Last modified on 13 November 2014, at 20:36

Fractals/Iterations in the complex plane/qpolynomials

Complex quadratic polynomial[1]

FormsEdit

z^2+cEdit

Complex quadratic polynomial of the form :

 f_c(z) = z^2 + c \,

belongs to the class of the functions :

 z^n + c  \,

How to compute iterationEdit

In Maxima CAS :

(%i28) z:zx+zy*%i;
(%o28) %i*zy+zx
(%i37) c:cx+cy*%i;
(%o37) %i*cy+cx
(%i38) realpart(z^2+c);
(%o38) -zy^2+zx^2+cx
(%i39) imagpart(z^2+c);
(%o39) 2*zx*zy+cy

Critical pointEdit

Critical orbits for various parabolic parameters on boundary of Main component of Mandelbrot set

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

f_c'(z_{cr}) = 0. \,

Since

f_c'(z) = \frac{d}{dz}f_c(z) = 2z

implies

 z_{cr} = 0\,

One can see that :

  • the only (finite) critical point of f_c \, is the point  z_{cr} = 0\,
  • critical point is the same for all c parameters

z_{cr} is an initial point for Mandelbrot set iteration.[2]

Dynamic planeEdit

1/2Edit

Julia set for parabolic parameter on the end fo 1/2 internal ray of main component of Mandelbrot set

First compute muliplier m of the fixed points using internal angle p/q and Maxima CAS:

\frac{p}{q} = \frac{1}{2}

(%i1) p:1$
(%i2) q:2$
(%i3) m:exp(2*%pi*%i*p/q);
(%o3)                                 - 1

Now compute parameter c of the function :

(%i1) GiveC(t,r):=
(
 [w,c],
 /* point of  unit circle   w:l(internalAngle,internalRadius); */
 w:r*%e^(%i*t*2*%pi),  /* point of circle */
 c:w/2-w*w/4, /* point on boundary of period 1 component of Mandelbrot set */
 float(rectform(c))    
)$

(%i3) c:GiveC(1/2,1);
(%o3) −0.75


Find fixed points z

(%i4) z1:z^2+c;
(%o4) z^2−0.75
(%i2) f:z^2+c;
(%o2)                              z^2  - 0.75
(%i3) d:diff(f,z,1);
(%o3)                                 2 z
(%i6) s:solve(z1=z);
(%o6)                              [z = 3/2, z = -1/2]
(%i7) s:map(rhs,s);
(%o7)                             [z = 3/2, z = -1/2]
(%i8) z:s[1];
(%o8)                                  3/2
(%i9) abs(float(rectform(ev(d))));
(%o9)                                 3.0
(%i10) z:s[2];
(%o10)                                - 1/2
(%i11) abs(float(rectform(ev(d))));
(%o11)                                1.0

So z=-1/2 is a parabolic fixed point.

z^2 + m*zEdit

Complex quadratic polynomial of the form :

 f_m(z) = z^2 + m z  \,

which has an indifferent fixed point[3] with multiplier[4]

\lambda = m = e^{2 \pi t i} \,

at the origin[5][6]

belongs to the class of the functions :

 z^n + m z  \,

How to compute iterationEdit

In Maxima CAS :

(%i1) z:zx+zy*%i;
(%o1) %i*zy+zx
(%i2) m:mx+my*%i;
(%o2) %i*my+mx
(%i3) z1:z^2+m*z;
(%o3) (%i*zy+zx)^2+(%i*my+mx)*(%i*zy+zx)
(%i4) realpart(z1);
(%o4) -zy^2-my*zy+zx^2+mx*zx
(%i5) imagpart(z1);
(%o5) 2*zx*zy+mx*zy+my*zx

Critical pointEdit

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

f_m'(z_{cr}) = 0. \,

Since

f_m'(z) = \frac{d}{dz}f_m(z) = 2z + m

implies

 z_{cr} = -\frac{m}{2}\,

One can see that :

  • critical point is related with m value and have to be computed for every m parameters

z_{cr} is an initial point for Mandelbrot set iteration.

Parameter plane lambda

parameter planeEdit

period 1 componentsEdit

(%i1) e1:z^2+m*z=z;
(%o1) z^2+m*z=z
(%i2) e2:2*z+m=w;
(%o2) 2*z+m=w
(%i3)  s:eliminate ([e1,e2], [z]);
(%o3) [-(m-w)*(w+m-2)]
(%i4) s:solve([s[1]], [m]);
(%o4) [m=2-w,m=w]

It means that there are 2 components of period 1 :

  • one with radius=1 and center=0 ( m=w )
  • second with radius=1 and center= -2 ( m=2-w)


How to compute boundary points of first component :

(%i1) m:exp(2*%pi*%i*p/q);
(%o1) %e^((2*%i*%pi*p)/q)
(%i2) realpart(m);
(%o2) cos((2*%pi*p)/q)
(%i3) imagpart(m);
(%o3) sin((2*%pi*p)/q)

Dynamic planeEdit

1/1Edit

Julia set for f(z) = z^2+z or f(z) = z^2 + 1/4 or f(z)= z-z^2

First compute parameter of the function :

p:1$
q:1$
m:exp(2*%pi*%i*p/q);

The result is m = 1 so f(z) = z^2 + z

Compute fixed points :

(%i3) solve(z=z^2+z);
(%o3) [z=0]
(%i4) multiplicities;
(%o4) [2]


Find it's stability index ( abs(multiplier)) :

(%i1) f:z^2+z;
(%o1) z^2+z
(%i2) d:diff(f,z,1);
(%o2) 2*z+1
(%i7) z:0;
(%o7) 0
(%i8) abs(float(rectform(ev(d))));
(%o8) 1.0


Iteration :

f(z):= z^2+z;

fn(n, z) :=
  if n=0 then z
  elseif n=1 then f(z)
  else f(fn(n-1, z));

1/3Edit

Critical orbit for f(z)=z^2 + mz where p over q=1 over 3

First compute parameter of the function :

/* Maxima CAS session */
(%i1) p:1;
      q:3;
      m:exp(2*%pi*%i*p/q);
(%o1) 1
(%o2) 3
(%o3) (sqrt(3)*%i)/2-1/2
(%i9) float(rectform(m));
(%o9) 0.86602540378444*%i-0.5


Then find fixed points :

/* Maxima CAS session */
(%i10) f:z^2+m*z;
(%o10) z^2+((sqrt(3)*%i)/2-1/2)*z
(%i11) z1:f;
(%o11) z^2+((sqrt(3)*%i)/2-1/2)*z
(%i12) solve(z1=z);
(%o12) [z=-(sqrt(3)*%i-3)/2,z=0]
(%i13) multiplicities;
(%o13) [1,1]

Compute multiplier of the fixed point :

(%i23) d:diff(f,z,1);
(%o23) 2*z+(sqrt(3)*%i)/2-1/2

Check stability of fixed points :

(%i12) s:solve(z1=z);
(%o12) [z=-(sqrt(3)*%i-3)/2,z=0]
(%i20) s:map(rectform,s);
(%o20) [3/2-(sqrt(3)*%i)/2,0]
(%i21) s:map('float,s);
(%o21) [1.5-0.86602540378444*%i,0.0]
(%i24) z:s[1];
(%o24) 1.5-0.86602540378444*%i;
(%i31) abs(float(rectform(ev(d))));
(%o31) 2.645751311064591

It means that fixed point z=1.5-0.86602540378444*%i is repelling.

Second point z=0 is parabolic :

(%i33) z:s[2];
(%o33) 0.0
(%i34) abs(float(rectform(ev(d))));
(%o34) 1.0

Find critical point :

(%i1) solve(2*z+(sqrt(3)*%i)/2-1/2);
(%o1) [z=-(sqrt(3)*%i-1)/4]
(%i2) s:solve(2*z+(sqrt(3)*%i)/2-1/2);
(%o2) [z=-(sqrt(3)*%i-1)/4]
(%i3) s:map(rhs,s);
(%o3) [-(sqrt(3)*%i-1)/4]
(%i4) s:map(rectform,s);
(%o4) [1/4-(sqrt(3)*%i)/4]
(%i5) s:map('float,s);
(%o5) [0.25-0.43301270189222*%i]
(%i6) abs(s[1]);
(%o6) 0.5

1/2Edit

First compute m = muliplier of the fixed points = parameter of the function f using internal angle p/q and Maxima CAS:

\frac{p}{q} = \frac{1}{2}

(%i1) p:1$
(%i2) q:2$
(%i3) m:exp(2*%pi*%i*p/q);
(%o3)                                 - 1

so function f is :

f_m = z^2 +mz = z^2 -z


How to compute iteration z_{n+1} = f_m(z_n)  ?

(%i29) z1;
(%o29)                              z^2  - z
(%i30) z:zx+zy*%i;
(%o30)                            %i zy + zx
(%i32) realpart(ev(z1));
(%o32)                         - zy^2  + zx^2  - zx
(%i33) imagpart(ev(z1));
(%o33)                           2 zx zy - zy

Then find fixed points of f :

 z_f : \{ z : f_m(z) = z \}

(%i4) z1:z^2+m*z;
(%o4)                               z^2  - z
(%i5) zf:solve(z1=z);                                                  
(%o5)                           [z = 0, z = 2]
(%i6) multiplicities;
(%o6)                               [1, 1]

Stability of the fixed points :

(%i7) f:z1;
(%o7)                               z^2  - z
(%i8) d:diff(f,z,1);
(%o8)                               2 z - 1
(%i9) z:zf[1];   
(%o9)                                z = 0
(%i10) abs(ev(d));
(%o10)                         abs(2 z - 1) = 1
(%i11) z:zf[2];
(%o11)                               z = 2
(%i12) abs(ev(d));
(%o12)                         abs(2 z - 1) = 3
(%i13) 

So fixed point :

  • z=0 is parabolic ( stability index = 1)
  • z=2 is repelling ( stability indexs = 3 , greater then 1 )


Find critical point z_{cr} :

(%i14) zcr:solve(d=0);
(%o14)                              [z = 1/2]
(%i15) multiplicities;
(%o15)                                [1]

Attracting vectors

Because q=2, thus we examine 2-th iteration of f :

(%i16) z1;
(%o16)                              z^2  - z
(%i17) z2:z1^2-z1;
(%o17)                        (z^2  - z)^2  - z^2  + z
(%i18) taylor(z2,z,0,20);
taylor: z = 2 cannot be a variable.
 -- an error. To debug this try: debugmode(true);
(%i19) remvalue(z);
(%o19)                                [z]
(%i20) z;
(%o20)                                 z
(%i21) taylor(z2,z,0,20);
(%o21)/T/                    z - 2 z^3  + z^4  + . . .

Next term after z is a :

-2z^3

so here :

  • degree of above term is k=3
  • number of attracting directions ( and petals) is n= k-1 = 2 ( also n = e*q)
  • the parabolic degeneracy e = n/q = 1
  • cooefficient of above term a = -2

Attracing vectore satisfy :

nav^n = -1

so here :

-4v^2 = -1

v^2 = \frac{1}{4}

One can solve it in Maxima CAS :

(%i22)  s:solve(z^2=1/4);
(%o22)                         [z = - 1/2, z =1/2]
(%i23) s:map(rhs,s);
(%o23)                             [-1/2, 1/2]
(%i24) carg_t(z):=
 block(
 [t],
 t:carg(z)/(2*%pi),  /* now in turns */
 if t<0 then t:t+1, /* map from (-1/2,1/2] to [0, 1) */
 return(t)
)$
(%i25)  s:map(carg_t,s);
(%o25)                              [1/2, 0]

So attracting vectors are :

  • V_{a1} = \overrightarrow{v_{a1}0} from z=-\frac{1}{2} to the origin
  • V_{a2} = \overrightarrow{v_{a2}0} from z=\frac{1}{2} to the origin


Critical point z=1/2 lie on attracting vector V_{a1} . Thus critical orbits tend straight to the origin under the iteration[7]

Repelling vectors satisfy :


nav^n = 1

so here :

-4v^2 = 1

v^2 = - \frac{1}{4}

One can solve it in Maxima CAS :

(%i26) s:solve(z^2=-1/4);
(%o26)                        [z = - %i/2, z = %i/2]
(%i27) s:map(rhs,s);
(%o27)                            [- %i/2, %i/2 ]
(%i28) s:map(carg_t,s);
(%o28)                              [3/4, 1/4]

1/7Edit

How to speed up computations ? Approximate f^7 by :

f_a^7(z) = (245.4962434402444i-234.5808769813032)*z^8 + z

How to compute f_a^7 :

(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z7:(245.4962434402444*%i-234.5808769813032)*z^8 + z;
(%o2) (245.4962434402444*%i-234.5808769813032)*(%i*y+x)^8+%i*y+x
(%i3) realpart(z7);
(%o3) -234.5808769813032*(y^8-28*x^2*y^6+70*x^4*y^4-28*x^6*y^2+x^8)-245.4962434402444*(-8*x*y^7+56*x^3*y^5-56*x^5*y^3+8*x^7*y)+x
(%i4) imagpart(z7);
(%o4) 245.4962434402444*(y^8-28*x^2*y^6+70*x^4*y^4-28*x^6*y^2+x^8)-234.5808769813032*(-8*x*y^7+56*x^3*y^5-56*x^5*y^3+8*x^7*y)+y

m*z*(1-z)Edit

Description

Critical pointsEdit

critical points :

  • z = 1/2
  • z = ∞

Parameter planeEdit

period 1 componentsEdit

(%i1) e1:m*z*(1-z)=z;
(%o1) m*(1-z)*z=z
(%i2) d:diff(m*z*(1-z),z,1);
(%o2) m*(1-z)-m*z
(%i3) e2:d=w;
(%o3) m*(1-z)-m*z=w
(%i4) s:eliminate ([e1,e2], [z]);
(%o4) [m*(m-w)*(w+m-2)]
(%i5) s:solve([s[1]], [m]);
(%o5) [m=2-w,m=w,m=0]

It means that there are 2 period 1 components :

  • discs of radius 1 and centre in 0
  • disc of radius 1 and centre = 2

z(1+ mz)Edit

dynamic planeEdit

z-z^2Edit

Description [9]


First compute m = muliplier of the fixed points = parameter of the function f using internal angle p/q and Maxima CAS:

\frac{p}{q} = \frac{1}{2}

(%i1) p:1$
(%i2) q:2$
(%i3) m:exp(2*%pi*%i*p/q);
(%o3)                                 - 1

so function f is :

f_m = z(1+mz) = z-z^2


How to compute iteration z_{n+1} = f_m(z_n)  ?

Find it using Maxima CAS :

(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z1:z-z^2;
(%o2) −(%i*y+x)^2+%i*y+x
(%i3) realpart(z1);
(%o3) y^2−x^2+x
(%i4) imagpart(z1);
(%o4) y−2*x*y

Then find fixed points of f :

 z_f : \{ z : f_m(z) = z \}

(%i6) remvalue(z);
(%o6) [z]
(%i7) zf:solve(z-z^2=z);
(%o7) [z=0]
(%i9) multiplicities;
(%o9) [2]
<pre>


Stability of the fixed points :

<pre>
(%i11) f:z-z^2;
(%o11) z−z^2
(%i12) d:diff(f,z,1);
(%o12) 1−2*z
(%i13) zf:solve(z-z^2=z);
(%o13) [z=0]
(%i14) z:zf[1];
(%o14) z=0
(%i15) abs(ev(d));
(%o15) abs(2*z−1)=1

It means that fixed point z=0 is a parabolic point ( stability indeks = 1 ).


Find critical point z_{cr} :

(%i16) zcr:solve(d=0);
(%o16) [z=1/2]

ReferencesEdit

  1. wikipedia : Complex quadratic polynomial
  2. Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  3. wikipedia : fixed point
  4. wikipedia : multiplier
  5. wikipedia : origin
  6. Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  7. Mark McClure in stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
  8. lambda map at Mu-Ency
  9. S Lapan : On the existence of attracting domains for maps tangent to the identity. Ph.D. Thesis, University of Michigan