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:
| 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. |