Complex quadratic polynomial

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


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.[1]
  • 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 the logistic map:
  • which has an indifferent fixed point with multiplier at the origin[2]
  • 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-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"[4]


Between forms[edit]

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 :[2]

When one wants change from to , the parameter transformation is[5]

and the transformation between the variables in and is

With doubling map[edit]

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



Here denotes the n-th iterate of the function :


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


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 :


  • 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


The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[6] 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:[7]

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[edit]

Critical points[edit]

complex plane[edit]

A critical point of is a point on 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.[8]

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.[9]

extended complex plane[edit]

In the Riemann sphere polynomial has 2d-2 critical points. Here zero and infinity are critical points.

Critical value[edit]

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


we have

So the parameter is the critical value of .

Critical level curves[edit]

A critical level curve the level curve which contain critical point. It acts as a sort of skeleton[10] of dynamical plane

Example : level curves cross at saddle point, which is a special type of critical point.

Critical limit set[edit]

Critical limit set is the set of forward orbit of all critical points

Critical orbit[edit]

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.[11][12][13]

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

Critical sector[edit]

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

Critical set[edit]

Critical set is a set of critical points

Critical polynomial[edit]


These polynomials are used for:

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

Critical curves[edit]

Critical curves

Diagrams of critical polynomials are called critical curves.[14]

These curves create the skeleton (the dark lines) of a bifurcation diagram.[15][16]

Spaces, planes[edit]

4D space[edit]

One can use the Julia-Mandelbrot 4-dimensional (4D) space for a global analysis of this dynamical system.[17]

w-plane and c-plane

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[18]
  • model plane[19]

2D Parameter plane[edit]

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.[21][22]

Multiplier map

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[edit]

"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[23]

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.[24][25]

Dynamical z-planes can be divided into two groups:

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

Riemann sphere[edit]

The extended complex plane plus a point at infinity


First derivative with respect to c[edit]

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[edit]

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[edit]

The Schwarzian derivative (SD for short) of f is:[26]

See also[edit]


  1. ^ Poirier, Alfredo (1993). "On postcritically finite polynomials, part 1: Critical portraits". arXiv:math/9305207.
  2. ^ a b "Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials" (PDF).
  3. ^ Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark
  4. ^ Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46
  5. ^ "Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$". Mathematics Stack Exchange.
  6. ^ 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
  7. ^ Weisstein, Eric W. "Quadratic Map".
  8. ^ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Archived 26 April 2012 at the Wayback Machine
  9. ^ "Convex Julia sets". MathOverflow.
  10. ^ Richards, Trevor (11 May 2015). "Conformal equivalence of analytic functions on compact sets". arXiv:1505.02671v1 [math.CV].
  11. ^ 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)
  12. ^ 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
  13. ^ "Khan Academy". Khan Academy.
  14. ^ 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
  15. ^ 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.
  16. ^ "M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint" (PDF). Archived from the original (PDF) on 2 October 2006.
  17. ^ "Julia-Mandelbrot Space, Mu-Ency at MROB".
  18. ^ 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
  19. ^ Holomorphic motions and puzzels by P Roesch
  20. ^ Rempe, Lasse; Schleicher, Dierk (12 May 2008). "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity". arXiv:0805.1658 [math.DS].
  21. ^ "Julia and Mandelbrot sets, alternate planes".
  22. ^ "Exponential Map, Mu-Ency at MROB".
  23. ^ Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000)
  24. ^ "The Mandelbrot Set is named after mathematician Benoit B".
  25. ^ Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia,
  26. ^ "Lecture Notes | Mathematical Exposition | Mathematics". MIT OpenCourseWare.

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