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Three-Coupled Oscillators


Reptillian
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Bifurcation diagram for two-parameters of torus-map or three-coupled oscillators.

 

Preview:

 

unknown.png

 

Download Plugin -> ThreeCoupledOscillators.zip

 

After install, it's under Effect->Texture.

 

------------------------

License : CeCiLL v2.0 - https://cecill.info/licences/Licence_CeCILL_V2-en.html

 

Codelab Source Code

Spoiler
// Name: Three-Coupled Oscillators
// Submenu: Texture
// Author: Reptorian
// Title: Three-Coupled Oscillators
// Version: 1
// Desc: Bifurcation diagram for two-parameters of torus-map or three-coupled oscillators
// Keywords: Texture, Oscillators
// URL: https://forums.getpaint.net/profile/85868-reptillian/
// Help:

/*

Source: 

[C. Baesens, J. Guckenheimer, S. Kim, R.S. MacKay
Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos 
Physica D: Nonlinear Phenomena, Volume 49, Issue 3, 1991, Pages 387-475, 
ISSN 0167-2789]

*/

#region UICode
IntSliderControl max_iter = 50; // [1,200] Maximum Iteration
DoubleSliderControl k_a_perc = 50; // [-100,100] K-A Tau (%)
DoubleSliderControl k_b_perc = -50; // [-100,100] K-B Tau (%)
AngleControl variable_angle = 45; // [-180,180] Angle
DoubleSliderControl axis_scale = 2; // [1,15] Axis-Scale
PanSliderControl point_xy = Pair.Create(0.000, 0.000); // Point
CheckboxControl antialiased_mode = true; // Use Anti-aliasing?
ListBoxControl color_order = 0; // {color_mode} Color Setting|0-1-2|0-2-1|1-0-2|1-2-0|2-0-1|2-1-0
CheckboxControl color_mode = true; // Color Mode
DoubleSliderControl mult = 1; // [0,10] Multiplier
DoubleSliderControl tau_add = 0; // [-100,100] Add by Tau (%)
#endregion

double tau = 2 * Math.PI;
double tc = .7 / (2 * Math.PI);

double rot_x(double a, double b, double sin_ang,double cos_ang){
    return a*cos_ang-b*sin_ang;
}

double rot_y(double a, double b, double sin_ang,double cos_ang){
    return a*sin_ang+b*cos_ang;
}

double oscil_a(double a,double b, double k_a){
    return a+k_a-tc*Math.Sin(tau*b);
}

double oscil_b(double a,double b, double k_b){
    return a+k_b-tc*Math.Sin(tau*b);
}

double norm2(double a,double b){
    return Math.Sqrt(a*a+b*b);
}

int[] col_order = new int[3];

void define_color_order(int r,int g,int b){
    col_order[0]=r;
    col_order[1]=g;
    col_order[2]=b;
}

void PreRender(Surface dst, Surface src)
{
    switch(color_order){
        case 0: define_color_order(0,1,2);
        break;
        case 1: define_color_order(0,2,1);
        break;
        case 2: define_color_order(1,0,2);
        break;
        case 3: define_color_order(1,2,0);
        break;
        case 4: define_color_order(2,0,1);
        break;
        case 5: define_color_order(2,1,0);
        break;
    }
}

void Render(Surface dst, Surface src, Rectangle rect)
{
    int ww = src.Width-1;
    int hh = src.Height-1;
    double d_ww=(double)(ww);
    double d_hh=(double)(hh);
    double sd = Math.Max(d_ww,d_hh)/Math.Min(d_ww,d_hh);
    double sx = ww>hh?sd:1;
    double sy = ww>hh?1:sd;
    double cx = d_ww/2;
    double cy = d_hh/2;
    double cxsx=cx/sx/axis_scale;
    double cysy=cy/sy/axis_scale;
    double ang = variable_angle/180*Math.PI;
    double sin_ang=Math.Sin(ang);
    double cos_ang=Math.Cos(ang);
    double k_a=k_a_perc/100*tau;
    double k_b=k_b_perc/100*tau;
    double ix,iy,txp,xp,yp,ev,disp;
    double shift_tau = tau_add/100*tau;
    int lim=antialiased_mode?2:1;
    byte r,g,b,avg;
    for (int y = rect.Top; y < rect.Bottom; y++)
    {
        if (IsCancelRequested) return;       
        
        for (int x = rect.Left; x < rect.Right; x++)
        {
            ev=0;
            double[] col = new double[3] {0,0,0};
            for (int m = 0 ; m < lim ; m++){
                disp=m*.5;
                iy=(y+disp-cy)/cysy;
                ix=(x+disp-cx)/cxsx;
                ix-=point_xy.First;
                iy-=point_xy.Second;
                xp=rot_x(ix,iy,sin_ang,cos_ang);
                yp=rot_y(ix,iy,sin_ang,cos_ang);
                for (int n = 0 ; n < max_iter ; n ++){
                    txp=xp;
                    xp=oscil_a(xp,yp,k_a);
                    yp=oscil_b(yp,txp,k_b);
                }
                ev=norm2(xp,yp)*mult+shift_tau;
                col[col_order[0]]+=(Math.Sin(ev)+1)/2;
                col[col_order[1]]+=(Math.Cos(ev)+1)/2;
                col[col_order[2]]+=(Math.Cos(ev+Math.PI/2)+1)/2;
            }
            col[col_order[0]]/=lim;
            col[col_order[1]]/=lim;
            col[col_order[2]]/=lim;
            r = (byte)((int)(Math.Round(col[0]*255)));
            g = (byte)((int)(Math.Round(col[1]*255)));
            b = (byte)((int)(Math.Round(col[2]*255)));
            if (color_mode){dst[x,y]=ColorBgra.FromBgr(b,g,r);}
            else{
                avg = (byte)((int)(3*((r+g+b)/3-85)));
                dst[x,y]=ColorBgra.FromBgr(avg,avg,avg);
            }       
        }
    }
}

 

G'MIC Source Code

Spoiler
#@cli rep_tco_anim: eq. to 'rep_three_coupled_oscillators_animation' : (+)
rep_tco_anim: rep_three_coupled_oscillators_animation $*
#@cli rep_three_coupled_oscillators_animation: _max_iter>0,-2pi<=_k_a_1<=2pi,-2pi<=_k_b_1<=2pi,-2pi<=_k_a_2<=2pi,-2pi<=_k_b_2<=2pi,_sublevel>=0,-360<=_ang_a<=360,-360<=_ang_b<=360,_ox_a,_oy_a,_ox_b,_oy_b,_axis_scale_a>0,_axis_scale_b>0
#@cli : Animation-based version of Bifurcation diagram for two-parameters of torus-map or three-coupled oscillators. [1]
#@cli :
#@cli : --- Citation ---
#@cli : 1) C. Baesens, J. Guckenheimer, S. Kim, R.S. MacKay
#@cli : Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos
#@cli : Physica D: Nonlinear Phenomena, Volume 49, Issue 3, 1991, Pages 387-475, ISSN 0167-2789
#@cli : ------
#@cli :
#@cli : _max_iter is the number of iterations within the bifurcation diagram.
#@cli : _k_a_1 is the a-variable that influence the output for first frame.
#@cli : _k_b_1 is the b-variable that influence the output for first frame.
#@cli : _k_a_2 is the a-variable that influence the output for last frame.
#@cli : _k_b_2 is the b-variable that influence the output for last frame.
#@cli : _sublevel defines the degree of subpixel processing. 0 means no subpixel sampling. Negative  number in here means it will not be resized (Useful for mapping before downsizing to original image size.).
#@cli : _ang_a defines the angle of function for first frame.
#@cli : _ang_b defines the angle of function for last frame.
#@cli : _ox_a defines the x-offset of function for first frame.
#@cli : _oy_a defines the y-offset of function for first fraime.
#@cli : _ox_b defines the x-offset of function for last frame.
#@cli : _oy_b defines the y-offset of function for last frame.
#@cli : _axis_scale_a defines the scale of each axis for first frame.
#@cli : _axis_scale_b defines the scale of each axis for last frame.
#@cli :
#@cli : Author: Reptorian.
#@cli : Default values: '_max_iter=50','_k_a_1=u(-2*pi,2*pi)','_k_b_1=u(-2*pi,2*pi)','_k_a_2=u(-2*pi,2*pi)','_k_b_2=u(-2*pi,2*pi)','_sublevel=1','_ang_a=0','_ang_b=0','_ox_a=0','_oy_a=0','_ox_b=0','_oy_b=1','_axis_scale_a=1','_axis_scale_b=1'
rep_three_coupled_oscillators_animation:
skip ${1=50},${2=},${3=},${4=},${5=},${6=1},${7=0},${8=180},{$9=0},${10=0},${11=0},${12=0},${13=1},${14=1}
check ${-max_d}>1
_rep_three_coupled_oscillators ${1-14}
#@cli rep_tco: eq. to 'rep_three_coupled_oscillators' : (+)
rep_tco: rep_three_coupled_oscillators $*
#@cli rep_three_coupled_oscillators: _max_iter>0,-2pi<=_k_a<=2pi,-2pi<=_k_b<=2pi,_sublevel>=0,-360<=_ang<=360,_ox,_oy,_axis_scale>0
#@cli : Bifurcation diagram for two-parameters of torus-map or three-coupled oscillators. [1]
#@cli :
#@cli : --- Citation ---
#@cli : 1) C. Baesens, J. Guckenheimer, S. Kim, R.S. MacKay
#@cli : Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos
#@cli : Physica D: Nonlinear Phenomena, Volume 49, Issue 3, 1991, Pages 387-475, ISSN 0167-2789
#@cli : ------
#@cli :
#@cli : _max_iter is the number of iterations within the bifurcation diagram.
#@cli : _k_a is a variable that influence the output.
#@cli : _k_b is a variable that influence the output.
#@cli : _sublevel defines the degree of subpixel processing. 0 means no subpixel sampling. Negative  number in here means it will not be resized (Useful for mapping before downsizing to original image size.).
#@cli : _ang defines the angle of function.
#@cli : _ox defines the x-offset of function.
#@cli : _oy defines the y-offset of function.
#@cli : _axis_scale defines the scale of each axis.
#@cli :
#@cli : Author: Reptorian.
#@cli : Default values: '_max_iter=50','_k_a=u(-2*pi,2*pi)','_k_b=u(-2*pi,2*pi)','_sublevel=1','_ang=0','_ox=0','_oy=0','_axis_scale=1'
rep_three_coupled_oscillators:
skip ${1=50},${2=},${3=},${4=1},${5=0},${6=0},${7=0},${8=1}
check ${-max_d}==1
_rep_three_coupled_oscillators ${1-8}
_rep_three_coupled_oscillators:
skip ${1=50},${2=},${3=},${4=},${5=}
check "abs($1)!=0&&$8!=0"
 
animation_mode=0
if ${-max_d}>1 animation_mode=1 if $!>1 error "One image only for animation mode!" fi fi

tau={pi*2}

if $animation_mode
 if narg($2) __k_a_1=$2
 else __k_a_1={u(-$tau,$tau)}
 fi
 
 if narg($3) __k_a_2=$3
 else __k_b_1={u(-$tau,$tau)}
 fi
 
 if narg($4) __k_b_1=$4
 else __k_a_2={u(-$tau,$tau)}
 fi
 
 if narg($5) __k_b_2=$5
 else __k_b_2={u(-$tau,$tau)}
 fi
 
 rescale_back={$6>0}
 
 __sub={abs($6)+1}
 m "generate_three_coupled: +_rep_three_coupled_oscillators_animation $1,$__k_a_1,$__k_b_1,$__k_a_2,$__k_b_2,$__sub,${7-14}"
 
else
 if narg($2) __k_a=$2
 else __k_a={u(-$tau,$tau)}
 fi

 if narg($3) __k_b=$3
 else __k_b={u(-$tau,$tau)}
 fi

 rescale_back={$4>0}
 __sub={abs($4)+1}
 
 m "generate_three_coupled: +_rep_three_coupled_oscillators_static_out $1,$__k_a,$__k_b,$__sub,${5-8}"
fi

repeat $! l[$>]

 generate_three_coupled
  
 if $rescale_back r. {w#0},{h#0},100%,100%,6 fi

 rm..
 
endl done

um generate_three_coupled
+_rep_three_coupled_oscillators_animation:
check "$13!=0&&$14!=0"

 {w*$6},{h*$6},{d},1,"*begin(
   const max_iter=abs($1);
   const ww=w-1;
   const hh=h-1;
   const dd=d-1;
   const sd=max(ww,hh)/min(ww,hh);
   const sx=w>h?sd:1;
   const sy=w>h?1:sd;
   const cx=ww/2;
   const cy=hh/2;
   const ox_a=($9+1)*cx;
   const oy_a=($10+1)*cx;
   const ox_b=(($11*-1)+1)*cy;
   const oy_b=(($12*-1)+1)*cy;
   const axis_scale_a=$13;
   const axis_scale_b=$14;
   const cxsx=cx/sx;
   const cysy=cy/sy;
   const cxsx_a=cxsx/axis_scale_a;
   const cxsx_b=cxsx/axis_scale_b;
   const cysy_a=cysy/axis_scale_a;
   const cysy_b=cysy/axis_scale_b;
   const ang_a=-$7;
   const ang_b=-$8;
   set_cos_ang=vectord();
   set_sin_ang=set_cos_ang;
   fill(set_cos_ang,p,cos(deg2rad(lerp(ang_a,ang_b,p/dd))));
   fill(set_sin_ang,p,sin(deg2rad(lerp(ang_a,ang_b,p/dd))));
   const tau=2*pi;
   const k_a_1=$2;
   const k_b_1=$3;
   const k_a_2=$4;
   const k_b_2=$5;
   const tc=.7/tau;
   rot_x(a,b,c)=a*set_cos_ang[c]-b*set_sin_ang[c];
   rot_y(a,b,c)=a*set_sin_ang[c]+b*set_cos_ang[c];
   oscil_a(a,b,c)=a+c-tc*sin(tau*b);
   oscil_b(a,b,c)=a+c-tc*sin(tau*b);
  );
  zp=z/dd;
  ix=(x-lerp(ox_a,ox_b,zp))/lerp(cxsx_a,cxsx_b,zp);
  iy=(y-lerp(oy_a,oy_b,zp))/lerp(cysy_a,cysy_b,zp);
  xp=rot_x(ix,iy,z);
  yp=rot_y(ix,iy,z);
  k_a=lerp(k_a_1,k_b_1,zp);
  k_b=lerp(k_a_2,k_b_2,zp);
  repeat(max_iter,
   txp=xp;
   xp=oscil_a(xp,yp,k_a);
   yp=oscil_b(yp,txp,k_b);
  );
  norm(xp,yp);"
+_rep_three_coupled_oscillators_static_out:
check "$8!=0"

 {w*$4},{h*$4},1,1,*"begin(
   const max_iter=abs($1);
   const ww=w-1;
   const hh=h-1;
   const sd=max(ww,hh)/min(ww,hh);
   const sx=w>h?sd:1;
   const sy=w>h?1:sd;
   const cx=ww/2;
   const cy=hh/2;
   const ox=($6+1)*cx;
   const oy=(($7*-1)+1)*cy;
   const axis_scale=abs($8);
   const cxsx=cx/sx/axis_scale;
   const cysy=cy/sy/axis_scale;
   const ang=-($5/180)*pi;
   const cos_ang=cos(ang);
   const sin_ang=sin(ang);
   const tau=2*pi;
   const k_a=$2;
   const k_b=$3;
   const tc=.7/tau;
   rot_x(a,b)=a*cos_ang-b*sin_ang;
   rot_y(a,b)=a*sin_ang+b*cos_ang;
   oscil_a(a,b)=a+k_a-tc*sin(tau*b);
   oscil_b(a,b)=a+k_b-tc*sin(tau*b);
  );
  ix=(x-ox)/cxsx;
  iy=(y-oy)/cysy;
  xp=rot_x(ix,iy);
  yp=rot_y(ix,iy);
  repeat(max_iter,
   txp=xp;
   xp=oscil_a(xp,yp);
   yp=oscil_b(yp,txp);
  );
  norm(xp,yp);"

 

 

Edited by Reptillian
  • Like 2
  • Upvote 2

G'MIC Filter Developer

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Could be a useful starting point for flames - thanks!

- I used the built-in dents effect to add some randomness - would the fractal algorithm work with some small random factor added in?... just a thought.

Concise code too!

threeoscillatorstestflames.png

  • Like 2

 

Red ochre Plugin pack.............. Diabolical Drawings ................Real Paintings

 

PdnForumSig2.jpg

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35 minutes ago, Red ochre said:

would the fractal algorithm work with some small random factor added in?

Unfortunately, it would not be easy to add it in. If it were possible to do post-processing Dents without resorting to gmic-pdn route or plain c# in visual studio (I only know how to do the former), then I could add it. This is why I release my source code other than belief in open-knowledge.

 

Thank you anyways.

G'MIC Filter Developer

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Well, I just played with this for two hours...it's addicting @Reptillian!  You can get endless textures with this and I like that you can choose 'no color' and do the coloring yourself.  Thank you for this.  🤩 👍

 

3coupledosillator_01.png

3coupledosillator_02.png

 

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