'Iterating x = x<**2 + c

'This equation may be thought of as a real-axis only version of Mandelbrot's 
'z = z**2 + c. The program StdLog2.bas does exactly the same for the above 
'equation as stdLog1.bas does for x = rx(1 - x). Typical values for c:

'for c = .5 and beyond, x skates off to infinity
'for c = .15, x climbs to a steady value
'for c = 0, x stays at zero
'for c = -.5, x climbs to a steady negative value
'for c = -1, x oscillates with period 2
'for c = -1.3, x oscillates with period 4
'for c = -1.4, x oscillates with period 8
'for c = -1.5, bounded chaotic oscillation
'for c = -1.7, a good demonstration of bounded chaos
'for c = -1.75, periodic oasis within chaos - period 3
'for c < -2, x rapidly goes wild into infinite chaos

'StdLog2.BAS

screen 9,3
color 7,0
th = 3                  'text horizontal bias
s = 1.5                 'horizontal scaling factor
n = 40                  'integral extent of the axes
h = 300                 'horizontal zero bias
v = 240                 'vertical zero bias
nn = 550                'extent of time-series horizontal axis
hh = 25                 'time-series horizontal bias
vv = 60                 'time-series vertical bias
ss = 50                 'inter-iteration delay

Lab:                    'INPUT THE VALUE OF c
  locate 16, th : print"Iterate the formula y = x * x + c  "
  locate 23, th : print"Press RETURN to exit program. "
  locate 17, th
  input"Enter c = ",r$  'input the growth factor for this run
  if c$ = "" then end   'exit if nothing entered
  c = val(r$)           'growth factor
  cls                   'clear the screen

  locate 14, th : print"FEIGENBAUM'S EXPERIMENT"
  locate 16, th : print"Iterating the formula y = x * x + c"
  locate 17, th + 2 : print"for c = "; c$
  locate 23, th : print"Press any key to stop program."
  locate 19, th : print"yMAX ="
  locate 20, th : print"yMIN ="

'DRAW AXES FOR THE TIME-SERIES DIAGRAM
  ts = 1                      'set time-series flag
  c = 3                       'set colour to cyan for axes
  t = nn : x = 0 : gosub p    'set pen to end of time axis
  t = 0 : gosub l             'draw line to time-series origin
  x =  2 : gosub p 
  x = -2 : gosub l            'draw time-series y-axis
  locate 2,  3 : print "y"
  locate 5, 74 : print "time"

'DRAW THE GRAPH AXES FOR THE FEIGENBAUM DIAGRAM
  ts = 0                      'clear time-series flag
  x = 2 : y = 0 : gosub p     'set pen to end of x-axis
  x = -2 : gosub l            'draw line to -1
  x = 0 : y = 2 : gosub p     'set pen to end of y-axis
  y = -2 : gosub l            'continue line to end of y-axis
  locate 11, 56 : print "y"
  locate 18, 73 : print "x";

'DRAW THE DIAGONAL BOUNCE-LINE
  c = 11
  x = -2 : y = -2 : gosub p   'set pen to origin
  x = +2 : y = +2 : gosub l   'draw diagonal line to point 1,1

'DRAW THE CURVE OF THE NON-LINEAR DIFFERENCE FUNCTION
  flag = 0
  for x = -2 to 2 step .01
    gosub f
    if y < 2 and y > -2 then
      if flag = 0 then
        flag = 1 : gosub p
      else gosub l
      end if
    end if
  next

'ITERATE NON-LINEAR DIFFERENCE FUNCTION UNTIL A KEY IS PRESSED
  oldx = 0
  t = 1                     'initialise the time
  u = 61                    'start of trace-clearing sweep
  ymax = 0                  'holds maximum excursion of y
  ymin = 1                  'holds minimum excursion of y
  x = .3                    'set initial value of input
  y = 0                     'for position of first plot
  gosub p                   'set plotting pen to start value
  e = 0                     'reset the out-of-range flag

  while inkey$ = ""         'while user has not pressed a key
    xx = x : yy = y         'store starting co-ordinates
    c = 15 : gosub w        'draw the two lines in bright colour
   'for ft = 0 to ss : next 'delay to aid visibility of trace
    sound 200 * (y + 3), 1
    gosub TimeSeries        'put plot on time-series graph
    x = xx : y = yy         'retrieve the starting co-ordinates
    c = 1 : gosub w         're-draw the lines in dim colour
  wend : goto LAB           'return for next c-value if key hit

'TIMES-SERIES GRAPH PLOTTING ROUTINE
TimeSeries:
     ts = 1                 'set time-series flag
     c = 10
     newx = x               'save new value of function output
     if t > nn then t = 1   'if end of sweep has been reached
     x = oldx : gosub p     'set to previous plot position
     t = t + 8              'increment the time axis
     x = newx : gosub l     'draw line to current plot
     oldx = x               'save current x for next pass
     if u > nn then u = 1
     uu = u + hh : uuu = uu + 10
     for i = uu to uuu
       line(i, vv - n - n) - (i, vv - 1), 0  'draw a trace-wiper 
       line (i, vv + n + n) - (i, vv + 1), 0 'in blue background
     next
     u = u + 8
     ts = 0                 'clear the time-series flag
     return

w:                          'DRAW VERT/HORZ LINE-PAIR (L-SHAPE)
  if e = 0 then             'provided y has not gone out of range
    gosub p                 'set the start of the line
    gosub f                 'iterate the non-linear function
    if y < -2 or y > 2 then 'if y is now out of range
      e = 1                 'set the out-of-range flag
      locate 21, th : print"Gone out of range: y ="y"     "
    else
      gosub l               'draw the up-line
      x = y : gosub l       'draw the across-line
    end if
  end if
return

p:   'set starting point for current line
  gosub q : pset(a, b), c : return

l:   'draw line to point x, y
  gosub q : line -(a, b), c : return 

q:'CONVERT REAL X/Y CO-ORDS TO SCALED INTEGRAL NUMBERS OF PIXELS
  if ts = 0 then
    a = s * (h + n * x)
    b = v - n * y
  else
    a = hh + t
    b = vv - n * x
  end if
return

f:'ITERATE DIFFERENCE FUNCTION. CHECK THAT RESULT IS IN RANGE
  y = x * x + c
  if y > ymax then ymax = y : locate 19, th + 8 : gosub num
  if y < ymin then ymin = y : locate 20, th + 8 : gosub num
return
num: y$ = str$(y) : PRINT y$ + space$(13 - len(y$))
return

'November 1997 Robert John Morton