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# Bifurcation diagram

A bifurcation diagram shows the possible values a function can have in function of a parameter of that function.

An example is the bifurcation diagram of the logistic map. In this case, the parameter r is show on the x-axis of the plot and the y-axis shows the possible population values of the logistic function.

The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the values of r where bifurcation occurs is called the Feigenbaum constant. The interesting thing about the diagram is that as periods go to infinity while r remains finite. When r is greater than 0.78***, there are no stable orbits and the orbits become chaotic. Hence the bifuraction diagram is a nice example of the importance of chaos theory in even very simple non-linear systems.

Period 3 occurs at 3.83, 5 at 3.74, and so on. Here are the values of c (Mandelbrot set) and r (logistic map) for which 0 and 0.5, respectively, are in stable cycles of the given period:
ncr
3 -1.75487766624669 3.83187405528332
5 -1.6254137251233 3.73891491297069
7 -1.5748891397523 3.70176915353796
9 -1.55528270076858 3.68721618093415
11 -1.54790376180395 3.68171867413713
13 -1.54520178169266 3.67970280568025
15 -1.54422856011953 3.6789763419034
17 -1.54388090052771 3.67871678273588
19 -1.54375717344627 3.67862440326842
21 -1.54371321190794 3.67859157910118
23 -1.54369760241228 3.67857992407341
25 -1.54369206143762 3.67857578682226
27 -1.54369009474707 3.67857431836197
29 -1.54368939672815 3.67857379717502
31 -1.54368914899106 3.67857361219815
33 -1.54368906106612 3.67857354654758
35 -1.54368902986056 3.67857352324744
37 -1.54368901878536 3.67857351497797
39 -1.54368901485465 3.67857351204304
41 -1.5436890134596 3.6785735110014
43 -1.54368901296448 3.67857351063172
45 -1.54368901278875 3.67857351050051
47 -1.54368901272639 3.67857351045394
49 -1.54368901270425 3.67857351043742
51 -1.5436890126964 3.67857351043155
53 -1.54368901269361 3.67857351042947
55 -1.54368901269262 3.67857351042873
57 -1.54368901269227 3.67857351042847
59 -1.54368901269215 3.67857351042837
61 -1.5436890126921 3.67857351042834
63 -1.54368901269208 3.67857351042833
65 -1.54368901269208 3.67857351042832
67 -1.54368901269208 3.67857351042832
69 -1.54368901269208 3.67857351042832
71 -1.54368901269208 3.67857351042832
.............................
6 -1.47601464272843 3.62755752951552
10 -1.44700884060748 3.60538583753537
14 -1.43641156477138 3.59723819837256
18 -1.43249708877703 3.59422210982563
22 -1.43109654904286 3.59314214731307
26 -1.4306095831777 3.5927665403408
30 -1.43044302383688 3.59263805714325
34 -1.43038649845399 3.59259445224585
38 -1.4303673801139 3.5925797037807
42 -1.43036092273569 3.59257472234509
46 -1.43035874289521 3.59257304074174
50 -1.43035800719415 3.59257247319657
54 -1.43035775891334 3.59257228166417
58 -1.43035767512727 3.59257221702869
62 -1.43035764685272 3.59257219521673
66 -1.4303576373112 3.59257218785607
70 -1.43035763409132 3.59257218537214
74 -1.43035763300474 3.59257218453392
78 -1.43035763263807 3.59257218425105
82 -1.43035763251433 3.5925721841556
86 -1.43035763247258 3.59257218412339
90 -1.43035763245848 3.59257218411252
94 -1.43035763245373 3.59257218410885
98 -1.43035763245212 3.59257218410761
102 -1.43035763245158 3.59257218410719
106 -1.4303576324514 3.59257218410705
110 -1.43035763245134 3.592572184107
114 -1.43035763245132 3.59257218410699
118 -1.43035763245131 3.59257218410698
122 -1.43035763245131 3.59257218410698
126 -1.43035763245131 3.59257218410698
130 -1.43035763245131 3.59257218410698
.............................
12 -1.41697773125021 3.58222983582036
20 -1.41094075752239 3.57754981136923
28 -1.40870040682787 3.57581086792324
36 -1.40786584608059 3.57516278792669
44 -1.40756521961828 3.5749292958202
52 -1.40746003476332 3.57484759530604
60 -1.40742384014022 3.57481948115997
68 -1.40741148396746 3.57480988344185
76 -1.40740728031309 3.57480661822444
84 -1.40740585222534 3.57480550894652
92 -1.40740536734078 3.57480513230868
100 -1.4074052027419 3.57480500445521
108 -1.40740514687184 3.5748049610577
116 -1.40740512790837 3.57480494632768
124 -1.40740512147185 3.57480494132806
132 -1.4074051192872 3.57480493963112
140 -1.40740511854569 3.57480493905515
148 -1.40740511829402 3.57480493885965
156 -1.40740511820859 3.5748049387933
164 -1.4074051181796 3.57480493877078
172 -1.40740511816976 3.57480493876314
180 -1.40740511816642 3.57480493876054
188 -1.40740511816528 3.57480493875966
196 -1.4074051181649 3.57480493875936
204 -1.40740511816477 3.57480493875926
212 -1.40740511816472 3.57480493875923
220 -1.40740511816471 3.57480493875921
228 -1.4074051181647 3.57480493875921
236 -1.4074051181647 3.57480493875921
244 -1.4074051181647 3.57480493875921
.............................
4096 -1.40115517044441 3.56994565735885
2048 -1.40115510202246 3.56994560411108
1024 -1.40115478254662 3.56994535548647
512 -1.40115329084992 3.56994419460807
256 -1.40114632582695 3.56993877423331
128 -1.40111380493978 3.56991346542235
64 -1.40096196294484 3.56979529374994
32 -1.40025308121478 3.56924353163711
16 -1.39694535970456 3.56666737985627
8 -1.38154748443206 3.55464086276882
4 -1.31070264133683 3.4985616993277
2 -1. 3.23606797749979
1 0. 2.

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