Image demonstrating Lorentz attractor

/*
Math reference:
Lorentz attractors and Chaos Theory
http://mathworld.wolfram.com/LorenzAttractor.html
initially for r=28, s = 10, b = 8/3.

The Lorenz attractor is generated by the differential equation written by

dx/dt=  -10x    +10y    
dy/dt=  28x -y  -xz
dz/dt=  -8/3z   +xy.    


By http://freedelta.free.fr 21-dec-2009
*/

 header("Content-type: image/jpeg");  
 
 define("TIMESTEP",0.0001);             /* Numerical timestep for dt        */
 define("XSCALE",4);                    /* Appropriate scaling factors for  */
 define("YSCALE",4);                    /*   each axis.                     */
 define("ZSCALE",4);
 define("TSCALE",20);                   /* Scaling factor for the time axis */
 define("INTERVAL",100);                /* Sample function every INTERVAL  points for output */
 
 
 
 $height = 530;
 $width = 620;  
 $im = imagecreate($width,$height); // Canvas width , height px  
 
 // Color definitions
 $white = imagecolorallocate($im,255,255,255); 
 $black = imagecolorallocate($im,0,0,0);   
 $red   = imagecolorallocate($im,255,0,0);  
 $green = imagecolorallocate($im,0,255,0);  
 $blue  = imagecolorallocate($im,0,0,255);  
 $indigo= imagecolorallocate($im,75,0,130);
                                     
 //   double 
 $x1=0.0;  $y1=0.0;  $z1=0.0;  $x=0.0;  $y=0.0;  $z=0.0;  $t=0.0;  $dt=0.0;  $t0=0.0;
 
 // int 
 $xp=0; $yp=0;  $tp=0;  $gxp=0;  $gyp=0;  $gzp=0;
 
 $x1=($y1=($z1=($t=0.0001)));        /* Initial conditions */
 $dt=TIMESTEP;                    
 
 // Plan lines and messages
 imagestring($im, 4 , 2, 5, "Demonstration of Lorentz attractor", $blue);
 imageline($im,320,0,320,479,$red);
 imageline($im,0,240,639,240,$black);   
 imagestring($im, 3, 2, 285, "X variable with respect to time", $black);
 imageline($im,0,300,639,300,$black);
 imagestring($im, 3 , 2, 365, "Y variable with respect to time", $black);
 imageline($im,0,380,639,380,$black);
 imagestring($im, 3, 2, 445, "Z variable with respect to time", $black);
 imageline($im,0,460,639,460,$black);
 
 $lineWidth=0.1;
 while($tp<639)
 {
     $x=$x1 + ( -10.0*$x1  + 10.0*$y1    ) * $dt;
     $y=$y1 + ( 28.0*$x1   - $y1 - $x1*$z1 ) * $dt;
     $z=$z1 + ( -(8/3)*$z1 + $x1*$y1      ) * $dt;
     $t=$t  +                              $dt;
 
     $xp=320+(int)floor(XSCALE*$x);
     $yp=240-(int)floor(YSCALE*$z);
     $tp=(int)floor(TSCALE*$t);
     if ($tp>639)
     {
         $tp=479;
     }
     
     $gxp=300-(int)floor(XSCALE*$x/4);
     $gyp=380-(int)floor(YSCALE*$y/4);
     $gzp=460-(int)floor(ZSCALE*$z/4);

 
           
     imageline($im, $xp, $yp, $xp+$lineWidth, $yp+$lineWidth, $blue);
     // x variable with respect to time
     imageline($im, $tp, $gxp, $tp+$lineWidth, $gxp+$lineWidth, $red);
     // y variable with respect to time
     imageline($im, $tp, $gyp, $tp+$lineWidth, $gyp+$lineWidth, $green);
     // z variable with respect to time
     imageline($im, $tp, $gzp, $tp+$lineWidth, $gzp+$lineWidth, $indigo);
 
     $x1=$x;
     $y1=$y;
     $z1=$z;
 
 }
 
 // Display image
 imagejpeg($im);  

It will output an image like this one: