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Bifurcation DiagramA bifurcation diagram shows the possible values a function can have as a function of a parameter of that function. An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the x-axis of the plot and the y-axis shows the possible population values of the logistic function. Bifurcation diagram of a logistic map 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 the periods go to infinity, r remains finite. When r is greater than 0.78***, there are no stable orbits and the orbits become chaotic. Hence the bifurcation diagram is a nice example of the importance of chaos theory in even very simple non-linear systems. Period 3 first occurs at 3.83, period 5 at 3.74, and so on. The table below lists the first values of c (Mandelbrot set) and r (logistic map) for which 0 and 0.5, respectively, converge to stable cycles of a given period. The periods are ordered in the fashion which appears in Sarkovskii's theorem: observe that in this ordering the sequences of c and r values are monotonic. Note, however, that these are not the only occurrences of these periods. Indeed, the bifurcation diagram is fractal, since any of the 'arms' in the periodic regions, when magnified, appear similar to the whole diagram. Inside the window containing the period 3 region, for example, one also finds periods 6 = 2 · 3, 9 = 3 · 3, and so forth. | n | c | r |
|---|
| 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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