877:
621:
749:
1294:
2008:
1173:
1011:
1112:
1072:
876:
620:
748:
2446:
1942:
1221:
2243:
3751:
2450:
2453:
2452:
2448:
2447:
2454:
1946:
1949:
1948:
1944:
1943:
1950:
1225:
1228:
1227:
1223:
1222:
2451:
1229:
1700:
1476:
3124:
In general, each period-multiplying route to chaos has its own pair of
Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.
3431:
3593:
2247:
2250:
2249:
2245:
2244:
1947:
2251:
3935:
1226:
3574:
2248:
4461:
3119:
2881:
2770:
3039:
2115:
1813:
4382:
2690:
2436:
3075:
2883:
As another example, period-4-pling has a pair of
Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define
3318:
2726:
1903:
3217:
3165:
1936:
141:
2197:
2144:
2001:
2528:
2492:
2045:
1555:
1516:
1331:
1287:
1202:
606:
2927:
2578:
2340:
217:
2802:
2366:
2285:
2237:
2449:
4273:
Order in Chaos, Proceedings of the
International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545, USA 24–28 May 1982
299:
351:
273:
2168:
1972:
1743:
1723:
913:
785:
657:
456:
389:
1141:
996:
868:
740:
580:
2981:
2954:
2632:
2605:
1101:
1040:
966:
940:
812:
710:
684:
531:
505:
418:
1945:
1560:
1336:
4010:
838:
1224:
1164:
1063:
3774:
3469:
2314:
1833:
1251:
554:
479:
319:
241:
3326:
3474:
3227:
This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by
3924:
Footnote on p. 46 of
Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."
2246:
4666:
1478:, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.
3986:
2807:
3746:{\displaystyle g(x)=\lim _{n\to \infty }{\frac {1}{F^{\left(2^{n}\right)}(0)}}F^{\left(2^{n}\right)}\left(xF^{\left(2^{n}\right)}(0)\right)}
2342:, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants
1748:
4280:
2170:
can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is
3236:
3872:
Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos
Theoretical Division Annual Report 1975-1976
3080:
2731:
2986:
2050:
582:, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain
3796:, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size
1702:, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.
4325:
2637:
2371:
1205:
609:
3044:
4765:
3253:
2695:
1838:
1293:
3170:
220:
2239:, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
3131:
2287:, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
1912:
72:
1143:, there are three intersection points, with the middle one unstable, and the two others having slope exactly
2007:
3852:
1172:
4430:
4125:
4070:
2173:
2120:
1977:
25:
2497:
2461:
2014:
1521:
1485:
1300:
1256:
1181:
585:
4760:
3584:
2886:
2537:
2319:
4690:
Mathar, Richard J. (2010). "Chebyshev series representation of
Feigenbaum's period-doubling function".
3232:
162:
4635:
4516:
4476:
4422:
4254:
4217:
4176:
4117:
4062:
2775:
2345:
2258:
2210:
4435:
4130:
4075:
3335:
4504:
3445:
3228:
37:
29:
1010:
219:, and we want to study what happens when we iterate the map many times. The map might fall into a
4718:
4691:
4540:
4492:
4448:
4233:
4192:
4151:
4096:
4004:
4413:
Lanford III, Oscar E. (1984). "A shorter proof of the existence of the
Feigenbaum fixed point".
533:, there are three intersection points, with the middle one unstable, and the two others stable.
278:
1705:
This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by
1103:, there are three intersection points, with the middle one unstable, and the two others stable.
324:
246:
4733:
4276:
3992:
3982:
3957:
3907:
4053:
Feigenbaum, M. (1978). "Quantitative universality for a class of nonlinear transformations".
2153:
1957:
1728:
1708:
1695:{\displaystyle f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots }
1471:{\displaystyle f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots }
885:
757:
629:
423:
356:
4678:
4643:
4600:
4565:
4524:
4484:
4440:
4393:
4300:
4262:
4225:
4184:
4135:
4080:
3947:
3899:
1120:
1111:
971:
843:
715:
559:
17:
4657:
4614:
4579:
4536:
4405:
4314:
4147:
4092:
2959:
2932:
2610:
2583:
1080:
1019:
945:
918:
790:
689:
662:
510:
484:
397:
4653:
4610:
4575:
4532:
4401:
4310:
4143:
4088:
1909:
between two period-doubling intervals converges to a limit, the first
Feigenbaum constant
4589:"Relationships between eigenfunctions associated with solutions of Feigenbaum's equation"
817:
4736:
4706:
4639:
4520:
4480:
4426:
4258:
4221:
4180:
4121:
4108:
Feigenbaum, M. (1979). "The universal metric properties of non-linear transformations".
4066:
4025:
1146:
1045:
712:. Before the period doubling bifurcation occurs. The orbit converges to the fixed point
3759:
3454:
2299:
1818:
1236:
539:
464:
304:
226:
4682:
4648:
4623:
1071:
507:, the intersection point splits to two, which is a period doubling. For example, when
4754:
4605:
4588:
4570:
4553:
4544:
4496:
4452:
4397:
4266:
4237:
4100:
4305:
4288:
4196:
4155:
3952:
4322:
Campanino, M.; Epstein, H.; Ruelle, D. (1982). "On
Feigenbaums functional equation
3887:
3847:
3789:
3433:
For a particular form of solution with a quadratic dependence of the solution near
48:
223:, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length
4205:
4164:
3871:
608:, the period doublings become infinite, and the map becomes chaotic. This is the
3888:"Dependence of universal constants upon multiplication period in nonlinear maps"
3996:
3793:
3961:
3911:
2534:
We can also consider period-tripling route to chaos by picking a sequence of
4741:
3903:
3785:
3754:
1905:. Further, as the period-doubling intervals become shorter and shorter, the
44:
4507:; Wittwer, Peter (1987). "A complete proof of the Feigenbaum Conjectures".
4245:
Feigenbaum, Mitchell J. (1983). "Universal
Behavior in Nonlinear Systems".
3976:
4667:"Continued fractions and solutions of the Feigenbaum-Cvitanović equation"
3426:{\displaystyle {\begin{cases}g(0)=1,\\g'(0)=0,\\g''(0)<0.\end{cases}}}
3569:{\displaystyle g(x)=1-1.52763x^{2}+0.104815x^{4}+0.026705x^{6}+O(x^{8})}
2530:, since all period-doubling routes to chaos are the same (universality).
4528:
4488:
4444:
4229:
4188:
4139:
4084:
3784:
The Feigenbaum scaling function provides a complete description of the
3167:, and the relation becomes exact as both numbers increase to infinity:
4665:
Tsygvintsev, Alexei V.; Mestel, Ben D.; Obaldestin, Andrew H. (2002).
3817:
of the attractor. The ratio of segments from two consecutive covers,
1557:
looks like a fractal. Furthermore, as we repeat the period-doublings
4723:
4696:
1906:
1204:, there are infinitely many intersections, and we have arrived at
2458:
Logistic map approaching the period-doubling chaos scaling limit
1482:
Looking at the images, one can notice that at the point of chaos
1166:, indicating that it is about to undergo another period-doubling.
3792:
at the end of the period-doubling cascade. The attractor is a
4554:"Relationships between the solutions of Feigenbaum's equation"
2494:
from below. At the limit, this has the same shape as that of
4275:, Eds. David Campbell, Harvey Rose; North-Holland Amsterdam
4165:"The transition to aperiodic behavior in turbulent systems"
3419:
1065:, indicating that it is about to undergo a period-doubling.
998:
becomes unstable, splitting into a periodic-2 stable cycle.
3940:
Bulletin (New Series) of the American Mathematical Society
4462:"New proofs of the existence of the Feigenbaum functions"
4289:"A computer-assisted proof of the Feigenbaum conjectures"
3936:"A computer-assisted proof of the Feigenbaum conjectures"
870:. is exactly 1, and a period doubling bifurcation occurs.
2634:
window of the bifurcation diagram. For example, we have
2199:, it converges. This is the second Feigenbaum constant.
3114:{\displaystyle \delta =981.6\dots ,\alpha =38.82\dots }
2876:{\displaystyle f(x)\mapsto -\alpha f(f(f(-x/\alpha )))}
2765:{\displaystyle \delta =55.26\dots ,\alpha =9.277\dots }
420:, we have a single intersection, with slope bounded in
3134:
2374:
43:
the scaling function that described the covers of the
4707:"Spectral properties of the period-doubling operator"
4328:
3762:
3596:
3477:
3457:
3329:
3256:
3235:, the equation is the mathematical expression of the
3173:
3083:
3047:
3034:{\displaystyle r_{1}=3.960102,r_{2}=3.9615554,\dots }
2989:
2962:
2935:
2889:
2810:
2778:
2734:
2698:
2640:
2613:
2586:
2540:
2500:
2464:
2348:
2322:
2302:
2261:
2213:
2176:
2156:
2123:
2053:
2017:
1980:
1960:
1915:
1841:
1821:
1751:
1731:
1711:
1563:
1524:
1488:
1339:
1303:
1259:
1239:
1184:
1149:
1123:
1083:
1048:
1022:
974:
948:
921:
888:
846:
820:
793:
760:
718:
692:
665:
632:
588:
562:
542:
513:
487:
467:
458:, indicating that it is a stable single fixed point.
426:
400:
359:
327:
307:
281:
249:
229:
165:
75:
3886:
Delbourgo, R.; Hart, W.; Kenny, B. G. (1985-01-01).
3077:. This has a different pair of Feigenbaum constants
2728:. This has a different pair of Feigenbaum constants
2110:{\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
1808:{\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
1042:, we have a single intersection, with slope exactly
4624:"A precise calculation of the Feigenbaum constants"
4206:"Analyticity properties of the Feigenbaum Function"
1815:then at the limit, we would end up with a function
4671:Comptes Rendus de l'Académie des Sciences, Série I
4377:{\displaystyle g\circ g(\lambda x)+\lambda g(x)=0}
4376:
3768:
3745:
3568:
3463:
3425:
3312:
3211:
3159:
3113:
3069:
3033:
2975:
2948:
2921:
2875:
2796:
2764:
2720:
2684:
2626:
2599:
2572:
2522:
2486:
2430:
2360:
2334:
2308:
2279:
2231:
2191:
2162:
2138:
2109:
2039:
1995:
1966:
1930:
1897:
1827:
1807:
1737:
1717:
1694:
1549:
1510:
1470:
1325:
1281:
1245:
1196:
1158:
1135:
1095:
1057:
1034:
990:
960:
934:
907:
862:
832:
806:
779:
734:
704:
678:
651:
600:
574:
548:
525:
499:
473:
450:
412:
383:
345:
313:
293:
267:
235:
211:
135:
2685:{\displaystyle r_{1}=3.8284,r_{2}=3.85361,\dots }
2438:is also the same function. This is an example of
2431:{\textstyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
2047:, as we repeat the functional equation iteration
1974:, the map does not converge to a limit, but when
3613:
3174:
2983:window of the bifurcation diagram. Then we have
2146:, we find that the map does converge to a limit.
3070:{\displaystyle r_{\infty }=3.96155658717\dots }
3313:{\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}
2721:{\displaystyle r_{\infty }=3.854077963\dots }
1898:{\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}
8:
3583:The Feigenbaum function can be derived by a
3239:of period doubling. It specifies a function
3212:{\displaystyle \lim \delta /\alpha ^{2}=2/3}
4009:: CS1 maint: location missing publisher (
3834:can be arranged to approximate a function
36:the solution to the Feigenbaum-Cvitanović
4722:
4695:
4647:
4604:
4569:
4434:
4327:
4304:
4129:
4074:
3951:
3761:
3717:
3708:
3684:
3675:
3647:
3638:
3628:
3616:
3595:
3557:
3538:
3522:
3506:
3476:
3456:
3330:
3328:
3296:
3255:
3223:Feigenbaum-Cvitanović functional equation
3201:
3189:
3180:
3172:
3160:{\textstyle 3\delta \approx 2\alpha ^{2}}
3151:
3133:
3082:
3052:
3046:
3013:
2994:
2988:
2967:
2961:
2940:
2934:
2907:
2894:
2888:
2856:
2809:
2788:
2783:
2777:
2733:
2703:
2697:
2664:
2645:
2639:
2618:
2612:
2591:
2585:
2558:
2545:
2539:
2505:
2499:
2469:
2463:
2414:
2373:
2347:
2321:
2301:
2271:
2266:
2260:
2223:
2218:
2212:
2175:
2155:
2122:
2093:
2052:
2022:
2016:
1979:
1959:
1914:
1881:
1840:
1820:
1791:
1750:
1730:
1710:
1680:
1673:
1668:
1655:
1648:
1643:
1630:
1623:
1618:
1605:
1598:
1593:
1580:
1573:
1568:
1562:
1541:
1534:
1529:
1523:
1493:
1487:
1456:
1449:
1444:
1431:
1424:
1419:
1406:
1399:
1394:
1381:
1374:
1369:
1356:
1349:
1344:
1338:
1308:
1302:
1264:
1258:
1238:
1183:
1148:
1122:
1082:
1047:
1021:
979:
973:
947:
926:
920:
893:
887:
851:
845:
819:
798:
792:
765:
759:
723:
717:
691:
670:
664:
637:
631:
587:
561:
541:
512:
486:
466:
425:
399:
358:
337:
332:
326:
306:
280:
259:
254:
248:
228:
170:
164:
121:
102:
80:
74:
2444:
2241:
2006:
1940:
1931:{\displaystyle \delta =4.6692016\cdots }
1292:
1219:
136:{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}
24:has been used to describe two different
3864:
1954:For the wrong values of scaling factor
840:. The tangent slope at the fixed point
4169:Communications in Mathematical Physics
4002:
321:points, and the slope of the graph of
4587:Stephenson, John; Wang, Yong (1991).
4552:Stephenson, John; Wang, Yong (1991).
7:
3881:
3879:
2192:{\displaystyle \alpha =2.5029\dots }
2139:{\displaystyle \alpha =2.5029\dots }
1996:{\displaystyle \alpha =2.5029\dots }
66:
3838:, the Feigenbaum scaling function.
2523:{\displaystyle r^{*}=3.5699\cdots }
2487:{\displaystyle r^{*}=3.84943\dots }
2040:{\displaystyle r^{*}=3.5699\cdots }
1550:{\displaystyle f_{r^{*}}^{\infty }}
1511:{\displaystyle r^{*}=3.5699\cdots }
1333:, as we repeat the period-doublings
1326:{\displaystyle r^{*}=3.5699\cdots }
1282:{\displaystyle r^{*}=3.5699\cdots }
1206:chaos via the period-doubling route
1197:{\displaystyle r\approx 3.56994567}
601:{\displaystyle r\approx 3.56994567}
3934:Iii, Oscar E. Lanford (May 1982).
3810:the set of segments forms a cover
3623:
3053:
2956:is the lowest value in the period-
2922:{\displaystyle r_{1},r_{2},\dots }
2789:
2704:
2607:is the lowest value in the period-
2573:{\displaystyle r_{1},r_{2},\dots }
2335:{\displaystyle r\approx 3.8494344}
2272:
2224:
1542:
243:, we would find that the graph of
14:
4649:10.1090/S0025-5718-1991-1079009-6
4204:Epstein, H.; Lascoux, J. (1981).
3590:The Feigenbaum function satisfies
1233:Approach to the scaling limit as
4163:Feigenbaum, Mitchell J. (1980).
1171:
1110:
1070:
1009:
875:
747:
619:
212:{\displaystyle f_{r}(x)=rx(1-x)}
4306:10.1090/S0273-0979-1982-15008-X
3953:10.1090/S0273-0979-1982-15008-X
2804:converges to the fixed point to
2797:{\displaystyle f_{r}^{\infty }}
2361:{\displaystyle \delta ,\alpha }
2280:{\displaystyle f_{r}^{\infty }}
2232:{\displaystyle f_{r}^{\infty }}
4365:
4359:
4347:
4338:
4287:Lanford III, Oscar E. (1982).
4110:Journal of Statistical Physics
4055:Journal of Statistical Physics
3735:
3729:
3665:
3659:
3620:
3606:
3600:
3563:
3550:
3487:
3481:
3407:
3401:
3377:
3371:
3347:
3341:
3307:
3304:
3287:
3281:
3266:
3260:
2870:
2867:
2864:
2847:
2841:
2835:
2823:
2820:
2814:
2425:
2422:
2405:
2399:
2387:
2384:
2378:
2104:
2101:
2084:
2078:
2066:
2063:
2057:
1892:
1889:
1872:
1866:
1851:
1845:
1802:
1799:
1782:
1776:
1764:
1761:
1755:
610:period-doubling route to chaos
445:
427:
378:
360:
285:
206:
194:
182:
176:
127:
108:
60:Period-doubling route to chaos
1:
4683:10.1016/S1631-073X(02)02330-0
4606:10.1016/0893-9659(91)90035-T
4571:10.1016/0893-9659(91)90031-P
4398:10.1016/0040-9383(82)90001-5
4267:10.1016/0167-2789(83)90112-4
28:introduced by the physicist
3978:Chaos and dynamical systems
3323:with the initial conditions
4782:
3975:Feldman, David P. (2019).
294:{\displaystyle x\mapsto x}
391:at those intersections.
346:{\displaystyle f_{r}^{n}}
268:{\displaystyle f_{r}^{n}}
3585:renormalization argument
3904:10.1103/PhysRevA.31.514
3776:at the onset of chaos.
2255:In the chaotic regime,
2207:In the chaotic regime,
2163:{\displaystyle \alpha }
1967:{\displaystyle \alpha }
1738:{\displaystyle \alpha }
1725:for a certain constant
1718:{\displaystyle \alpha }
908:{\displaystyle x_{n+2}}
780:{\displaystyle x_{n+2}}
652:{\displaystyle x_{n+2}}
451:{\displaystyle (-1,+1)}
384:{\displaystyle (-1,+1)}
4622:Briggs, Keith (1991).
4378:
3770:
3747:
3570:
3465:
3427:
3314:
3213:
3161:
3115:
3071:
3035:
2977:
2950:
2923:
2877:
2798:
2766:
2722:
2686:
2628:
2601:
2574:
2531:
2524:
2488:
2432:
2362:
2336:
2310:
2288:
2281:
2233:
2193:
2164:
2147:
2140:
2111:
2041:
2011:At the point of chaos
2004:
1997:
1968:
1932:
1899:
1829:
1809:
1739:
1719:
1696:
1551:
1512:
1479:
1472:
1327:
1297:At the point of chaos
1290:
1283:
1247:
1198:
1160:
1137:
1136:{\displaystyle r=3.45}
1097:
1059:
1036:
992:
991:{\displaystyle x_{f2}}
962:
936:
909:
864:
863:{\displaystyle x_{f2}}
834:
808:
781:
736:
735:{\displaystyle x_{f2}}
706:
680:
653:
602:
576:
575:{\displaystyle r=3.45}
550:
527:
501:
475:
452:
414:
385:
347:
315:
295:
269:
237:
213:
137:
4737:"Feigenbaum Function"
4705:Varin, V. P. (2011).
4379:
4030:mathworld.wolfram.com
4026:"Feigenbaum Function"
3853:Presentation function
3771:
3748:
3571:
3466:
3428:
3315:
3214:
3162:
3116:
3072:
3036:
2978:
2976:{\displaystyle 4^{n}}
2951:
2949:{\displaystyle r_{n}}
2924:
2878:
2799:
2767:
2723:
2687:
2629:
2627:{\displaystyle 3^{n}}
2602:
2600:{\displaystyle r_{n}}
2575:
2525:
2489:
2457:
2433:
2363:
2337:
2311:
2282:
2254:
2234:
2194:
2165:
2141:
2112:
2042:
2010:
1998:
1969:
1953:
1933:
1900:
1830:
1810:
1740:
1720:
1697:
1552:
1513:
1473:
1328:
1296:
1284:
1248:
1232:
1199:
1161:
1138:
1098:
1096:{\displaystyle r=3.4}
1060:
1037:
1035:{\displaystyle r=3.0}
993:
963:
961:{\displaystyle a=3.3}
937:
935:{\displaystyle x_{n}}
910:
882:Relationship between
865:
835:
809:
807:{\displaystyle x_{n}}
782:
754:Relationship between
737:
707:
705:{\displaystyle a=2.7}
681:
679:{\displaystyle x_{n}}
654:
626:Relationship between
603:
577:
551:
528:
526:{\displaystyle r=3.4}
502:
500:{\displaystyle r=3.0}
476:
453:
415:
413:{\displaystyle r=3.0}
386:
348:
316:
296:
270:
238:
214:
138:
64:In the logistic map,
4505:Eckmann, Jean-Pierre
4460:Epstein, H. (1986).
4326:
3760:
3594:
3475:
3455:
3451:The power series of
3446:Feigenbaum constants
3327:
3254:
3171:
3132:
3081:
3045:
2987:
2960:
2933:
2887:
2808:
2776:
2732:
2696:
2638:
2611:
2584:
2538:
2498:
2462:
2372:
2346:
2320:
2300:
2292:Other scaling limits
2259:
2211:
2174:
2154:
2121:
2051:
2015:
1978:
1958:
1913:
1839:
1819:
1749:
1729:
1709:
1561:
1522:
1486:
1337:
1301:
1257:
1237:
1182:
1147:
1121:
1081:
1046:
1020:
972:
946:
919:
886:
844:
818:
791:
758:
716:
690:
663:
630:
586:
560:
540:
511:
485:
481:increases to beyond
465:
424:
398:
357:
325:
305:
279:
247:
227:
163:
73:
4640:1991MaCom..57..435B
4521:1987JSP....46..455E
4481:1986CMaPh.106..395E
4427:1984CMaPh..96..521L
4293:Bull. Am. Math. Soc
4259:1983PhyD....7...16F
4222:1981CMaPh..81..437E
4181:1980CMaPh..77...65F
4122:1979JSP....21..669F
4067:1978JSP....19...25F
4024:Weisstein, Eric W.
3753:for any map on the
3229:Mitchell Feigenbaum
2793:
2276:
2228:
1685:
1660:
1635:
1610:
1585:
1546:
1461:
1436:
1411:
1386:
1361:
833:{\displaystyle a=3}
342:
264:
159:we have a function
38:functional equation
30:Mitchell Feigenbaum
22:Feigenbaum function
4734:Weisstein, Eric W.
4529:10.1007/BF01013368
4489:10.1007/BF01207254
4469:Commun. Math. Phys
4445:10.1007/BF01212533
4415:Commun. Math. Phys
4374:
4230:10.1007/BF01209078
4210:Commun. Math. Phys
4189:10.1007/BF01205039
4140:10.1007/BF01107909
4085:10.1007/BF01020332
3766:
3743:
3627:
3566:
3461:
3423:
3418:
3310:
3233:Predrag Cvitanović
3209:
3157:
3111:
3067:
3031:
2973:
2946:
2919:
2873:
2794:
2779:
2762:
2718:
2682:
2624:
2597:
2570:
2532:
2520:
2484:
2428:
2358:
2332:
2306:
2289:
2277:
2262:
2229:
2214:
2189:
2160:
2148:
2136:
2107:
2037:
2005:
1993:
1964:
1928:
1895:
1825:
1805:
1735:
1715:
1692:
1664:
1639:
1614:
1589:
1564:
1547:
1525:
1508:
1480:
1468:
1440:
1415:
1390:
1365:
1340:
1323:
1291:
1279:
1243:
1194:
1159:{\displaystyle +1}
1156:
1133:
1093:
1058:{\displaystyle +1}
1055:
1032:
988:
968:. The fixed point
958:
932:
905:
860:
830:
804:
777:
732:
702:
676:
649:
598:
572:
546:
523:
497:
471:
448:
410:
394:For example, when
381:
343:
328:
311:
291:
265:
250:
233:
209:
133:
4766:Dynamical systems
3988:978-0-691-18939-0
3892:Physical Review A
3769:{\displaystyle F}
3669:
3612:
3464:{\displaystyle g}
3041:, with the limit
2692:, with the limit
2455:
2309:{\displaystyle r}
2252:
1951:
1828:{\displaystyle g}
1246:{\displaystyle r}
1230:
549:{\displaystyle r}
474:{\displaystyle r}
314:{\displaystyle n}
275:and the graph of
236:{\displaystyle n}
157:
156:
18:dynamical systems
4773:
4747:
4746:
4728:
4726:
4701:
4699:
4686:
4661:
4651:
4634:(195): 435–439.
4618:
4608:
4593:Appl. Math. Lett
4583:
4573:
4558:Appl. Math. Lett
4548:
4500:
4466:
4456:
4438:
4409:
4383:
4381:
4380:
4375:
4318:
4308:
4270:
4241:
4200:
4159:
4133:
4104:
4078:
4040:
4039:
4037:
4036:
4021:
4015:
4014:
4008:
4000:
3972:
3966:
3965:
3955:
3931:
3925:
3922:
3916:
3915:
3883:
3874:
3869:
3780:Scaling function
3775:
3773:
3772:
3767:
3752:
3750:
3749:
3744:
3742:
3738:
3728:
3727:
3726:
3722:
3721:
3695:
3694:
3693:
3689:
3688:
3670:
3668:
3658:
3657:
3656:
3652:
3651:
3629:
3626:
3575:
3573:
3572:
3567:
3562:
3561:
3543:
3542:
3527:
3526:
3511:
3510:
3471:is approximately
3470:
3468:
3467:
3462:
3443:
3432:
3430:
3429:
3424:
3422:
3421:
3400:
3370:
3319:
3317:
3316:
3311:
3300:
3247:by the relation
3246:
3243:and a parameter
3218:
3216:
3215:
3210:
3205:
3194:
3193:
3184:
3166:
3164:
3163:
3158:
3156:
3155:
3120:
3118:
3117:
3112:
3076:
3074:
3073:
3068:
3057:
3056:
3040:
3038:
3037:
3032:
3018:
3017:
2999:
2998:
2982:
2980:
2979:
2974:
2972:
2971:
2955:
2953:
2952:
2947:
2945:
2944:
2928:
2926:
2925:
2920:
2912:
2911:
2899:
2898:
2882:
2880:
2879:
2874:
2860:
2803:
2801:
2800:
2795:
2792:
2787:
2771:
2769:
2768:
2763:
2727:
2725:
2724:
2719:
2708:
2707:
2691:
2689:
2688:
2683:
2669:
2668:
2650:
2649:
2633:
2631:
2630:
2625:
2623:
2622:
2606:
2604:
2603:
2598:
2596:
2595:
2579:
2577:
2576:
2571:
2563:
2562:
2550:
2549:
2529:
2527:
2526:
2521:
2510:
2509:
2493:
2491:
2490:
2485:
2474:
2473:
2456:
2437:
2435:
2434:
2429:
2418:
2367:
2365:
2364:
2359:
2341:
2339:
2338:
2333:
2315:
2313:
2312:
2307:
2286:
2284:
2283:
2278:
2275:
2270:
2253:
2238:
2236:
2235:
2230:
2227:
2222:
2198:
2196:
2195:
2190:
2169:
2167:
2166:
2161:
2145:
2143:
2142:
2137:
2116:
2114:
2113:
2108:
2097:
2046:
2044:
2043:
2038:
2027:
2026:
2002:
2000:
1999:
1994:
1973:
1971:
1970:
1965:
1952:
1937:
1935:
1934:
1929:
1904:
1902:
1901:
1896:
1885:
1834:
1832:
1831:
1826:
1814:
1812:
1811:
1806:
1795:
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1701:
1699:
1698:
1693:
1684:
1679:
1678:
1677:
1659:
1654:
1653:
1652:
1634:
1629:
1628:
1627:
1609:
1604:
1603:
1602:
1584:
1579:
1578:
1577:
1556:
1554:
1553:
1548:
1545:
1540:
1539:
1538:
1517:
1515:
1514:
1509:
1498:
1497:
1477:
1475:
1474:
1469:
1460:
1455:
1454:
1453:
1435:
1430:
1429:
1428:
1410:
1405:
1404:
1403:
1385:
1380:
1379:
1378:
1360:
1355:
1354:
1353:
1332:
1330:
1329:
1324:
1313:
1312:
1288:
1286:
1285:
1280:
1269:
1268:
1252:
1250:
1249:
1244:
1231:
1203:
1201:
1200:
1195:
1175:
1165:
1163:
1162:
1157:
1142:
1140:
1139:
1134:
1114:
1102:
1100:
1099:
1094:
1074:
1064:
1062:
1061:
1056:
1041:
1039:
1038:
1033:
1013:
997:
995:
994:
989:
987:
986:
967:
965:
964:
959:
941:
939:
938:
933:
931:
930:
914:
912:
911:
906:
904:
903:
879:
869:
867:
866:
861:
859:
858:
839:
837:
836:
831:
813:
811:
810:
805:
803:
802:
786:
784:
783:
778:
776:
775:
751:
741:
739:
738:
733:
731:
730:
711:
709:
708:
703:
685:
683:
682:
677:
675:
674:
658:
656:
655:
650:
648:
647:
623:
607:
605:
604:
599:
581:
579:
578:
573:
555:
553:
552:
547:
532:
530:
529:
524:
506:
504:
503:
498:
480:
478:
477:
472:
457:
455:
454:
449:
419:
417:
416:
411:
390:
388:
387:
382:
352:
350:
349:
344:
341:
336:
320:
318:
317:
312:
300:
298:
297:
292:
274:
272:
271:
266:
263:
258:
242:
240:
239:
234:
218:
216:
215:
210:
175:
174:
151:
142:
140:
139:
134:
126:
125:
107:
106:
91:
90:
67:
16:In the study of
4781:
4780:
4776:
4775:
4774:
4772:
4771:
4770:
4751:
4750:
4732:
4731:
4704:
4689:
4664:
4621:
4586:
4551:
4503:
4464:
4459:
4436:10.1.1.434.1465
4412:
4324:
4323:
4321:
4286:
4244:
4203:
4162:
4131:10.1.1.418.7733
4107:
4076:10.1.1.418.9339
4052:
4049:
4044:
4043:
4034:
4032:
4023:
4022:
4018:
4001:
3989:
3974:
3973:
3969:
3933:
3932:
3928:
3923:
3919:
3885:
3884:
3877:
3870:
3866:
3861:
3844:
3833:
3822:
3815:
3808:
3803:. For a fixed
3801:
3782:
3758:
3757:
3713:
3709:
3704:
3700:
3696:
3680:
3676:
3671:
3643:
3639:
3634:
3633:
3592:
3591:
3581:
3579:Renormalization
3553:
3534:
3518:
3502:
3473:
3472:
3453:
3452:
3434:
3417:
3416:
3393:
3390:
3389:
3363:
3360:
3359:
3331:
3325:
3324:
3252:
3251:
3244:
3225:
3185:
3169:
3168:
3147:
3130:
3129:
3079:
3078:
3048:
3043:
3042:
3009:
2990:
2985:
2984:
2963:
2958:
2957:
2936:
2931:
2930:
2903:
2890:
2885:
2884:
2806:
2805:
2774:
2773:
2730:
2729:
2699:
2694:
2693:
2660:
2641:
2636:
2635:
2614:
2609:
2608:
2587:
2582:
2581:
2554:
2541:
2536:
2535:
2501:
2496:
2495:
2465:
2460:
2459:
2445:
2370:
2369:
2368:. The limit of
2344:
2343:
2318:
2317:
2298:
2297:
2294:
2257:
2256:
2242:
2209:
2208:
2205:
2172:
2171:
2152:
2151:
2119:
2118:
2049:
2048:
2018:
2013:
2012:
2003:, it converges.
1976:
1975:
1956:
1955:
1941:
1911:
1910:
1837:
1836:
1835:that satisfies
1817:
1816:
1747:
1746:
1727:
1726:
1707:
1706:
1669:
1644:
1619:
1594:
1569:
1559:
1558:
1530:
1520:
1519:
1518:, the curve of
1489:
1484:
1483:
1445:
1420:
1395:
1370:
1345:
1335:
1334:
1304:
1299:
1298:
1260:
1255:
1254:
1235:
1234:
1220:
1218:
1213:
1212:
1211:
1210:
1209:
1180:
1179:
1176:
1168:
1167:
1145:
1144:
1119:
1118:
1115:
1106:
1105:
1104:
1079:
1078:
1075:
1067:
1066:
1044:
1043:
1018:
1017:
1014:
1003:
1002:
1001:
1000:
999:
975:
970:
969:
944:
943:
922:
917:
916:
889:
884:
883:
880:
872:
871:
847:
842:
841:
816:
815:
794:
789:
788:
761:
756:
755:
752:
744:
743:
719:
714:
713:
688:
687:
666:
661:
660:
633:
628:
627:
624:
584:
583:
558:
557:
538:
537:
509:
508:
483:
482:
463:
462:
422:
421:
396:
395:
355:
354:
323:
322:
303:
302:
277:
276:
245:
244:
225:
224:
166:
161:
160:
149:
117:
98:
76:
71:
70:
62:
57:
12:
11:
5:
4779:
4777:
4769:
4768:
4763:
4753:
4752:
4749:
4748:
4729:
4702:
4687:
4677:(8): 683–688.
4662:
4619:
4584:
4549:
4501:
4475:(3): 395–426.
4457:
4421:(4): 521–538.
4410:
4392:(2): 125–129.
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4319:
4299:(3): 427–434.
4284:
4253:(1–3): 16–39.
4242:
4216:(3): 437–453.
4201:
4160:
4116:(6): 669–706.
4105:
4048:
4045:
4042:
4041:
4016:
3987:
3967:
3946:(3): 427–434.
3926:
3917:
3898:(1): 514–516.
3875:
3863:
3862:
3860:
3857:
3856:
3855:
3850:
3843:
3840:
3828:
3820:
3813:
3806:
3799:
3781:
3778:
3765:
3741:
3737:
3734:
3731:
3725:
3720:
3716:
3712:
3707:
3703:
3699:
3692:
3687:
3683:
3679:
3674:
3667:
3664:
3661:
3655:
3650:
3646:
3642:
3637:
3632:
3625:
3622:
3619:
3615:
3611:
3608:
3605:
3602:
3599:
3580:
3577:
3565:
3560:
3556:
3552:
3549:
3546:
3541:
3537:
3533:
3530:
3525:
3521:
3517:
3514:
3509:
3505:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3460:
3444:is one of the
3420:
3415:
3412:
3409:
3406:
3403:
3399:
3396:
3392:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3369:
3366:
3362:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3336:
3334:
3321:
3320:
3309:
3306:
3303:
3299:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3224:
3221:
3208:
3204:
3200:
3197:
3192:
3188:
3183:
3179:
3176:
3154:
3150:
3146:
3143:
3140:
3137:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3066:
3063:
3060:
3055:
3051:
3030:
3027:
3024:
3021:
3016:
3012:
3008:
3005:
3002:
2997:
2993:
2970:
2966:
2943:
2939:
2918:
2915:
2910:
2906:
2902:
2897:
2893:
2872:
2869:
2866:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2791:
2786:
2782:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2717:
2714:
2711:
2706:
2702:
2681:
2678:
2675:
2672:
2667:
2663:
2659:
2656:
2653:
2648:
2644:
2621:
2617:
2594:
2590:
2569:
2566:
2561:
2557:
2553:
2548:
2544:
2519:
2516:
2513:
2508:
2504:
2483:
2480:
2477:
2472:
2468:
2427:
2424:
2421:
2417:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2357:
2354:
2351:
2331:
2328:
2325:
2305:
2293:
2290:
2274:
2269:
2265:
2226:
2221:
2217:
2204:
2203:Chaotic regime
2201:
2188:
2185:
2182:
2179:
2159:
2135:
2132:
2129:
2126:
2106:
2103:
2100:
2096:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2036:
2033:
2030:
2025:
2021:
1992:
1989:
1986:
1983:
1963:
1927:
1924:
1921:
1918:
1894:
1891:
1888:
1884:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1824:
1804:
1801:
1798:
1794:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1734:
1714:
1691:
1688:
1683:
1676:
1672:
1667:
1663:
1658:
1651:
1647:
1642:
1638:
1633:
1626:
1622:
1617:
1613:
1608:
1601:
1597:
1592:
1588:
1583:
1576:
1572:
1567:
1544:
1537:
1533:
1528:
1507:
1504:
1501:
1496:
1492:
1467:
1464:
1459:
1452:
1448:
1443:
1439:
1434:
1427:
1423:
1418:
1414:
1409:
1402:
1398:
1393:
1389:
1384:
1377:
1373:
1368:
1364:
1359:
1352:
1348:
1343:
1322:
1319:
1316:
1311:
1307:
1278:
1275:
1272:
1267:
1263:
1242:
1217:
1214:
1193:
1190:
1187:
1177:
1170:
1169:
1155:
1152:
1132:
1129:
1126:
1116:
1109:
1108:
1107:
1092:
1089:
1086:
1076:
1069:
1068:
1054:
1051:
1031:
1028:
1025:
1015:
1008:
1007:
1006:
1005:
1004:
985:
982:
978:
957:
954:
951:
929:
925:
902:
899:
896:
892:
881:
874:
873:
857:
854:
850:
829:
826:
823:
801:
797:
774:
771:
768:
764:
753:
746:
745:
729:
726:
722:
701:
698:
695:
673:
669:
646:
643:
640:
636:
625:
618:
617:
616:
615:
614:
597:
594:
591:
571:
568:
565:
545:
522:
519:
516:
496:
493:
490:
470:
447:
444:
441:
438:
435:
432:
429:
409:
406:
403:
380:
377:
374:
371:
368:
365:
362:
353:is bounded in
340:
335:
331:
310:
301:intersects at
290:
287:
284:
262:
257:
253:
232:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
173:
169:
155:
154:
145:
143:
132:
129:
124:
120:
116:
113:
110:
105:
101:
97:
94:
89:
86:
83:
79:
61:
58:
56:
53:
52:
51:
41:
13:
10:
9:
6:
4:
3:
2:
4778:
4767:
4764:
4762:
4759:
4758:
4756:
4744:
4743:
4738:
4735:
4730:
4725:
4720:
4716:
4712:
4711:KIAM Preprint
4708:
4703:
4698:
4693:
4688:
4684:
4680:
4676:
4672:
4668:
4663:
4659:
4655:
4650:
4645:
4641:
4637:
4633:
4629:
4625:
4620:
4616:
4612:
4607:
4602:
4598:
4594:
4590:
4585:
4581:
4577:
4572:
4567:
4563:
4559:
4555:
4550:
4546:
4542:
4538:
4534:
4530:
4526:
4522:
4518:
4514:
4510:
4509:J. Stat. Phys
4506:
4502:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4470:
4463:
4458:
4454:
4450:
4446:
4442:
4437:
4432:
4428:
4424:
4420:
4416:
4411:
4407:
4403:
4399:
4395:
4391:
4387:
4371:
4368:
4362:
4356:
4353:
4350:
4344:
4341:
4335:
4332:
4329:
4320:
4316:
4312:
4307:
4302:
4298:
4294:
4290:
4285:
4282:
4281:0-444-86727-9
4278:
4274:
4268:
4264:
4260:
4256:
4252:
4248:
4243:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4211:
4207:
4202:
4198:
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4166:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4132:
4127:
4123:
4119:
4115:
4111:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4077:
4072:
4068:
4064:
4060:
4056:
4051:
4050:
4046:
4031:
4027:
4020:
4017:
4012:
4006:
3998:
3994:
3990:
3984:
3981:. Princeton.
3980:
3979:
3971:
3968:
3963:
3959:
3954:
3949:
3945:
3941:
3937:
3930:
3927:
3921:
3918:
3913:
3909:
3905:
3901:
3897:
3893:
3889:
3882:
3880:
3876:
3873:
3868:
3865:
3858:
3854:
3851:
3849:
3846:
3845:
3841:
3839:
3837:
3831:
3827:
3823:
3816:
3809:
3802:
3795:
3791:
3787:
3779:
3777:
3763:
3756:
3739:
3732:
3723:
3718:
3714:
3710:
3705:
3701:
3697:
3690:
3685:
3681:
3677:
3672:
3662:
3653:
3648:
3644:
3640:
3635:
3630:
3617:
3609:
3603:
3597:
3588:
3586:
3578:
3576:
3558:
3554:
3547:
3544:
3539:
3535:
3531:
3528:
3523:
3519:
3515:
3512:
3507:
3503:
3499:
3496:
3493:
3490:
3484:
3478:
3458:
3449:
3447:
3441:
3437:
3413:
3410:
3404:
3397:
3394:
3386:
3383:
3380:
3374:
3367:
3364:
3356:
3353:
3350:
3344:
3338:
3332:
3301:
3297:
3293:
3290:
3284:
3278:
3275:
3272:
3269:
3263:
3257:
3250:
3249:
3248:
3242:
3238:
3234:
3230:
3222:
3220:
3206:
3202:
3198:
3195:
3190:
3186:
3181:
3177:
3152:
3148:
3144:
3141:
3138:
3135:
3126:
3122:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3064:
3062:3.96155658717
3061:
3058:
3049:
3028:
3025:
3022:
3019:
3014:
3010:
3006:
3003:
3000:
2995:
2991:
2968:
2964:
2941:
2937:
2916:
2913:
2908:
2904:
2900:
2895:
2891:
2861:
2857:
2853:
2850:
2844:
2838:
2832:
2829:
2826:
2817:
2811:
2784:
2780:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2715:
2712:
2709:
2700:
2679:
2676:
2673:
2670:
2665:
2661:
2657:
2654:
2651:
2646:
2642:
2619:
2615:
2592:
2588:
2567:
2564:
2559:
2555:
2551:
2546:
2542:
2517:
2514:
2511:
2506:
2502:
2481:
2478:
2475:
2470:
2466:
2443:
2441:
2419:
2415:
2411:
2408:
2402:
2396:
2393:
2390:
2381:
2375:
2355:
2352:
2349:
2329:
2326:
2323:
2303:
2291:
2267:
2263:
2240:
2219:
2215:
2202:
2200:
2186:
2183:
2180:
2177:
2157:
2150:The constant
2133:
2130:
2127:
2124:
2098:
2094:
2090:
2087:
2081:
2075:
2072:
2069:
2060:
2054:
2034:
2031:
2028:
2023:
2019:
2009:
1990:
1987:
1984:
1981:
1961:
1939:
1925:
1922:
1919:
1916:
1908:
1886:
1882:
1878:
1875:
1869:
1863:
1860:
1857:
1854:
1848:
1842:
1822:
1796:
1792:
1788:
1785:
1779:
1773:
1770:
1767:
1758:
1752:
1732:
1712:
1703:
1689:
1686:
1681:
1674:
1670:
1665:
1661:
1656:
1649:
1645:
1640:
1636:
1631:
1624:
1620:
1615:
1611:
1606:
1599:
1595:
1590:
1586:
1581:
1574:
1570:
1565:
1535:
1531:
1526:
1505:
1502:
1499:
1494:
1490:
1465:
1462:
1457:
1450:
1446:
1441:
1437:
1432:
1425:
1421:
1416:
1412:
1407:
1400:
1396:
1391:
1387:
1382:
1375:
1371:
1366:
1362:
1357:
1350:
1346:
1341:
1320:
1317:
1314:
1309:
1305:
1295:
1276:
1273:
1270:
1265:
1261:
1240:
1216:Scaling limit
1215:
1207:
1191:
1188:
1185:
1174:
1153:
1150:
1130:
1127:
1124:
1113:
1090:
1087:
1084:
1073:
1052:
1049:
1029:
1026:
1023:
1012:
983:
980:
976:
955:
952:
949:
927:
923:
900:
897:
894:
890:
878:
855:
852:
848:
827:
824:
821:
799:
795:
772:
769:
766:
762:
750:
727:
724:
720:
699:
696:
693:
671:
667:
644:
641:
638:
634:
622:
613:
611:
595:
592:
589:
569:
566:
563:
543:
534:
520:
517:
514:
494:
491:
488:
468:
459:
442:
439:
436:
433:
430:
407:
404:
401:
392:
375:
372:
369:
366:
363:
338:
333:
329:
308:
288:
282:
260:
255:
251:
230:
222:
203:
200:
197:
191:
188:
185:
179:
171:
167:
153:
146:
144:
130:
122:
118:
114:
111:
103:
99:
95:
92:
87:
84:
81:
77:
69:
68:
65:
59:
54:
50:
46:
42:
39:
35:
34:
33:
31:
27:
23:
19:
4761:Chaos theory
4740:
4714:
4710:
4674:
4670:
4631:
4627:
4599:(3): 53–56.
4596:
4592:
4564:(3): 37–39.
4561:
4557:
4515:(3/4): 455.
4512:
4508:
4472:
4468:
4418:
4414:
4389:
4385:
4296:
4292:
4272:
4250:
4246:
4213:
4209:
4175:(1): 65–86.
4172:
4168:
4113:
4109:
4061:(1): 25–52.
4058:
4054:
4047:Bibliography
4033:. Retrieved
4029:
4019:
3977:
3970:
3943:
3939:
3929:
3920:
3895:
3891:
3867:
3848:Logistic map
3835:
3829:
3825:
3818:
3811:
3804:
3797:
3790:logistic map
3783:
3589:
3582:
3450:
3439:
3435:
3322:
3240:
3237:universality
3226:
3127:
3123:
2533:
2440:universality
2439:
2295:
2206:
2149:
1704:
1481:
535:
460:
393:
158:
147:
63:
49:logistic map
21:
15:
3442:= 2.5029...
3128:Generally,
2713:3.854077963
2316:approaches
1289:from below.
1253:approaches
556:approaches
221:fixed point
4755:Categories
4628:Math. Comp
4035:2023-05-07
3997:1103440222
3794:Cantor set
2929:such that
2580:such that
1192:3.56994567
596:3.56994567
4742:MathWorld
4724:1202.4672
4697:1008.4608
4545:121353606
4497:119901937
4453:121613330
4431:CiteSeerX
4354:λ
4342:λ
4333:∘
4271:Bound as
4238:119924349
4126:CiteSeerX
4101:124498882
4071:CiteSeerX
4005:cite book
3962:0273-0979
3912:0556-2791
3786:attractor
3755:real line
3624:∞
3621:→
3497:−
3302:α
3291:−
3276:α
3273:−
3187:α
3178:δ
3149:α
3142:≈
3139:δ
3109:…
3100:α
3094:…
3085:δ
3065:…
3054:∞
3029:…
3023:3.9615554
2917:…
2862:α
2851:−
2830:α
2827:−
2824:↦
2790:∞
2760:…
2751:α
2745:…
2736:δ
2716:…
2705:∞
2680:…
2568:…
2518:⋯
2507:∗
2482:…
2471:∗
2420:α
2409:−
2394:α
2391:−
2388:↦
2356:α
2350:δ
2330:3.8494344
2327:≈
2273:∞
2225:∞
2187:…
2178:α
2158:α
2134:…
2125:α
2099:α
2088:−
2073:α
2070:−
2067:↦
2035:⋯
2024:∗
1991:…
1982:α
1962:α
1926:⋯
1923:4.6692016
1917:δ
1887:α
1876:−
1861:α
1858:−
1797:α
1786:−
1771:α
1768:−
1765:↦
1733:α
1713:α
1690:…
1675:∗
1650:∗
1625:∗
1600:∗
1575:∗
1543:∞
1536:∗
1506:⋯
1495:∗
1466:…
1451:∗
1426:∗
1401:∗
1376:∗
1351:∗
1321:⋯
1310:∗
1277:⋯
1266:∗
1189:≈
593:≈
431:−
364:−
286:↦
201:−
115:−
45:attractor
26:functions
20:the term
4386:Topology
4197:18314876
4156:17956295
3842:See also
3532:0.026705
3516:0.104815
3398:″
3368:′
3004:3.960102
4658:1079009
4636:Bibcode
4615:1101875
4580:1101871
4537:0883539
4517:Bibcode
4477:Bibcode
4423:Bibcode
4406:0641996
4315:0648529
4255:Bibcode
4247:Physica
4218:Bibcode
4177:Bibcode
4148:0555919
4118:Bibcode
4093:0501179
4063:Bibcode
3788:of the
3500:1.52763
2674:3.85361
2479:3.84943
47:of the
4656:
4613:
4578:
4543:
4535:
4495:
4451:
4433:
4404:
4313:
4279:
4236:
4195:
4154:
4146:
4128:
4099:
4091:
4073:
3995:
3985:
3960:
3910:
3836:σ
3826:Δ
3819:Δ
3812:Δ
3440:α
3245:α
2772:. And
2655:3.8284
2515:3.5699
2184:2.5029
2131:2.5029
2032:3.5699
1988:2.5029
1503:3.5699
1318:3.5699
1274:3.5699
4719:arXiv
4692:arXiv
4541:S2CID
4493:S2CID
4465:(PDF)
4449:S2CID
4234:S2CID
4193:S2CID
4152:S2CID
4097:S2CID
3859:Notes
3438:= 0,
3106:38.82
3091:981.6
2757:9.277
2742:55.26
2296:When
2117:with
1907:ratio
1178:When
1117:When
1077:When
1016:When
942:when
814:when
686:when
40:; and
4277:ISBN
4011:link
3993:OCLC
3983:ISBN
3958:ISSN
3908:ISSN
3824:and
3411:<
3231:and
1131:3.45
915:and
787:and
659:and
570:3.45
55:Idea
4679:doi
4675:334
4644:doi
4601:doi
4566:doi
4525:doi
4485:doi
4473:106
4441:doi
4394:doi
4384:".
4301:doi
4263:doi
4226:doi
4185:doi
4136:doi
4081:doi
3948:doi
3900:doi
3614:lim
3175:lim
1091:3.4
1030:3.0
956:3.3
700:2.7
536:As
521:3.4
495:3.0
461:As
408:3.0
4757::
4739:.
4717:.
4713:.
4709:.
4673:.
4669:.
4654:MR
4652:.
4642:.
4632:57
4630:.
4626:.
4611:MR
4609:.
4595:.
4591:.
4576:MR
4574:.
4560:.
4556:.
4539:.
4533:MR
4531:.
4523:.
4513:46
4511:.
4491:.
4483:.
4471:.
4467:.
4447:.
4439:.
4429:.
4419:96
4417:.
4402:MR
4400:.
4390:21
4388:.
4311:MR
4309:.
4295:.
4291:.
4261:.
4251:7D
4249:.
4232:.
4224:.
4214:81
4212:.
4208:.
4191:.
4183:.
4173:77
4171:.
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