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A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.



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[1]
  • The monic and centered form,  

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

The lambda form   is:

  • the simplest non-trival 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"[3]

Quadratic polynomials have the following properties, regardless of the form:


Between formsEdit

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  :[5]


When one wants change from   to   the parameter transformation is[6]


and the transformation between the variables in   and   is


With doubling mapEdit

There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.


The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[7] 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:[8]


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.


Here   denotes the n-th iteration of the function   (and not exponentiation of the function):




Because of the possible confusion with exponentiation, some authors write   for the nth iterate of the function  

Critical itemsEdit

Critical pointEdit

A critical point of   is a point   in the dynamical plane such that the derivative vanishes:






we see that the only (finite) critical point of   is the point  .

  is an initial point for Mandelbrot set iteration.[9]

Critical valueEdit

A critical value   of   is the image of a critical point:




we have


So the parameter   is the critical value of  

Critical orbitEdit

Dynamical plane with critical orbit falling into 3-period cycle
Dynamical plane with Julia set and critical orbit.
Dynamical plane : changes of critical orbit along internal ray of main cardioid for angle 1/6
Critical orbit tending to weakly attracting fixed point with abs(multiplier)=0.99993612384259

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.[10][11][12]


This orbit falls into an attracting periodic cycle if one exists.

Critical sectorEdit

The critical sector is a sector of the dynamical plane containing the critical point.

Critical polynomialEdit




These polynomials are used for:

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

Critical curvesEdit

Diagrams of critical polynomials are called critical curves.[13]

These curves create the skeleton (the dark lines) of a bifurcation diagram.[14][15]

Spaces, planesEdit

4D spaceEdit

One can use the Julia-Mandelbrot 4-dimensional ( 4D) space for a global analysis of this dynamical system.[16]

w-plane and c-plane

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:

  • the conjugation plane[17]
  • model plane[18]

2D Parameter planeEdit

Gamma parameter plane for complex logistic map  
Multiplier map

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:

There are many different subtypes of the parameter plane.[20][21]

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 planeEdit

 "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 K a u k o[22]

On the dynamical plane one can find:

The dynamical plane consists of:

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.[23][24]

Dynamical z-planes can be divided in two groups :

  •   plane for   ( see complex squaring map )
  •   planes ( all other planes for   )

Riemann sphereEdit

The extended complex plane plus a point at infinity


First derivative with respect to cEdit

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 zEdit

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.

Schwarzian derivativeEdit

The Schwarzian derivative (SD for short) of f is:[25]


See alsoEdit


  1. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  2. ^ Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark
  3. ^ Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46
  4. ^ Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
  5. ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  6. ^ stackexchange questions : Show that the familiar logistic map ...
  7. ^ 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
  8. ^ Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Web Resource
  9. ^ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Archived 26 April 2012 at the Wayback Machine
  10. ^ M. Romera Archived 22 June 2008 at the Wayback Machine, G. Pastor Archived 1 May 2008 at the Wayback Machine, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Archived 11 December 2009 at the Wayback Machine Fractalia Archived 19 September 2008 at the Wayback Machine 6, No. 21, 10-12 (1997)
  11. ^ 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
  12. ^ Khan Academy : Mandelbrot Spirals 2
  13. ^ 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
  14. ^ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. Archived from the original on 5 December 2009. Retrieved 2 December 2009.
  15. ^ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint Archived 2 October 2006 at the Wayback Machine
  16. ^ Julia-Mandelbrot Space at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
  17. ^ Carleson, 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., ISBN 978-0-387-97942-7
  18. ^ Holomorphic motions and puzzels by P Roesch
  19. ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  20. ^ Alternate Parameter Planes by David E. Joyce
  21. ^ exponentialmap by Robert Munafo
  22. ^ Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000)
  23. ^ Mandelbrot set by Saratov group of theoretical nonlinear dynamics
  24. ^ Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,
  25. ^ The Schwarzian Derivative & the Critical Orbit by Wes McKinney 18.091 20 April 2005

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