• August 14, 2022, 09:41:13 PM

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

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#### marcm200

• 3f
• Posts: 1150
##### 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).

• Fractal Freak
• Posts: 695
##### 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

#### marcm200

• 3f
• Posts: 1150
##### 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*2071/3072z0  = e^2*i*pi*711/1024

• Fractal Freak
• Posts: 695
##### 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*2071/3072z0  = 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

#### marcm200

• 3f
• Posts: 1150
##### 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)-0.34266071731199431 )  (-0.9394592236021899*%i)-0.3426607173119943  ||=  0.99999999999999992 )  0.2747576642246148*%i-0.9615135079393496  ||=  1.03 )  0.6267772150216261*%i+0.779198513043842  ||=  1.0roll=  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.

#### xenodreambuie

• Fractal Furball
• Posts: 208
##### 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 »

• Fractal Freak
• Posts: 695
##### 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

#### claude

• 3f
• Posts: 2260
##### 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

• Fractal Freak
• Posts: 695
##### 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 ?

#### xenodreambuie

• Fractal Furball
• Posts: 208
##### 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+swABLFKrSoVyxf0wUx3z+nbwP94ernZbhZf2u2u26znduaVOj25btb37ephc9/OKb182LfbZo8Vu7lTOUyw67nxwP7eLZYXTbeaazVToubj5pmymsXgTk8+rLGj22xv291cB7z//K1ZtfMw09l677VzKlmjsfK3zeZh/k7NbHZYdtnKxn/N9czaHJT1WB+sC84e3v57TpuTX6cnV6vuEYfbdzAlZ/nStc/n6wXMlvVwYrl5vm0XPJs+ww9U3ray+lWFlx8emsV/M7bZPjytmg+/zPPpyU2zbR40GB2vIGhwkj+Bdm7+DLr5u1f2+yPzFQyvKY2v7k+vLc2vKdVqrp2a+ZwG/GJ9f7fsvv1Yt7vdvMYZ2PnqcdnMwd37l/0v7/uHdr3r9j+xzMZkZSXAT9+/79q9YM6riv26bR6HzZdP29/buaV0s9mvilwW3rUv+6dtO/haFjV7jQzsn01de74CPDz26Ocf7bM+3i+QOYautu1/bqtH8vxxs15U8bp7GXIH4sWqXd+PAXjwa/P7xEVB7n4+9sfAmpvNc7vtFx2B1919XXaJ5B2rlmQeGGA99Qy7wRjAyrCeeRN6rGdY9ryvmwdi8VzVHg5X4MHRd/oVtHoqJr/tccLdQbqVzlBTesA2T4tlzRvf48f8AfjSbDu2lXGUgEuUxutuN/tm37svYZc41cwesPOH/XgPoCGaaay8f9Hsh9rocTSXtZ4fiaaKn5fd933/7nxR3blodt36anV/3ex+DLLwewRw7+Db5c9vq7aHnNLSl9pDgPD8t21zLwHE43XbrH+T7ltlecX4lpV9TvRUABpqriBXXbu6/9itS8Dg6j+fHr5CnaNws2x2bX1x99zt9qPOUMBR/Rt/gLeNGJYxomb5gFd/jrkd4RIMNX7RnwWPpbm3PLm4VvivL2vQ6rujoht7er25n+JHfFRsHBAgd0/br8ieC3IxASddaMDJ6zE6TiCRh/Rz41WT3CtgX1NmEI+LqgelqoaKki7RIZYlsvL0GXE+CA+bzX5JxkREX7xij6nydbtdtH1XLxuWbbMtflXkKL1GeobqOz35dPd+3XxdtewOn+6u6z3i0x3yDFnyuBMHMDQkcTtEGBeGM3NGs2c8CJ+nf2k+8Ue/dakOMUaj/RmZGezlMx1jb5CLQkheAB0JeAy8SGVGE7DaWesIhAlgbQFiMCEL4D0BrU0ytBK4wmdtbKDSWHRolZzXAiT64X3M3lJHchMgGwIJk9TQ0xyrDgMzALTS/QpFpVoVP5xJmZ5qlQkoiz+iQ2taCdCQUyCQKhBCpA5TzLqMKxp1mDgBLM066ApCHgA3BajUZJO8oxVnpwCVCmW6uO6p1ESrtOcKT8bgllWMnA6q12GLlUCzKnpo4eECzaKXGuwRIIrZBK0uJcqRMi4mgUaFgipzfRK6UvDRSQDlljqRi9fZJseQGGWmQIlzdCZI2syMrvlmo69AiZFKmlQDYIw8IOUk8MbYKUCzTm6/jlYsrTgbUqTjxhVurPE+aQI1m1xMmVYcubH4NyqJovGmemqR+wRCWRFwTMlZExgSm1OMmjoCXUc+Zu/KCrrukouBhIPoKUCz8KJSyMw9BkLxFPVk6XouWR3xm9VlVeHDZs2AzawuK6yJjBAAOuaDRxKLY9aU6sJJAnMW554CqVhJOSZacbrqyCVZrCtKoxG1AvhSf8oHI4mAK24tJg+fhSBbSjboFPlu5nQBnMue+eFc6LdoicjM+XRYIUppHEBw3gZuqUq9hEYRiLWFob949kRdzTp8KhEoOWaTQpkLULPQuKzkFABsPZwKBahNDm1Re+qwagpUCjUagCOQJ0Cpcid8SA4DiBOgJB0CGxxr2JekEyAbCYMvVY4mh1+aQG2lAVnJCRBqNYRkAg8Xc8lTk1WklUIhqhpdiytyLVOHOFEpKUTVglYJXOBUwASwic0nSIkcyaIxoq8Etq9A+say6Is4pZOEnwXOkLEsxzQKPZe5FUguekQoUy0UbtHGXaQ/bKAGxFX7lVlnFA8U2C0xjTDq+J4DyKqE8reUhRKPWaMs9ZHEsSz+Iq+tLecXq0ey+BstEomx56jEeQzmGO2xUWIVWhPtsVGG7FNgTwvk12c4xDYQ2ScD6kazgUUmKFhWigUcOacCMttyTEWOKY8aUJx0kVMKJQT3xX7kkBrL4i+4M5FJE6UMzhDtZDjkI1PViUNsOpHtEZlrAws3sjtifeCbWWSeYhZjugjfkVkJll3K1M9JI5/IaPMil0EzkjlofMbcpX3ylxzygM03ZlUGj0kcujGLP1nY1kUW/dl4dDbJj8SOh+PkwJRP5CcnzGRhDnI+lsvE0NKF2VSTrRXgo4pcUUaIYUjFxVRGiJb/Z5FoAWBOYuBigFoC1UgsSZk8k1I5i45JI2yQyWY0JjllEi6O5MAscTBOhRwfI5ksCsva8tSpZA14Y9WkMq4xB1x5z0mBo3nWD2SyjNsf+y5kjvOMsU/9mYMkYfRkRjWzqsGqzqSATVzep+i5XpfrQY6abTUbNZHZJVDUmImU07FcSAfB1jNs2cYJUIYOkjiYCpQRi4JNxQfPsFj5Dxl2/+zLiMVEK9exXCYIekVkWs0yaUc/xJx0BaBZ5FrMllbK4LIRvBdPaws9AMkWxyImZCAQJ0AZ07iuGesLEMaA7q/auAAbNN4zfjKP7//M9Dfe6pEUntMVF1b9lg1aBfQChtnUmfhXO6Afs4Uj94020IZVobi07bd8nODOy2T0b7RhcX9TQj6G0JtsWFzHeQ2dldn2lxswVag/ag5gOyzSr+u3pafncm/4P366oSpQfO6MH+f/45fi8Pn3ZqsK9j52i+W+Wy9GH7F/APZtI50=}{Title: Blaschke macrame by Garth Thornton}

#### xenodreambuie

• Fractal Furball
• Posts: 208
##### 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).

• Fractal Freak
• Posts: 695
##### 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

• Fractal Freak
• Posts: 695
##### 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

#### xenodreambuie

• Fractal Furball
• Posts: 208
##### 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;}

#### xenodreambuie

• Fractal Furball
• Posts: 208
##### 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.

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