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: 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).Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark 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"Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46
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,Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264 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 that
we see that the only (finite) critical point of is the point .
is an initial point for Mandelbrot set iteration.Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
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, a complex quadratic polynomial has 2d-2 critical points. In this model, 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.M. Romera , G. Pastor , and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104–116
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.The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640–653
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 planeCarleson, Lennart, Gamelin, Theodore W.: Complex Dynamics Series: Universitext, Subseries: Universitext: Tracts in Mathematics, 1st ed. 1993. Corr. 2nd printing, 1996, IX, 192 p. 28 illus., model planeHolomorphic motions and puzzels by P Roesch
2D Parameter plane
The phase space of a quadratic map is called its parameter plane. Here:
is constant and 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 External rays Equipotential lines
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 KaukoTrees of visible components in the Mandelbrot set by Virpi Kauko, FUNDAMENTA MATHEMATICAE 164 (2000)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, 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.Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,
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