Complex quadratic polynomial#Parameter plane

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.{{cite arXiv | eprint=math/9305207 | last1=Poirier | first1=Alfredo | title=On postcritically finite polynomials, part 1: Critical portraits | year=1993 }}
• 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: $f(x)\; =\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0$ where $a\_2\; \backslash ne\; 0$
• The factored form used for the logistic map: $f\_r(x)\; =\; r\; x\; (1-x)$
• $f\_\{\backslash theta\}(x)\; =\; x^2\; +\backslash lambda\; x$ which has an indifferent fixed point with multiplier $\backslash lambda\; =\; e^\{2\; \backslash pi\; \backslash theta\; i\}$ at the origin{{Cite web|url=https://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf|title=Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.}}
• The monic and centered form, $f\_c(x)\; =\; x^2\; +c$

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).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 $f\_\{\backslash lambda\}(z)\; =\; z^2\; +\backslash lambda\; z$ is:

• the simplest non-trivial perturbation of unperturbated system $z\; \backslash mapsto\; \backslash lambda\; z$
• "the first family of dynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46

Since $f\_c(x)$ 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 $\backslash theta$ to $c$:

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

When one wants change from $r$ to $c$, the parameter transformation is{{Cite web|url=https://math.stackexchange.com/q/290564|title=Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$.|website=Mathematics Stack Exchange}}

:$$

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 $z\_\{t+1\}=z\_t^2+c$ and $x\_\{t+1\}=rx\_t(1-x\_t)$ is

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

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

Here $f^n$ denotes the n-th iterate of the function $f$:

:$f\_c^n(z)\; =\; f\_c^1(f\_c^\{n-1\}(z))$

so

:$z\_n\; =\; f\_c^n(z\_0).$

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

The monic and centered form $f\_c(x)\; =\; x^2\; +c$ can be marked by:

• the parameter $c$
• the external angle $\backslash theta$ 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 :

:$f\_c\; =\; f\_\{\backslash theta\}$

:$c\; =\; c(\{\backslash theta\})$

Examples:

• c is the landing point of the 1/6 external ray of the Mandelbrot set, and is $z\; \backslash to\; z^2+i$ (where i^2=-1)
• c is the landing point the 5/14 external ray and is $z\; \backslash to\; z^2+\; c$ with $c\; =\; -1.23922555538957\; +\; 0.412602181602004*i$

Paritition of dynamic plane of quadratic polynomial for 1 4.svg|1/4

Paritition of dynamic plane of quadratic polynomial for 1 6.svg|1/6

Paritition of dynamic plane of quadratic polynomial for 9 56.svg|9/56

Paritition of dynamic plane of quadratic polynomial for 129 over 16256.svg|129/16256

The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/2004MandLocConn.pdf 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 $z$ and parameter $c$:

:$f\_c(z)\; =\; z^2\; +c.$

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

: $z\_\{n+1\}\; =\; f\_c(z\_n)$

it is named the quadratic map:{{Cite web|url=https://mathworld.wolfram.com/|title=Quadratic Map|first=Eric W.|last=Weisstein|website=mathworld.wolfram.com}}

:$f\_c\; :\; z\; \backslash to\; z^2\; +\; c.$

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.

A critical point of $f\_c$ is a point $z\_\{cr\}$ on the dynamical plane such that the derivative vanishes:

: $f\_c\text{'}(z\_\{cr\})\; =\; 0.$

Since

: $f\_c\text{'}(z)\; =\; \backslash frac\{d\}\{dz\}f\_c(z)\; =\; 2z$

implies

: $z\_\{cr\}\; =\; 0,$

we see that the only (finite) critical point of $f\_c$ is the point $z\_\{cr\}\; =\; 0$.

$z\_0$ is an initial point for Mandelbrot set iteration.[http://mathesim.degruyter.de/jws_en/show_simulation.php?id=1052&type=RoessMa&lang=en Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations] {{webarchive|url=https://web.archive.org/web/20120426002245/http://mathesim.degruyter.de/jws_en/show_simulation.php?id=1052&type=RoessMa&lang=en |date=26 April 2012 }}

For the quadratic family $f\_c(z)=z^2+c$ 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.{{Cite web|url=https://mathoverflow.net/questions/356342/convex-julia-sets|title=Convex Julia sets|website=MathOverflow}}

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

A critical value $z\_\{cv\}$ of $f\_c$ is the image of a critical point:

: $z\_\{cv\}\; =\; f\_c(z\_\{cr\})$

Since

: $z\_\{cr\}\; =\; 0$

we have

: $z\_\{cv\}\; =\; c$

So the parameter $c$ is the critical value of $f\_c(z)$.

A critical level curve the level curve which contain critical point. It acts as a sort of skeleton{{Cite arXiv|title=Conformal equivalence of analytic functions on compact sets|first=Trevor|last=Richards|date=11 May 2015|class=math.CV |eprint=1505.02671v1}} of dynamical plane

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

Julia set for z^2+0.7i*z.png|attracting

IntLSM_J.jpg| attracting

ILSMJ.png| attracting

Level sets of attraction time to parabolic fixed point in the fat basilica Julia set.png|parabolic

Quadratic Julia set with Internal level sets for internal ray 0.ogv| Video for c along internal ray 0

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

File:Cr orbit 3.png

File:Miimcr.png

File:6furcation.gif

File:Critical orbit 3d.png

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.[http://www.iec.csic.es/~miguel/miguel.html M. Romera] {{webarchive|url=https://web.archive.org/web/20080622042735/http://www.iec.csic.es/~miguel/miguel.html |date=22 June 2008 }}, [http://www.iec.csic.es/~gerardo/Gerardo.html G. Pastor] {{webarchive|url=https://web.archive.org/web/20080501044137/http://www.iec.csic.es/~gerardo/gerardo.html |date=1 May 2008 }}, and F. Montoya : [http://www.iec.csic.es/~miguel/Preprint4.ps Multifurcations in nonhyperbolic fixed points of the Mandelbrot map.] {{webarchive|url=https://web.archive.org/web/20091211153845/http://iec.csic.es/~miguel/Preprint4.ps |date=11 December 2009 }} [http://users.utcluj.ro/~mdanca/fractalia/ Fractalia] {{webarchive|url=https://web.archive.org/web/20080919234345/http://users.utcluj.ro/~mdanca/fractalia/ |date=19 September 2008 }} 6, No. 21, 10-12 (1997)[https://web.archive.org/web/20041031082406/http://myweb.cwpost.liu.edu/aburns/index.html 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{{Cite web|url=https://www.khanacademy.org/computer-programming/mandelbrot-spirals-2/1030775610|title=Khan Academy|website=Khan Academy}}

:$z\_0\; =\; z\_\{cr\}\; =\; 0$

:$z\_1\; =\; f\_c(z\_0)\; =\; c$

:$z\_2\; =\; f\_c(z\_1)\; =\; c^2\; +c$

:$z\_3\; =\; f\_c(z\_2)\; =\; (c^2\; +\; c)^2\; +\; c$

::$\backslash \; \backslash vdots$

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.

Critical set is a set of critical points

:$P\_n(c)\; =\; f\_c^n(z\_\{cr\})\; =\; f\_c^n(0)$

so

:$P\_0(c)=\; 0$

:$P\_1(c)\; =\; c$

:$P\_2(c)\; =\; c^2\; +\; c$

:$P\_3(c)\; =\; (c^2\; +\; c)^2\; +\; c$

These polynomials are used for:

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

::$\backslash text\{centers\}\; =\; \backslash \{\; c\; :\; P\_n(c)\; =\; 0\; \backslash \}$

• finding roots of Mandelbrot set components of period n (local minimum of $P\_n(c)$)
• Misiurewicz points

::$M\_\{n,k\}\; =\; \backslash \{\; c\; :\; P\_k(c)\; =\; P\_\{k+n\}(c)\; \backslash \}$

File:LogisticMap EarlyIterates.png

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.{{Cite book

| last = Hao

| first = Bailin

| author-link = Bailin Hao

| title = Elementary Symbolic Dynamics and Chaos in Dissipative Systems

| publisher = World Scientific

| year = 1989

| url = http://power.itp.ac.cn/~hao/

| isbn = 9971-5-0682-3

| access-date = 2 December 2009

| archive-url = https://web.archive.org/web/20091205014855/http://power.itp.ac.cn/~hao/

| archive-date = 5 December 2009

| url-status = dead

}}{{Cite web|url=http://www.iec.csic.es/~gerardo/publica/Romera96b.pdf|archiveurl=https://web.archive.org/web/20061002202249/http://www.iec.csic.es/~gerardo/publica/Romera96b.pdf|url-status=dead|title=M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint|archivedate=2 October 2006}}

One can use the Julia-Mandelbrot 4-dimensional (4D) space for a global analysis of this dynamical system.{{Cite web|url=http://www.mrob.com/pub/muency/juliamandelbrotspace.html|title=Julia-Mandelbrot Space, Mu-Ency at MROB|website=www.mrob.com}}

File:Iray.png

In this space there are two basic types of 2D planes:

• the dynamical (dynamic) plane, $f\_c$-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 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., {{ISBN|978-0-387-97942-7}}
• model planeHolomorphic motions and puzzels by P Roesch

[http://matwbn.icm.edu.pl/ksiazki/fm/fm164/fm16413.pdf Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA 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, $c$ is a constant and $z$ is a variable.

The two-dimensional dynamical plane can be treated as a Poincaré cross-section of three-dimensional space of continuous dynamical system.{{Cite web|url=http://www.sgtnd.narod.ru/science/complex/eng/main.htm|title=The Mandelbrot Set is named after mathematician Benoit B|website=www.sgtnd.narod.ru}}[http://www.scholarpedia.org/article/Periodic_orbit#Stability_of_a_Periodic_Orbit|Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia],

Dynamical z-planes can be divided into two groups:

• $f\_0$ plane for $c\; =\; 0$ (see complex squaring map)
• $f\_c$ planes (all other planes for $c\; \backslash ne\; 0$)

The extended complex plane plus a point at infinity

• the Riemann sphere

On the parameter plane:

• $c$ is a variable
• $z\_0\; =\; 0$ is constant

The first derivative of $f\_c^n(z\_0)$ with respect to c is

: $z\_n\text{'}\; =\; \backslash frac\{d\}\{dc\}\; f\_c^n(z\_0).$

This derivative can be found by iteration starting with

: $z\_0\text{'}\; =\; \backslash frac\{d\}\{dc\}\; f\_c^0(z\_0)\; =\; 1$

and then replacing at every consecutive step

: $z\_\{n+1\}\text{'}\; =\; \backslash frac\{d\}\{dc\}\; f\_c^\{n+1\}(z\_0)\; =\; 2\backslash cdot\{\}f\_c^n(z)\backslash cdot\backslash frac\{d\}\{dc\}\; f\_c^n(z\_0)\; +\; 1\; =\; 2\; \backslash cdot\; z\_n\; \backslash cdot\; z\_n\text{'}\; +1.$

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.

On the dynamical plane:

• $z$ is a variable;
• $c$ is a constant.

At a fixed point $z\_0$,

: $f\_c\text{'}(z\_0)\; =\; \backslash frac\{d\}\{dz\}f\_c(z\_0)\; =\; 2z\_0\; .$

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

: $(f\_c^p)\text{'}(z\_0)\; =\; \backslash frac\{d\}\{dz\}f\_c^p(z\_0)\; =\; \backslash prod\_\{i=0\}^\{p-1\}\; f\_c\text{'}(z\_i)\; =\; 2^p\; \backslash prod\_\{i=0\}^\{p-1\}\; z\_i\; =\; \backslash lambda$

is often represented by $\backslash lambda$ 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 $z\text{'}\_n$, can be found by iteration starting with

: $z\text{'}\_0\; =\; 1,$

and then using

:$z\text{'}\_n=\; 2*z\_\{n-1\}*z\text{'}\_\{n-1\}.$

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

The Schwarzian derivative (SD for short) of f is:{{Cite web|url=https://ocw.mit.edu/courses/18-091-mathematical-exposition-spring-2005/pages/lecture-notes/|title=Lecture Notes | Mathematical Exposition | Mathematics|website=MIT OpenCourseWare}}

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

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

{{Reflist}}

{{Commons category}}

• Monica Nevins and Thomas D. Rogers, "[http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/nevins.pdf Quadratic maps as dynamical systems on the p-adic numbers]{{Dead link|date=June 2024 |bot=InternetArchiveBot |fix-attempted=yes }}"
• [https://web.archive.org/web/20040828174339/http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002]
• More about Quadratic Maps : [https://mathworld.wolfram.com/QuadraticMap.html Quadratic Map]

{{Chaos theory}}

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Category:Complex dynamics

Category:Fractals

Category:Polynomials