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 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.[1]
• 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: ${\displaystyle f(x)=a_{2}x^{2}+a_{1}x+a_{0}\qquad \,}$ where ${\displaystyle \qquad a_{2}\neq 0}$
• The factored form used for logistic map ${\displaystyle f_{r}(x)=rx(1-x)\,}$
• ${\displaystyle f_{\theta }(x)=x^{2}+\lambda x\,}$ which has an indifferent fixed point with multiplier ${\displaystyle \lambda =e^{2\pi \theta i}\,}$ at the origin[2]
• The monic and centered form, ${\displaystyle f_{c}(x)=x^{2}+c\,}$

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).[3]

The lambda form ${\displaystyle f_{\lambda }(z)=z^{2}+\lambda z\,}$ is:

• the simplest non-trivial perturbation of unperturbated system ${\displaystyle z\mapsto \lambda z}$
• "the first family of dynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"[4]

Conjugation

Between forms

Since ${\displaystyle f_{c}(x)\,}$ 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 ${\displaystyle \theta \,}$ to ${\displaystyle c\,}$:[5]

${\displaystyle c=c(\theta )={\frac {e^{2\pi \theta i}}{2}}\left(1-{\frac {e^{2\pi \theta i}}{2}}\right).}$

When one wants change from ${\displaystyle r\,}$ to ${\displaystyle c,\,}$ the parameter transformation is[6]

${\displaystyle c=c(r)\,=\,{\frac {1-(r-1)^{2}}{4}}=-{\frac {r}{2}}\left({\frac {r-2}{2}}\right)}$

and the transformation between the variables in ${\displaystyle z_{t+1}=z_{t}^{2}+c}$ and ${\displaystyle x_{t+1}=rx_{t}(1-x_{t})}$ is

${\displaystyle z=r\left({\frac {1}{2}}-x\right).}$

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 ${\displaystyle f^{n}\,}$ denotes the n-th iteration of the function ${\displaystyle f\,}$ (and not exponentiation of the function):

${\displaystyle f_{c}^{n}(z)=f_{c}^{1}(f_{c}^{n-1}(z))\,}$

so

${\displaystyle z_{n}=f_{c}^{n}(z_{0}).\,}$

Because of the possible confusion with exponentiation, some authors write ${\displaystyle f^{\circ n}\,}$ for the nth iterate of the function ${\displaystyle f.\,}$

Parameter

The monic and centered form ${\displaystyle f_{c}(x)=x^{2}+c\,}$ can be marked by:

• the parameter ${\displaystyle c}$
• the external angle ${\displaystyle \theta }$ of the ray that lands:
  • at c in M on the parameter plane
  • at z = c in J(f) on the dynamic plane

so :

${\displaystyle f_{c}=f_{\theta }\,}$

${\displaystyle c=c({\theta })}$

Map

The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[7] is typically used with variable ${\displaystyle z\,}$ and parameter ${\displaystyle c\,}$:

${\displaystyle f_{c}(z)=z^{2}+c.\,}$

When it is used as an evolution function of the discrete nonlinear dynamical system

${\displaystyle z_{n+1}=f_{c}(z_{n})\,}$

it is named the quadratic map:[8]

${\displaystyle f_{c}:z\to z^{2}+c.\,}$

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 point

A critical point of ${\displaystyle f_{c}\,}$ is a point ${\displaystyle z_{cr}\,}$ in the dynamical plane such that the derivative vanishes:

${\displaystyle f_{c}'(z_{cr})=0.\,}$

Since

${\displaystyle f_{c}'(z)={\frac {d}{dz}}f_{c}(z)=2z}$

implies

${\displaystyle z_{cr}=0\,}$

we see that the only (finite) critical point of ${\displaystyle f_{c}\,}$ is the point ${\displaystyle z_{cr}=0\,}$.

${\displaystyle z_{0}}$ is an initial point for Mandelbrot set iteration.[9]

Critical value

A critical value ${\displaystyle z_{cv}\ }$ of ${\displaystyle f_{c}\,}$ is the image of a critical point:

${\displaystyle z_{cv}=f_{c}(z_{cr})\,}$

Since

${\displaystyle z_{cr}=0\,}$

we have

${\displaystyle z_{cv}=c.\,}$

So the parameter ${\displaystyle c\,}$ is the critical value of ${\displaystyle f_{c}(z).\,}$

Critical orbit

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]

${\displaystyle z_{0}=z_{cr}=0\,}$

${\displaystyle z_{1}=f_{c}(z_{0})=c\,}$

${\displaystyle z_{2}=f_{c}(z_{1})=c^{2}+c\,}$

${\displaystyle z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,}$

${\displaystyle ...\,}$

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 polynomial

${\displaystyle P_{n}(c)=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,}$

so

${\displaystyle P_{0}(c)=0\,}$

${\displaystyle P_{1}(c)=c\,}$

${\displaystyle P_{2}(c)=c^{2}+c\,}$

${\displaystyle P_{3}(c)=(c^{2}+c)^{2}+c\,}$

These polynomials are used for:

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

${\displaystyle centers=\{c:P_{n}(c)=0\}\,}$

• finding roots of Mandelbrot set components of period n (local minimum of ${\displaystyle P_{n}(c)\,}$)
• Misiurewicz points

${\displaystyle M_{n,k}=\{c:P_{k}(c)=P_{k+n}(c)\}\,}$

Critical curves

Critical curves

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

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

Spaces, planes

4D space

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, ${\displaystyle f_{c}\,}$-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 plane

Gamma parameter plane for complex logistic map ${\displaystyle z_{n+1}=\gamma z_{n}\left(1-z_{n}\right),}$

Multiplier map

The phase space of a quadratic map is called its parameter plane. Here:

${\displaystyle z_{0}=z_{cr}\,}$ is constant and ${\displaystyle c\,}$ 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[19] with internal rays
• exterior of Mandelbrot set with
  • external rays
  • equipotential lines

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 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 K a u k o[22]

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, ${\displaystyle c\,}$ is a constant and ${\displaystyle z\,}$ 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 :

• ${\displaystyle f_{0}}$ plane for ${\displaystyle c=0}$ ( see complex squaring map )
• ${\displaystyle f_{c}}$ planes ( all other planes for ${\displaystyle c\neq 0}$ )

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:

• ${\displaystyle c}$ is a variable
• ${\displaystyle z_{0}=0}$ is constant

The first derivative of ${\displaystyle f_{c}^{n}(z_{0})}$ with respect to c is

${\displaystyle z_{n}'={\frac {d}{dc}}f_{c}^{n}(z_{0}).}$

This derivative can be found by iteration starting with

${\displaystyle z_{0}'={\frac {d}{dc}}f_{c}^{0}(z_{0})=1}$

and then replacing at every consecutive step

${\displaystyle z_{n+1}'={\frac {d}{dc}}f_{c}^{n+1}(z_{0})=2\cdot {}f_{c}^{n}(z)\cdot {\frac {d}{dc}}f_{c}^{n}(z_{0})+1=2\cdot z_{n}\cdot z_{n}'+1.}$

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:

• ${\displaystyle z}$ is a variable;
• ${\displaystyle c}$ is a constant.

At a fixed point ${\displaystyle z_{0}\,,}$

${\displaystyle f_{c}'(z_{0})={\frac {d}{dz}}f_{c}(z_{0})=2z_{0}.}$

At a periodic point z₀ of period p the first derivative of a function

${\displaystyle (f_{c}^{p})'(z_{0})={\frac {d}{dz}}f_{c}^{p}(z_{0})=\prod _{i=0}^{p-1}f_{c}'(z_{i})=2^{p}\prod _{i=0}^{p-1}z_{i}=\lambda }$

is often represented by ${\displaystyle \lambda }$ 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 ${\displaystyle z'_{n},\,}$ can be found by iteration starting with

${\displaystyle z'_{0}=1,\,}$

and then using

${\displaystyle z'_{n}=2*z_{n-1}*z'_{n-1}.\,}$

This derivative is used for computing the external distance to the Julia set.

Schwarzian derivative

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

${\displaystyle (Sf)(z)={\frac {f'''(z)}{f'(z)}}-{\frac {3}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}}$.

See also

• Misiurewicz point
• Periodic points of complex quadratic mappings
• Mandelbrot set
• Julia set
• Milnor–Thurston kneading theory
• Tent map
• Logistic map

References

1. Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
2. Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
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. 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

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