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:
When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms:
f(x)=a2x2+a1x+a0
a2\ne0
fr(x)=rx(1-x)
f\theta(x)=x2+λx
λ=e2
fc(x)=x2+c
The monic and centered form has been studied extensively, and has the following properties:
The lambda form
fλ(z)=z2+λz
z\mapstoλz
Since
fc(x)
When one wants change from
\theta
c
c=c(\theta)=
| e2 | |
| 2 |
\left(1-
| e2 | |
| 2 |
\right).
When one wants change from
r
c
c=c(r)=
| 1-(r-1)2 | |
| 4 |
=-
| r | \left( | |
| 2 |
| r-2 | |
| 2 |
\right)
and the transformation between the variables in
zt+1
| 2+c | |
| =z | |
| t |
xt+1=rxt(1-xt)
| z=r\left( | 1 |
| 2 |
-x\right).
There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.
Here
fn
f
| n(z) | |
| f | |
| c |
=
| n-1 | |
| f | |
| c |
(z))
so
zn=
| n(z | |
| f | |
| 0). |
Because of the possible confusion with exponentiation, some authors write
f\circ
f
The monic and centered form
fc(x)=x2+c
c
\theta
so :
fc=f\theta
c=c({\theta})
Examples:
z\toz2+i
z\toz2+c
c=-1.23922555538957+0.412602181602004*i
The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,[6] is typically used with variable
z
c
fc(z)=z2+c.
When it is used as an evolution function of the discrete nonlinear dynamical system
zn+1=fc(zn)
it is named the quadratic map:[7]
fc:z\toz2+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
fc
zcr
fc'(zcr)=0.
Since
fc'(z)=
| d | |
| dz |
fc(z)=2z
implies
zcr=0,
we see that the only (finite) critical point of
fc
zcr=0
z0
For the quadratic family
| 2+c | |
| f | |
| c(z)=z |
In the Riemann sphere polynomial has 2d-2 critical points. Here zero and infinity are critical points.
A critical value
zcv
fc
zcv=fc(zcr)
Since
zcr=0
we have
zcv=c
So the parameter
c
fc(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 at saddle point, which is a special type of critical point.
Critical limit set is the set of forward orbit of all critical points
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]
z0=zcr=0
z1=fc(z0)=c
z2=fc(z1)=c2+c
z3=fc(z2)=(c2+c)2+c
\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
Pn(c)=
| n(z | |
| f | |
| cr |
)=
| n(0) | |
| f | |
| c |
so
P0(c)=0
P1(c)=c
P2(c)=c2+c
P3(c)=(c2+c)2+c
These polynomials are used for:
centers=\{c:Pn(c)=0\}
Pn(c)
Mn,k=\{c:Pk(c)=Pk+n(c)\}
Diagrams of critical polynomials are called critical curves.[14]
These curves create the skeleton (the dark lines) of a bifurcation 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:
fc
There is also another plane used to analyze such dynamical systems w-plane:
The phase space of a quadratic map is called its parameter plane. Here:
z0=zcr
c
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]
See also :
"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,
c
z
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:
f0
c=0
fc
c\ne0
The extended complex plane plus a point at infinity
On the parameter plane:
c
z0=0
The first derivative of
| n(z | |
| f | |
| 0) |
zn'=
| d | |
| dc |
| n(z | |
| f | |
| 0). |
This derivative can be found by iteration starting with
z0'=
| d | |
| dc |
| 0(z | |
| f | |
| 0) |
=1
and then replacing at every consecutive step
zn+1'=
| d | |
| dc |
| n+1 | |
| f | |
| c |
(z0)=
| ||||
| 2 ⋅ {}f | ||||
| c |
| n(z | |
| f | |
| 0) |
+1=2 ⋅ zn ⋅ zn'+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
c
At a fixed point
z0
fc'(z0)=
| d | |
| dz |
fc(z0)=2z0.
At a periodic point z0 of period p the first derivative of a function
| p)'(z | |
| (f | |
| 0) |
=
| d | |
| dz |
| p(z | |
| f | |
| 0) |
=
| p-1 | |
| \prod | |
| i=0 |
fc'(zi)=2p
| p-1 | |
| \prod | |
| i=0 |
zi=λ
is often represented by
λ
At a nonperiodic point, the derivative, denoted by
z'n
z'0=1,
and then using
z'n=2*zn-1*z'n-1.
This derivative is used for computing the external distance to the Julia set.
The Schwarzian derivative (SD for short) of f is:[26]
(Sf)(z)=
| f'''(z) | |
| f'(z) |
-
| 3 | |
| 2 |
\left(
| f''(z) | |
| f'(z) |
\right)2.