameter is the critical value of .
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 of )
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:
the dynamical (dynamic) plane, -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 constant and 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 KaukoOn 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, is a constant and 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:
plane for (see complex squaring map)
planes (all other planes for )
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:
is a variable
is constant
The first derivative of 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
is often represented by 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 , 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
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 : Quadratic Map
Complex dynamics
Fractals
Polynomials