Acomplex quadratic polynomial is aquadratic polynomial whosecoefficients andvariable arecomplex numbers.

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

• It is a unicritical polynomial, i.e. it has onefinite 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 bepostcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic.^[1]
• It is aunimodal function,
• It is arational function,
• It is anentire function.

When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms:

• The general form:${\displaystyle f(x)=a_2{x}^2+a_{1}x+a_0}$ where${\displaystyle a_2\ne 0}$
• The factored form used for thelogistic map:${\displaystyle f_r(x)=rx(1-x)}$
• ${\displaystyle f_{\theta }(x)=x^2+\lambda x}$ which has an indifferentfixed point withmultiplier${\displaystyle \lambda =e^{2\pi \theta i}}$ at theorigin^[2]
• Themonic and centered form,${\displaystyle f_c(x)=x^2+c}$

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

• It is the simplest form of anonlinearfunction with one coefficient (parameter),
• It is a centered polynomial (the sum of its critical points is zero).^[3]
• it is abinomial

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 ofdynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"^[4]

Since${\displaystyle f_c(x)}$ isaffineconjugate to the general form of the quadratic polynomial it is often used to studycomplex dynamics and to create images ofMandelbrot,Julia andFatou sets.

When one wants change from${\displaystyle \theta }$ to${\displaystyle c}$:^[2]

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

When one wants change from${\displaystyle r}$ to${\displaystyle c}$, the parameter transformation is^[5]

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

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

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

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

Here${\displaystyle f^n}$ denotes then-thiterate of the function${\displaystyle f}$:

${\displaystyle f_c^n(z)=f_c^1(f_c^{n-1}(z))}$

so

${\displaystyle z_n=f_c^n(z_0).}$

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

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:
  • atc in Mandelbrot set on the parameter plane
  • on the critical value:z =c in Julia set on the dynamic plane

so :

${\displaystyle f_c=f_{\theta }}$

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

Examples:

• c is the landing point of the 1/6external ray of theMandelbrot set, and is${\displaystyle z\to z^2+i}$ (where i^2=-1)
• c is the landing point the 5/14external ray and is${\displaystyle z\to z^2+c}$ with${\displaystyle c=-1.23922555538957+0.412602181602004\ast i}$

• 
  1/4
• 
  1/6
• 
  9/56
• 
  129/16256

The monic and centered form, sometimes called theDouady-Hubbard family of quadratic polynomials,^[6] is typically used with variable${\displaystyle z}$ andparameter${\displaystyle c}$:

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

When it is used as anevolution function of thediscrete nonlinear dynamical system

${\displaystyle z_{n+1}=f_c(z_n)}$

it is named thequadraticmap:^[7]

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

TheMandelbrot set is the set of values of the parameterc for which the initial conditionz₀ = 0 does not cause the iterates to diverge to infinity.

Acritical point of${\displaystyle f_c}$ is a point${\displaystyle z_{cr}}$ onthe dynamical plane such that thederivative vanishes:

${\displaystyle f_c^{\prime }(z_{cr})=0.}$

Since

${\displaystyle f_c^{\prime }(z)=\frac{d}{dz}{f}_c(z)=2z}$

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 forMandelbrot set iteration.^[8]

For the quadratic family${\displaystyle f_c(z)=z^2+c}$ the critical point z = 0 is thecenter of symmetry of theJulia set Jc, so it is aconvex combination of two points in Jc.^[9]

In theRiemann sphere polynomial has 2d-2 critical points. Here zero andinfinity are critical points.

Acritical 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${\displaystyle f_c(z)}$.

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 atsaddle point, which is a special type of critical point.

• 
  attracting
• 
  attracting
• 
  attracting
• 
  parabolic
• Video for c along internal ray 0

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

Theforward orbit of a critical point is called acritical orbit. Critical orbits are very important because every attractingperiodic orbit attracts a critical point, so studying the critical orbits helps us understand the dynamics in theFatou set.^[11][12][13]

${\displaystyle z_0=z_{cr}=0}$

${\displaystyle z_1=f_c(z_0)=c}$

${\displaystyle z_2=f_c(z_1)=c^2+c}$

${\displaystyle z_3=f_c(z_2)=(c^2+c{)}^2+c}$

${\displaystyle ⋮}$

This orbit falls into anattracting periodic cycle if one exists.

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

Critical set is a set of critical points

${\displaystyle P_n(c)=f_c^n(z_{cr})=f_c^n(0)}$

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 periodn. Centers areroots ofn-th critical polynomials

${\displaystyle \text{centers}=\{c:P_n(c)=0\}}$

• finding roots of Mandelbrot set components of periodn (local minimum of${\displaystyle P_n(c)}$)
• Misiurewicz points

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

Diagrams of critical polynomials are calledcritical curves.^[14]

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

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

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

• the dynamical (dynamic) plane,${\displaystyle f_c}$-plane orc-plane
• the parameter plane orz-plane

There is also another plane used to analyze such dynamical systemsw-plane:

• the conjugation plane^[18]
• model plane^[19]

• Parameter plane types
• 
  r parameter plane (logistic map)
• 
  c parameter plane

Thephase space of a quadratic map is called itsparameter 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:

• TheMandelbrot set
  • Thebifurcation locus = boundary ofMandelbrot set with
    • root points
  • Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set^[20] with internal rays
• exterior of Mandelbrot set with
  • external rays
  • equipotential lines

There are many different subtypes of the parameter plane.^[21][22]

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

"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:

• TheJulia set
• TheFilled Julia set
• TheFatou 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 aPoincaré cross-section of three-dimensional space of continuous dynamical system.^[24][25]

Dynamicalz-planes can be divided into two groups:

• ${\displaystyle f_0}$ plane for${\displaystyle c=0}$ (seecomplex squaring map)
• ${\displaystyle f_c}$ planes (all other planes for${\displaystyle c\ne 0}$)

The extended complex plane plus apoint at infinity

• the Riemann sphere

On the parameter plane:

• ${\displaystyle c}$ is a variable
• ${\displaystyle z_0=0}$ is constant

The firstderivative of${\displaystyle f_c^n(z_0)}$ with respect toc is

${\displaystyle z_n^{\prime }=\frac{d}{dc}{f}_c^n(z_0).}$

This derivative can be found byiteration starting with

${\displaystyle z_0^{\prime }=\frac{d}{dc}{f}_c^0(z_0)=1}$

and then replacing at every consecutive step

${\displaystyle z_{n+1}^{\prime }=\frac{d}{dc}{f}_c^{n+1}(z_0)=2\cdot f_c^n(z)\cdot \frac{d}{dc}{f}_c^n(z_0)+1=2\cdot z_n\cdot z_n^{\prime }+1.}$

This can easily be verified by using thechain rule for the derivative.

This derivative is used in thedistance estimation method for drawing a Mandelbrot set.

On the dynamical plane:

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

At afixed point${\displaystyle z_0}$,

${\displaystyle f_c^{\prime }(z_0)=\frac{d}{dz}{f}_c(z_0)=2z_0.}$

At aperiodic pointz₀ of periodp the first derivative of a function

${\displaystyle (f_c^p{)}^{\prime }(z_0)=\frac{d}{dz}{f}_c^p(z_0)=\prod _{i=0}^{p-1}{f}_c^{\prime }(z_i)=2^p\prod _{i=0}^{p-1}{z}_i=\lambda }$

is often represented by${\displaystyle \lambda }$ and referred to as the multiplier or the Lyapunov characteristic number. Itslogarithm is known as the Lyapunov exponent. Absolute value of multiplier is used to check thestability ofperiodic (also fixed) points.

At anonperiodic point, the derivative, denoted by${\displaystyle z_n^{\prime }}$, can be found byiteration starting with

${\displaystyle z_0^{\prime }=1,}$

and then using

${\displaystyle z_n^{\prime }=2\ast z_{n-1}\ast z_{n-1}^{\prime }.}$

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

TheSchwarzian derivative (SD for short) off is:^[26]

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

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

Wikimedia Commons has media related toComplex quadratic polynomials.

• Monica Nevins and Thomas D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers^{[permanent dead link]}"
• Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
• More about Quadratic Maps :Quadratic Map

• Complex dynamics
• Fractals
• Polynomials

• Webarchive template wayback links
• Articles with short description
• Short description matches Wikidata
• Use dmy dates from June 2023
• Commons category link from Wikidata
• All articles with dead external links
• Articles with dead external links from June 2024
• Articles with permanently dead external links