Complex quadratic polynomial
Quadratic polynomials have the following properties, regardless of the form:
- It is a unicritical polynomial, i.e. it has one critical point,
- 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.
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 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"
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.
Because of the possible confusion with exponentiation, some authors write for the nth iteration of the function .
The monic and centered form can be marked by:
- the parameter
- the external angle of the ray that lands:
- at c in M on the parameter plane
- at z = c in J(f) on the dynamic plane
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.
we see that the only (finite) critical point of is the point .
A critical value of is the image of a critical point:
So the parameter is the critical value of .
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.
The critical sector is a sector of the dynamical plane containing the critical point.
These polynomials are used for:
- finding centers of these Mandelbrot set components of period n. Centers are roots of n-th critical polynomials
Diagrams of critical polynomials are called critical curves.
In this space there are 2 basic types of 2-D 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:
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
- exterior of Mandelbrot set with
- external rays
- equipotential lines
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 dynamical plane consists of:
Here, is a constant and is a variable.
Dynamical z-planes can be divided in two groups :
- plane for (see complex squaring map)
- planes (all other planes for )
The extended complex plane plus a point at infinity
First derivative with respect to c
On the parameter plane:
- is a variable
- is constant
The first derivative of with respect to c is
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. It 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.
- Misiurewicz point
- Periodic points of complex quadratic mappings
- Mandelbrot set
- Julia set
- Milnor–Thurston kneading theory
- Tent map
- Logistic map
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- stackexchange questions : Show that the familiar logistic map ...
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- Khan Academy : Mandelbrot Spirals 2
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- Julia-Mandelbrot Space at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
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- Holomorphic motions and puzzels by P Roesch
- Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity[permanent dead link]
- Alternate Parameter Planes by David E. Joyce
- exponentialmap by Robert Munafo
- Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000)
- Mandelbrot set by Saratov group of theoretical nonlinear dynamics
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- The Schwarzian Derivative & the Critical Orbit by Wes McKinney 18.091 20 April 2005
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