• August 14, 2022, 09:41:13 PM

Author Topic:  can you identify this fractal ?  (Read 773 times)

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Offline marcm200

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Re: can you identify this fractal ?
« Reply #15 on: June 12, 2022, 10:53:54 AM »
I can't compute multiplier in Maxima ( long expression)

You can use the numerical rolling derivative along the orbit, {f[3](z0)}' = f'(z0)*f'(z1)*f'(z2) with z1=f(z0), z2=f(z1), instead of a very long explicit expression for period-3 (I received "stack overflow" errors with maxima there myself).

Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #16 on: June 12, 2022, 03:38:47 PM »
derivative of iterated function]
is the example of problems with translation of math equation to code



Maxima CAS code
Code: [Select]

kill(all);
display2d:false;
ratprint : false; /* remove "rat :replaced " */



rho : 0.7883464276266063*%i - 0.6152315905806267;

f(z):= rho * z^2 * (z-3)/(1-3*z);

define(dz(z), diff(f(z),z,1));

GiveStability(z0, p):=block(
[z,d],
/* initial values */
d :1,
z:z0,
for i: 1 thru p step 1 do (

d : float(rectform(dz(z)*d)),
z: float(rectform(f(z)))
),

return (abs(d))





)$




period:3;



/* period 3 points */
e: f(f(f(z))) = z$

load (to_poly_solve);
s: to_poly_solve (e,z);

s: args(s); /*  https://stackoverflow.com/questions/12834709/create-a-union-into-a-list-in-maxima */
s:flatten(s);
s:map(rhs,s);

r:[];
for z in s do if (abs(abs(z) -1) < 0.001) then  r:cons(z,r);



for z in r do print("z= ",z, "Stability = ", GiveStability(z,period));

result
Code: [Select]
z=  0.7341202224500978*%i-0.6790195129669069 Stability =  0.7847847465597322
z=  0.7109880439912231*%i+0.7032040964766471 Stability =  1.207052759700951
z=  0.6575907313229205*%i-0.7533753580242649 Stability =  1.207052759700943
z=  0.5478342670335127*%i+0.8365868848265807 Stability =  0.7847847465597244
z=  (-0.8495833981750556*%i)-0.5274543103865237 Stability = 
   0.7847847465597284
z=  (-0.9203524011916424*%i)-0.391090088880784 Stability =  1.20705275970095


non is strictly parabolic

Offline marcm200

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Re: can you identify this fractal ?
« Reply #17 on: June 12, 2022, 04:32:53 PM »
Next brute-force output:
Code: [Select]
rho = e^2*i*pi*2071/3072
z0  = e^2*i*pi*711/1024





Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #18 on: June 12, 2022, 05:59:26 PM »
Next brute-force output:
Code: [Select]
rho = e^2*i*pi*2071/3072
z0  = e^2*i*pi*711/1024
for rho : float(rectform(%e^(2*%i*%pi*2071/3072)));
Code: [Select]
[0,4.202617505905039-2.560021705859437*%i,
        4.260423514245568-1.47963830121871*%i,
        5.069206415503641-1.217944363855348*%i,
        (-0.9541132419824905*%i)-0.3460783579385921,
        (-0.9262315347119798*%i)-0.3359650348611799,
        3.130488176173231-0.2484425269695521*%i,
        3.162112638273884-0.2132260533913495*%i,
        3.038157478131937-0.1908605614151723*%i,
        0.1735492524275814-0.1057174136421102*%i,
        0.2094548093767481-0.0727433217106854*%i,
        0.1865033573407289-0.04480991585218228*%i,
        0.3174396455772014-0.02519271843450431*%i,
        0.3148128123005833-0.02122830575264191*%i,
        0.3278529908989091-0.02059610355973053*%i,
        0.03598354818471924*%i+0.3140805599617254,
        0.07546684462895094*%i+0.2117423954858107,
        0.243183694860436*%i-0.2205383359164248,
        0.267088302846746*%i-0.9433019080743703,
        0.2778827553786301*%i-0.9814257328975492,
        0.3118955108215367*%i-0.3327227650045798,
        0.3600464190403506*%i+3.142646754240181,
        0.6112317102899262*%i+0.7542980172749191,
        0.648473750694549*%i+0.8002570157424302,
        1.493503169711316*%i+4.190422169838659,
        1.49961842497606*%i-1.599757519739826,
        2.256391936922672*%i-2.046275854266266]


non of these point in on unit circle

Offline marcm200

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Re: can you identify this fractal ?
« Reply #19 on: June 12, 2022, 06:47:05 PM »
I don't understand the length and starting value of your output. z0 is by definition on the unit circle, and I get a parabolic 3-cycle using:

Code: [Select]
kill(all);
numer:false;
display2d:false;

rho:float(rectform( exp(2*%i*%pi*2071/3072) ));
z0:float(rectform( exp( %i*2*%pi* 711/1024 ) ));

p(z) := rho * z^2 * (z-3)/(1-3*z)$

roll:1$
dd:diff(p(z),z,1)$
pp:float(z0)$
for i from 1 thru 3 do (
print(i,") ",pp," ||= ",cabs(pp)),
w:rectform(subst(pp,z,dd)),
roll:rectform(roll*w),
pp:rectform(p(pp))
)$
print("roll= ",roll);
mult:cabs(roll)$
print("|mult|= ",mult);

with output

Code: [Select]
(%i3) rho:float(rectform(exp((2*%i*%pi*2071)/3072)))
(%o3) (-0.8885797065021895*%i)-0.4587222527766477
(%i4) z0:float(rectform(exp((%i*2*%pi*711)/1024)))
(%o4) (-0.9394592236021899*%i)-0.3426607173119943

1 )  (-0.9394592236021899*%i)-0.3426607173119943  ||=  0.9999999999999999
2 )  0.2747576642246148*%i-0.9615135079393496  ||=  1.0
3 )  0.6267772150216261*%i+0.779198513043842  ||=  1.0

roll=  4.467721079608467E-4*%i+0.9926380240606019

|mult|=  0.9926381246034491

But a quick look at the image shows no resemblance to the one in the article. So I either get a prabolic 3-cycle, or one that looks like the one published, but not both. So for me I consider the problem unsolveable at the moment.

Offline xenodreambuie

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Re: can you identify this fractal ?
« Reply #20 on: June 13, 2022, 10:18:53 AM »
This is a nice formula, that Jux can already do with its Rational 1 formula. The basin at 0 is super-attracting. The basin at 1 can have a wide range of periods depending on rho. Here's one that shows the unit circle well, with rho= 140.598 degrees, period 351.
« Last Edit: June 14, 2022, 05:18:06 AM by xenodreambuie »

Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #21 on: June 13, 2022, 04:03:15 PM »
I have tried to draw/ comput eMandelbrot set . 2 critical points. Mandelbrot set is when trajectories of all critical pointa sre bounded

Code: [Select]

unsigned char ComputeColorOfMandel(complex double rho)
{


int interior = 0;
unsigned  char iColor ;
int i; // number of iteration
complex double z;
 
 
  z = 0.0; // first critical point
 
  for (i = 0; i < IterMax; ++i)
    {



   
       if ( cabs(z) > ER ){ interior-- ; break; } //
      // 
      if ( cabs2(z) < AR2 ){ interior++; break; } // basin of zf = 0
     
     

     
     
      z = f(z); //  iteration: z(n+1) = f(zn)


    }
   
    if ( i == IterMax) {interior++;} // second basin
 
  z = 1.0; // second critical point
 
  for (i = 0; i < IterMax; ++i)
    {



   
       if ( cabs(z) > ER ){ interior-- ; break; } //
      // 
      if ( cabs2(z) < AR2 ){ interior++; break; } // basin of zf = 0
     
     

     
     
      z = f(z); //  iteration: z(n+1) = f(zn)


    }

   if ( i == IterMax) {interior++;}
   
   if (interior==2)
    { iColor = iColorOfInterior;}
    else { iColor = iColorOfExterior;}
 
 
  return iColor;


}


Not works : all points have the same color

Offline claude

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Re: can you identify this fractal ?
« Reply #22 on: June 13, 2022, 04:13:22 PM »
maybe you want cabs(z - critical point) < period detection radius? instead of just cabs(z) ?  not sure, haven't tried

Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #23 on: June 13, 2022, 05:53:53 PM »
This is a nice formula, that Jux can already do with its Rational 1 formula. The basin at 0 is super-attracting. The basin at 1 can have a wide range of periods depending on rho. Here's one that shows the unit circle well, with rho= 240.598 degrees, period 351.
How to set it to give rho * z^2 * (z-3)/(1-3z)
?
Maybe you can give parameter file ?

Offline xenodreambuie

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Re: can you identify this fractal ?
« Reply #24 on: June 13, 2022, 09:18:25 PM »
How to set it to give rho * z^2 * (z-3)/(1-3z)
?
Maybe you can give parameter file ?
Just express it as (z^3-3z^2)/(1-3z), with Numerator Angle set to rho...

Code: [Select]
JuxV3.500{
eJytWNty28gRfVeV/kE/YNbcLw98kLRW4pQVq2SVN9k32IJJlClSIamV/Pd7+swABLFKrSoVyxf0
wUx3z+nbwP94ernZbhZf2u2u26znduaVOj25btb37ephc9/OKb182LfbZo8Vu7lTOUyw67nxwP7e
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bHZYdtnKxn/N9czaHJT1WB+sC84e3v57TpuTX6cnV6vuEYfbdzAlZ/nStc/n6wXMlvVwYrl5vm0X
PJs+ww9U3ray+lWFlx8emsV/M7bZPjytmg+/zPPpyU2zbR40GB2vIGhwkj+Bdm7+DLr5u1f2+yPz
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zMZkZSXAT9+/79q9YM6riv26bR6HzZdP29/buaV0s9mvilwW3rUv+6dtO/haFjV7jQzsn01de74C
PDz26Ocf7bM+3i+QOYautu1/bqtH8vxxs15U8bp7GXIH4sWqXd+PAXjwa/P7xEVB7n4+9sfAmpvN
c7vtFx2B1919XXaJ5B2rlmQeGGA99Qy7wRjAyrCeeRN6rGdY9ryvmwdi8VzVHg5X4MHRd/oVtHoq
Jr/tccLdQbqVzlBTesA2T4tlzRvf48f8AfjSbDu2lXGUgEuUxutuN/tm37svYZc41cwesPOH/XgP
oCGaaay8f9Hsh9rocTSXtZ4fiaaKn5fd933/7nxR3blodt36anV/3ex+DLLwewRw7+Db5c9vq7aH
nNLSl9pDgPD8t21zLwHE43XbrH+T7ltlecX4lpV9TvRUABpqriBXXbu6/9itS8Dg6j+fHr5CnaNw
s2x2bX1x99zt9qPOUMBR/Rt/gLeNGJYxomb5gFd/jrkd4RIMNX7RnwWPpbm3PLm4VvivL2vQ6ruj
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ki7RIZYlsvL0GXE+CA+bzX5JxkREX7xij6nydbtdtH1XLxuWbbMtflXkKL1GeobqOz35dPd+3Xxd
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ZqsK9j52i+W+Wy9GH7F/APZtI50=}
{Title: Blaschke macrame by Garth Thornton}

Offline xenodreambuie

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Re: can you identify this fractal ?
« Reply #25 on: June 13, 2022, 09:51:29 PM »
I have tried to draw/ compute Mandelbrot set . 2 critical points. Mandelbrot set is when trajectories of all critical pointa sre bounded

Not works : all points have the same color

The first loop exits immediately as interior because z=0 before you start iterating. Move the tests after z=f(z).

Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #26 on: June 14, 2022, 03:13:58 AM »
The first loop exits immediately as interior because z=0 before you start iterating. Move the tests after z=f(z).

good point.
Code: [Select]
unsigned char ComputeColorOfMandel(complex double rho)
{


int interior = 0;
unsigned  char iColor ;
int i; // number of iteration
complex double z;
 
  // The first loop exits immediately as interior because z=0 before you start iterating. Move the tests after z=f(z).
 /*
  z = 0.0; // first critical point
 
  for (i = 0; i < IterMax; ++i)
    {



   
       if ( cabs(z) > ER ){  break; } // escaping basin
      // 
      if ( cabs(z) < AR ){ interior++; break; } // basin of zf = 0
     
     

     
     
      z = f(z); //  iteration: z(n+1) = f(zn)


    }
   
    if ( i == IterMax) {interior++;} // basin of 1.0
  */
 
  z = 1.0; // second critical point
 
  for (i = 0; i < IterMax; ++i)
    {



   
       if ( cabs(z) > ER ){  break; } // escaping basin
      // 
      // 
      if ( cabs(z) < AR ){ interior++; break; } // basin of zf = 0
     
     

     
     
      z = f(z); //  iteration: z(n+1) = f(zn)


    }

   if ( i == IterMax) {interior++;}
   
  iColor = ((2+interior)*53) % 255;
 
  return iColor;


}

still all the same color

Offline Adam Majewski

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Re: can you identify this fractal ?
« Reply #27 on: June 14, 2022, 03:23:05 AM »
Just express it as (z^3-3z^2)/(1-3z), with Numerator Angle set to rho...

Thx. Great.
So Jux shows that this Mandelbrot set for this function

Offline xenodreambuie

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Re: can you identify this fractal ?
« Reply #28 on: June 14, 2022, 05:00:23 AM »
I think you forgot to move the tests - should be like this:

Code: [Select]
unsigned char ComputeColorOfMandel(complex double rho)
{


int interior = 0;
unsigned  char iColor ;
int i; // number of iteration
complex double z;
 
 /*
  z = 0.0; // first critical point
 
  for (i = 0; i < IterMax; ++i)
    {

   
      z = f(z); //  iteration: z(n+1) = f(zn)

   
      if ( cabs(z) > ER ){  break; } // escaping basin
      // 
      if ( cabs(z) < AR ){ interior++; break; } // basin of zf = 0
     


    }
   
    if ( i == IterMax) {interior++;} // basin of 1.0
  */
 
  z = 1.0; // second critical point
 
  for (i = 0; i < IterMax; ++i)
    {


      z = f(z); //  iteration: z(n+1) = f(zn)

   
       if ( cabs(z) > ER ){  break; } // escaping basin
      // 
      // 
      if ( cabs(z) < AR ){ interior++; break; } // basin of zf = 0
     
     

    }

   if ( i == IterMax) {interior++;}
   
  iColor = ((2+interior)*53) % 255;
 
  return iColor;


}


Offline xenodreambuie

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Re: can you identify this fractal ?
« Reply #29 on: June 14, 2022, 05:17:08 AM »
Thx. Great.
So Jux shows that this Mandelbrot set for this function

Yes, that's with rho 140.598 (I mistyped it as 240.598 earlier); the function you're interested in has rho near 120 degrees, and in the Mandelbrot panel you can choose which critical point to use and see both Mandelbrot sets.