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home Balloting Simple trig Logistic map The Attractor of Henon Number doubling Barnsley's Fern The Sierpinski Triangle |

Logistic MapAlso called the logistic difference equation or the quadratic difference equation. Mathematician Paul Stein called the complexity of this iterated equation "frightening". X = rX(1 - X) (Complete program code at bottom of page) This was Mitchell Feigenbaum's break-through equation on his road to discovering universality across different chaotic systems. Iterating this equation produces regions of distinct values, involving period doubling, as well as regions of chaos. It is the rate of convergence of period doubling that is universal. Explanation: A distinct set of values occur at a certain value of r (r1). The number of distinct values suddenly doubles at a higher value of r (r2). Take the difference between r2 and r1. Divide that difference by the difference between r3 (where the next doubling occurs) and r2. That number (the convergence ratio) approaches 4.6692016... as r increases. That convergence ratio appears in many other chaotic systems involving very different equations. There are four ways to graph this iterated equation in a static manner: 1. In a bifurcation diagram, the values of X are plotted against the values of the parameter r. One such diagram is shown at the top of this page. 2. Plotting X against the previous value of X for a given value of r produces a parabolic curve, always in segments corresponding to the range of values that X acquires for that value of r. Of special interest to investigators is the question of whether a point lands to the left or to the right of the previous point. Various LRLRRRL etc patterns are found to repeat. Below the diagrams section, there is a table showing some LRRR patterning and a table showing L shift : R shift ratios for various values of the parameter r. 3. Plotting each value of X against each iteration produces a straight line graph which reveals how many iterations are needed for convergence to distinct values for X for a given value of r. (Anywhere from just a few iterations to four million iterations prove to be sufficient, depending on the value of r.) 4. Graphing in a circular manner: Plotting the sine and cosine of an incrementally increasing angle times the ever evolving X value reveals hidden patterns amidst the chaotic zones. All four types of graphs are displayed further down on this page. (Types 3 and 4 are my own creations.) The initial value (0 - 1) for X makes no difference in the appearance of any of the graphs. For some values of r, the equation is sensitive to the initial value of X, but the resulting values are not significantly different from one initial value of X to another. The range of values for r is 0 - 4. Here are some period doubling events for approximate values of r: distinct values r (range 0 - 4) _______________ _________________ 1 1 2 3 4 3.4494899 8 3.5440905 16 3.5644074 32 3.5687595 64 3.5696917 128 3.5698914 256 3.5699341 512 3.5699433 1024 3.5699453 2048 3.5699456 . . 2^n 3.5699468376 and so on, without end as r approaches 3.56994692, at which point chaos begins to take over. The highest value of r below 3.56994692 for which I discovered distinct values in the form 2^n was 3.5699468376. There must have been millions of distinct values there. Distinct values also appear when r > 3.56994692, in between periods of chaos. For instance, one such set is found at r = 3.57004996211. Distinct values also occur in groups of 6, 12, 18, 24... and 5, 10... in this higher region. (See notes below the diagrams section for general information about generating a bifurcation diagram. Program code for all four diagram types follows that.) ================================================================================ This algorithm is infinitely deep. Similar, yet never identical, patterns repeat as one looks infinitely deeper. After displaying the following four bifurcation diagrams exploring the infinite depth of this algorithm, we'll look at some graphs for single values of the parameter r. ======================================================================================= To appreciate just how abruptly chaos ends and distinct value lines begin, see the following diagram. Shown below are the final two X values out of twenty thousand iterations for the middle distinct value line for each value of r. See the graphs immediately above. As r approaches 3.8284272, the X values are not yet stable (though they are becoming bunched quite tightly), and they are not falling downwards in an overall sense: r X 3.8284268 .514396041025985 3.8284268 .514396756770264 3.8284269 .514516155621660 3.8284269 .514517495144266 3.8284270 .514118995580905 3.8284270 .514121155779832 3.8284271 .514407341511282 3.8284271 .514407484278966 Chaos ends (for awhile) at r = 3.824271 The .514... values are falling now and they are locked in on a single value for each iteration of the current parameter r: 3.8284272 .514288397179200 3.8284272 .514288397179200 3.8284273 .514253214008774 3.8284273 .514253214008774 3.8284274 .514227367177942 3.8284274 .514227367177942 3.8284275 .514205928117646 3.8284275 .514205928117646 The same is true for the other two distinct X value lines at this value of r, as well as for any r value splitting event. ================================================================================ The following five images show the straight graphing (X plotted against iterations in blue) and the parabolic graphing (X plotted against the previous X in red) superimposed onto a bifurcation diagram: Here is the only full parabola, that generated at r = 4: Here is some output showing RRRL patterning: Note that the Rs and Ls indicate a right or left shift from the previous plotted point, not the direction of the point from the central axis. X value shift .3186, " R" .7685129016, " R" .6297689087271950, " R" .8253865073602450, "L" .5101976177307690, " R" .8846318704178640, "L" .3612864678763470, " R" .8168852882604630, " R" .5295263478576550, " R" .881913809528948, "L" .368662121002446, " R" .823936279853100, " R" .513531114546066, " R" .884351857644560, "L" .362048719319123, " R" .817631832321111, " R" .527849467601022, " R" .882254401326362, "L" .367740767239850, " R" .823076533537926, " R" .515500299308996, " R" .884149482153514, "L" .362599280778603, " R" .818168489945347, " R" .526641594500876, " R" .882487398066276, "L" .367110001734535 " R" .822484479197956, " R" .516853312794537, " R" .883994519101387, "L" ================================================== The ratio of total L shifts to total R shifts for a given value of r can take on many values, such as 0:1 1:1 1:2 1:2.333... 1:3 1:4 1:4.333... 1:5 1:6 1:7 and, I surmise, other ratios as well. (One can already detect a pattern -- with the .333.. decimal portion following powers of 2 in the right side of the ratios.) These ratios remain uniform over specific ranges of r. These ratios do not correlate with the number of distinct value lines. I found examples of the L:R ratio changing in the midst of a range of a given number of distinct values. Between r = 3.56994687 and 3.56994693, the L:R ratio alternates between two distinct values: 1:5.04723799963414 and 1:5.04724714191982 This is the exact region of r where period doubling of the 2^n variety ends and chaos begins. This may be the only region where recurring values of L:R ratios are found that are not tidy ratios. Here is a tiny snapshot of the L:R ratios going through some changes through adjacent ranges of r: total total L:R init X r L shifts R shifts ratio iterations ------ ------ -------- -------- ------- ---------- ".90"," 3.7416," ",200000," ",800000," ",0," ",4," ",1000000," ",0 ".90"," 3.7417," ",200000," ",800000," ",0," ",4," ",1000000," ",0 ".90"," 3.7418," ",200000," ",800000," ",0," ",4," ",1000000," ",0 ".90"," 3.7419," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7420," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7421," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7422," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7423," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7424," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7425," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7426," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7427," ",300000," ",700000," ",0," ",2.3333," ",1000000," ",0 ".90"," 3.7428," ",250000," ",750000," ",0," ",3," ",1000000," ",0 ".90"," 3.7429," ",250000," ",750000," ",0," ",3," ",1000000," ",0 Interesting patterns appear when graphing chaotic zones in a circular manner, running sufficient iterations to complete hundreds of orbits. (These same patterns are generated by at least two other very different algorithms. One is the Balloting algorithm seen on another page of this website. Another is a repeating line drawing routine used to color a plane in a CAD program. That image is also displayed on the Balloting page. These patterns cannot be generated by simply orbiting random numbers -- an algorithm is required.) Here is a graph generated at r = 3.59 Shrinking the image to 1/2 size, previously unseen strong patterns appear: At 1/3 size: Another one at r = 3.64 1/2 size: 1/3 size: Here is r = 3.64 again, superimposed on another pattern which we discuss on another page: Finally, at r = 3.9 we see another chaos zone:Notes about generating bifurcation diagrams:The correct way to generate this diagram is to use a nested loop. The outside loop iterates values of incremented r and resets the value of X to its initial value. The inside loop iterates the equation. Amazingly, one can also generate a bifurcation diagram by using just a single loop, whereby r is incremented for each iteration of the equation. I would have thought that this would not generate anything but garbage. However, the resulting diagram is nearly indistinguishable from the one obtained using the nested loop algorithm. Looking closely, one can detect a slight difference in the dots comprising the chaotic zones, and examining the data read out reveals that period doubling events are slightly off. For example, period two occured at about 3.004, as opposed to 3.The code:(X in some of the program code is not the same X that appears in the equation appearing on this page or in the discussion on this page. Rather, it is the horizontal axis plot value.) ============================================================ My VisualBASIC program code for the bifurcation diagram: Private Sub bifurbtn_Click() viewport.ScaleMode = pixel viewport.DrawWidth = 1 'viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252) 'top 'viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom 'viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252) 'left 'viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right 'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center yshift = Val(yshiftbox.Text) 'xshift = val(xshiftbox.text) amp = Val(dsamp.Text) If amp = 0 Then amp = 1 ITER = Val(itera.Text) If ITER = 0 Or ITER > 1000 Then ITER = 1000 reset# = Val(numb.Text) If reset# = 0 Then reset# = 0.9 expand = Val(expandbox.Text) If expand = 0 Then expand = 1 ri# = Val(parameter.Text) If ri# = 0 Then ri# = 2.8 rf# = Val(rfbox.Text) If rf# = 0 Or rf# > 3.999 Then rf# = 3.999 range = CInt((rf# - ri#) * 583.333 * expand) X = (583.33 * (ri# - 2.8)) + 80 rinc# = 0.0017143 If expand > 1 Then X = 80: rinc# = rinc# / expand r# = ri# For g = 1 To range num# = reset# r# = r# + rinc# X = X + 1 For a = 1 To ITER num# = r# * num# * (1 - num#) Y# = (-num# * 700 * amp) + 760 + yshift viewport.PSet (X, Y#), RGB(0, 0, 0) 100: Next a Next g totalpoints = g * a xplotbox.Text = " " xplotbox.Text = Str$(g) totalptsbox.Text = " " totalptsbox.Text = Str$(totalpoints) rincbox.Text = " " rincbox.Text = Str$(rinc#) finalrbox.Text = " " finalrbox.Text = Str$(r#) finalnum.Text = " " finalnum.Text = Str$(num#) End Sub ============================================================ My VisualBASIC program code for bifurcation listing values: Private Sub quick2loopbtn_Click() ITER = Val(itera.Text) reset# = Val(numb.Text) If reset# = 0 Then reset# = 0.9 rinc# = 0.0000001 ' or whatever r# = 3.8284151 ' or whatever For g = 1 To 80 num# = reset# r# = r# + 0.0000001 For a = 1 To 4000000 ' or whatever num# = r# * num# * (1 - num#) If a <= 999990 Then GoTo 100 Open "ds\" + "list2" + ".txt" For Append As #1 Write #1, r#; " "; num# Close #1 100: Next a Open "ds\" + "list2" + ".txt" For Append As #1 Write #1, " " Close #1 Next g g = g - 1: a = a - 1 totalpoints = g * a xplotbox.Text = " " xplotbox.Text = Str$(g) totalptsbox.Text = " " totalptsbox.Text = Str$(totalpoints) rincbox.Text = " " rincbox.Text = Str$(rinc) finalrbox.Text = " " finalrbox.Text = Str$(r) finalnum.Text = " " finalnum.Text = Str$(num#) End Sub ============================================================ My VisualBASIC program code for the parabola version: Private Sub parabtn_Click() viewport.ScaleMode = pixel viewport.DrawWidth = 1 viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252) 'top viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252) 'left viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right 'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center dw = Val(drawwidthbox.Text) If dw = 0 Then dw = 1 viewport.DrawWidth = dw ITER = Val(itera.Text) If ITER = 0 Then ITER = 100000 r = Val(parameter.Text) para$ = parameter.Text X# = Val(numb.Text) If X# = 0 Then X# = 0.9 RR = 0 LL = 0 pt5 = 0 For a = 1 To ITER Y# = r * X# * (1 - X#) If a < 50000 Then GoTo 50 TEMPX# = (700 * X#) + 80 TEMPY# = (-700 * Y#) + 760 viewport.PSet (TEMPX#, TEMPY#), RGB(255, 0, 0) viewport.PSet (TEMPX#, 770), RGB(255, 0, 0) ' shows previous X values viewport.PSet (70, TEMPY#), RGB(255, 0, 0) ' shows X values 50: X# = Y# Next a statusbox.Text = "done" End Sub ============================================================ My VisualBASIC program code for the parbola L:R listing version: Private Sub quickparabtn_Click() ITER = Val(itera.Text) If ITER = 0 Then ITER = 20000 r# = Val(parameter.Text) para$ = parameter.Text rinc# = Val(rincbox.Text) loops = Val(gbox.Text) resetx# = Val(numb.Text) init$ = numb.Text If X# = 0 Then X# = 0.9: init$ = ".90" RR = 0 LL = 0 pt5 = 0 oops = 0 For g = 1 To loops X# = resetx# r# = r# + rinc# For a = 1 To ITER Y# = r# * X# * (1 - X#) 'If a < 999000 Then GoTo 50 'Open "ds\" + "xy-list" + ".txt" For Append As #1 'Write #1, init$; " "; r#; " "; X#; " "; Y# 'Close #1 If X# = 0.5 Then pt5 = pt5 + 1: GoTo 50 If Y# < X# And X# > 0.5 Then sh$ = " R": RR = RR + 1 If Y# > X# And X# > 0.5 Then sh$ = "L": LL = LL + 1 If Y# < X# And X# < 0.5 Then sh$ = "L": LL = LL + 1 If Y# > X# And X# < 0.5 Then sh$ = " R": RR = RR + 1 50: X# = Y# Next a If LL = 0 Then LL = 1: oops = oops + 1 ' prevents division by zero LRRATIO# = RR / LL Open "ds\" + "lr-ratio" + ".txt" For Append As #1 Write #1, init$; " "; r; " "; LL; " "; RR; " "; pt5; " "; LRRATIO#; " "; ITER; " "; oops; " "; X#; " "; Y# Close #1 RR = 0 LL = 0 pt5 = 0 oops = 0 Next g statusbox.Text = "done" End Sub ============================================================ My VisualBASIC program code for the straight graph version: Private Sub logdifstrbtn_Click() viewport.ScaleMode = pixel viewport.DrawWidth = 1 viewport.Line (80, 60)-(780, 60), RGB(148, 234, 252) 'top viewport.Line (80, 760)-(780, 760), RGB(148, 234, 252) 'bottom viewport.Line (80, 60)-(80, 760), RGB(148, 234, 252) 'left viewport.Line (780, 60)-(780, 760), RGB(148, 234, 252) 'right 'viewport.Line (430, 60)-(430, 760), RGB(148, 234, 252) 'center r# = Val(parameter.Text) amp = Val(dsamp.Text) If amp = 0 Then amp = 1 inc = Val(dsinc.Text) If inc = 0 Then inc = 0.007 num# = Val(numb.Text) If num# = 0 Then num# = 0.9 ITER = Val(itera.Text) If ITER = 0 Then ITER = 100000: inc = 0.007 yshift = Val(yshiftbox.Text) X = 80 For a = 1 To ITER num# = r# * num# * (1 - num#) X = X + inc Y# = (-num# * 700 * amp) + 760 + yshift viewport.PSet (X, Y#), RGB(0, 0, 255) Next a finalnum.Text = " " finalnum.Text = Str$(num#) End Sub ============================================================ My VisualBASIC program code for the circular graph version: Private Sub logisticrunbtn_Click() R = Val(parameter.Text) num# = Val(numb.Text) ra = Val(radiusmult.Text) * 450 ITER = Val(itera.Text) anginc# = Val(angleinc.Text) * (3.14159265358979 / 180) For a = 1 To ITER num# = R * num# * (1 - num#) angle# = angle# + anginc# X# = num# * (Cos(angle#) * ra) + 430 Y# = num# * 1.2284 * (Sin(angle#) * ra) + 530 viewport.PSet (X#, Y#), RGB(0, 0, 0) Next a finalnum.Text = " " finalnum.Text = Str$(num#) End Sub ============================================================= home Balloting Simple trig Logistic map The Attractor of Henon Number doubling Barnsley's Fern The Sierpinski Triangle