715:
980:
427:
1141:
919:
548:
490:
855:
774:
308:
369:
1047:
813:
1172:
604:
171:
108:
1231:
817:
there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and
865:
one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).
945:
861:
rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of
50:
is a point which the system returns to after a certain number of function iterations or a certain amount of time.
928:
17:
381:
1084:
881:
505:
444:
874:
820:
742:
255:
590:
342:
1010:
785:
1152:
131:
1221:
333:
78:
47:
59:
8:
1259:
1243:
936:
739:
between 1 and 3, the value 0 is still periodic but is not attracting, while the value
573:; and if both the stable and unstable manifold have nonzero dimension, it is called a
710:{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),\qquad 0\leq x_{t}\leq 1,\qquad 0\leq r\leq 4}
566:
235:
188:
66:
35:
1226:
1183:
39:
727:
between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence
1216:
558:
329:
1253:
597:
578:
1206:
31:
1239:
337:
27:
Point which a function/system returns to after some time or iterations
1211:
732:
554:
434:
1238:
This article incorporates material from hyperbolic fixed point on
561:
of a periodic point or fixed point is zero, the point is called a
196:
719:
exhibits periodicity for various values of the parameter
825:
747:
1155:
1087:
1013:
948:
884:
823:
788:
745:
607:
508:
447:
384:
345:
258:
134:
81:
371:is defined, then one says that a periodic point is
1166:
1135:
1041:
974:
913:
849:
807:
776:is an attracting periodic point of period 1. With
768:
709:
542:
484:
421:
363:
302:
165:
102:
234:(this is not to be confused with the notion of a
1251:
1244:Creative Commons Attribution/Share-Alike License
975:{\displaystyle \Phi :\mathbb {R} \times X\to X}
1232:Periodic points of complex quadratic mappings
121:is called periodic point if there exists an
1157:
1038:
956:
889:
218:is a periodic point with the same period
1196:are periodic with the same prime period.
422:{\displaystyle |f_{n}^{\prime }|\neq 1,}
1136:{\displaystyle \Phi (t,x)=\Phi (t+T,x)}
914:{\displaystyle (\mathbb {R} ,X,\Phi ),}
543:{\displaystyle |f_{n}^{\prime }|>1.}
485:{\displaystyle |f_{n}^{\prime }|<1,}
321:. All periodic points are preperiodic.
14:
1252:
53:
868:
202:satisfying the above is called the
24:
1109:
1088:
1014:
949:
902:
850:{\displaystyle {\tfrac {r-1}{r}}.}
524:
463:
400:
356:
25:
1271:
769:{\displaystyle {\tfrac {r-1}{r}}}
303:{\displaystyle f_{n}(x)=f_{m}(x)}
691:
665:
589:A period-one point is called a
364:{\displaystyle f_{n}^{\prime }}
1242:, which is licensed under the
1130:
1112:
1103:
1091:
1042:{\displaystyle \Phi (T,x)=x\,}
1029:
1017:
966:
905:
885:
808:{\displaystyle 1+{\sqrt {6}},}
659:
640:
530:
510:
469:
449:
406:
386:
297:
291:
275:
269:
154:
148:
91:
13:
1:
1167:{\displaystyle \mathbb {R} .}
1067:
1056:with this property is called
780:greater than 3 but less than
875:real global dynamical system
166:{\displaystyle \ f_{n}(x)=x}
7:
1200:
584:
10:
1276:
857:As the value of parameter
565:; if the dimension of its
103:{\displaystyle f:X\to X,}
569:is zero, it is called a
241:If there exist distinct
195:. The smallest positive
1182:then all points on the
1178:Given a periodic point
1073:Given a periodic point
334:differentiable manifold
1168:
1137:
1052:The smallest positive
1043:
976:
915:
851:
809:
770:
711:
544:
486:
423:
365:
304:
167:
104:
1169:
1138:
1044:
977:
916:
852:
810:
771:
712:
545:
487:
424:
366:
305:
168:
105:
1222:Sharkovsky's theorem
1153:
1085:
1011:
946:
882:
821:
786:
743:
605:
506:
445:
382:
343:
256:
214:. If every point in
132:
79:
528:
467:
404:
360:
1164:
1133:
1039:
972:
937:evolution function
911:
847:
842:
805:
766:
764:
707:
540:
514:
482:
453:
419:
390:
361:
346:
300:
163:
100:
54:Iterated functions
36:iterated functions
34:, in the study of
841:
800:
763:
735:all orbits). For
567:unstable manifold
319:preperiodic point
236:periodic function
137:
40:dynamical systems
16:(Redirected from
1267:
1227:Stationary point
1195:
1191:
1181:
1175:
1173:
1171:
1170:
1165:
1160:
1146:
1142:
1140:
1139:
1134:
1080:
1076:
1063:
1055:
1048:
1046:
1045:
1040:
1003:
992:
988:
981:
979:
978:
973:
959:
934:
926:
922:
920:
918:
917:
912:
892:
869:Dynamical system
864:
860:
856:
854:
853:
848:
843:
837:
826:
816:
814:
812:
811:
806:
801:
796:
779:
775:
773:
772:
767:
765:
759:
748:
738:
730:
726:
722:
716:
714:
713:
708:
681:
680:
658:
657:
639:
638:
623:
622:
549:
547:
546:
541:
533:
527:
522:
513:
491:
489:
488:
483:
472:
466:
461:
452:
428:
426:
425:
420:
409:
403:
398:
389:
370:
368:
367:
362:
359:
354:
327:
316:
309:
307:
306:
301:
290:
289:
268:
267:
248:
244:
233:
225:
221:
217:
213:
201:
194:
186:
182:
172:
170:
169:
164:
147:
146:
135:
124:
120:
116:
109:
107:
106:
101:
71:
64:
21:
1275:
1274:
1270:
1269:
1268:
1266:
1265:
1264:
1250:
1249:
1203:
1193:
1190:
1186:
1179:
1156:
1154:
1151:
1150:
1148:
1144:
1086:
1083:
1082:
1078:
1074:
1070:
1061:
1053:
1012:
1009:
1008:
1001:
990:
986:
955:
947:
944:
943:
932:
924:
888:
883:
880:
879:
877:
871:
862:
858:
827:
824:
822:
819:
818:
795:
787:
784:
783:
781:
777:
749:
746:
744:
741:
740:
736:
728:
724:
720:
676:
672:
653:
649:
634:
630:
612:
608:
606:
603:
602:
587:
559:stable manifold
529:
523:
518:
509:
507:
504:
503:
468:
462:
457:
448:
446:
443:
442:
405:
399:
394:
385:
383:
380:
379:
355:
350:
344:
341:
340:
325:
314:
285:
281:
263:
259:
257:
254:
253:
246:
242:
231:
223:
219:
215:
211:
199:
192:
184:
181:
177:
142:
138:
133:
130:
129:
122:
118:
114:
80:
77:
76:
69:
62:
56:
28:
23:
22:
15:
12:
11:
5:
1273:
1263:
1262:
1235:
1234:
1229:
1224:
1219:
1214:
1209:
1202:
1199:
1198:
1197:
1188:
1176:
1163:
1159:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1069:
1066:
1050:
1049:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
983:
982:
971:
968:
965:
962:
958:
954:
951:
910:
907:
904:
901:
898:
895:
891:
887:
870:
867:
846:
840:
836:
833:
830:
804:
799:
794:
791:
762:
758:
755:
752:
706:
703:
700:
697:
694:
690:
687:
684:
679:
675:
671:
668:
664:
661:
656:
652:
648:
645:
642:
637:
633:
629:
626:
621:
618:
615:
611:
586:
583:
551:
550:
539:
536:
532:
526:
521:
517:
512:
493:
492:
481:
478:
475:
471:
465:
460:
456:
451:
430:
429:
418:
415:
412:
408:
402:
397:
393:
388:
358:
353:
349:
336:, so that the
330:diffeomorphism
311:
310:
299:
296:
293:
288:
284:
280:
277:
274:
271:
266:
262:
179:
174:
173:
162:
159:
156:
153:
150:
145:
141:
125:>0 so that
111:
110:
99:
96:
93:
90:
87:
84:
55:
52:
44:periodic point
26:
18:Periodic orbit
9:
6:
4:
3:
2:
1272:
1261:
1258:
1257:
1255:
1248:
1247:
1245:
1241:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1204:
1185:
1177:
1161:
1127:
1124:
1121:
1118:
1115:
1106:
1100:
1097:
1094:
1072:
1071:
1065:
1060:of the point
1059:
1035:
1032:
1026:
1023:
1020:
1007:
1006:
1005:
1000:
996:
969:
963:
960:
952:
942:
941:
940:
938:
930:
908:
899:
896:
893:
876:
866:
844:
838:
834:
831:
828:
802:
797:
792:
789:
760:
756:
753:
750:
734:
717:
704:
701:
698:
695:
692:
688:
685:
682:
677:
673:
669:
666:
662:
654:
650:
646:
643:
635:
631:
627:
624:
619:
616:
613:
609:
600:
599:
594:
592:
582:
580:
576:
572:
568:
564:
560:
556:
537:
534:
519:
515:
502:
501:
500:
498:
479:
476:
473:
458:
454:
441:
440:
439:
437:
436:
416:
413:
410:
395:
391:
378:
377:
376:
374:
351:
347:
339:
335:
331:
322:
320:
294:
286:
282:
278:
272:
264:
260:
252:
251:
250:
239:
237:
229:
210:of the point
209:
205:
198:
190:
160:
157:
151:
143:
139:
128:
127:
126:
97:
94:
88:
85:
82:
75:
74:
73:
72:into itself,
68:
61:
51:
49:
45:
41:
37:
33:
19:
1237:
1236:
1077:with period
1058:prime period
1057:
1051:
998:
994:
984:
872:
718:
601:
598:logistic map
595:
588:
579:saddle point
574:
570:
562:
552:
496:
494:
433:
431:
372:
323:
318:
317:is called a
312:
240:
230:with period
227:
208:least period
207:
204:prime period
203:
175:
112:
57:
43:
29:
1207:Limit cycle
929:phase space
729:0, 0, 0, …,
591:fixed point
432:that it is
32:mathematics
1260:Limit sets
1240:PlanetMath
1217:Stable set
1068:Properties
993:is called
495:and it is
435:attractive
373:hyperbolic
338:derivative
249:such that
226:is called
1212:Limit set
1110:Φ
1089:Φ
1015:Φ
967:→
961:×
950:Φ
903:Φ
832:−
754:−
702:≤
696:≤
683:≤
670:≤
647:−
555:dimension
525:′
497:repelling
464:′
411:≠
401:′
357:′
92:→
1254:Category
1201:See also
1192:through
1143:for all
995:periodic
985:a point
873:Given a
733:attracts
585:Examples
228:periodic
113:a point
58:Given a
48:function
1174:
1149:
1081:, then
921:
878:
815:
782:
557:of the
553:If the
222:, then
197:integer
189:iterate
183:is the
65:from a
60:mapping
1187:γ
999:period
731:which
723:. For
575:saddle
563:source
176:where
136:
1184:orbit
997:with
923:with
332:of a
328:is a
313:then
46:of a
935:the
931:and
927:the
596:The
571:sink
535:>
474:<
245:and
42:, a
38:and
1147:in
1004:if
989:in
577:or
499:if
438:if
375:if
324:If
238:).
206:or
191:of
187:th
117:in
67:set
30:In
1256::
1064:.
939:,
593:.
581:.
538:1.
1246:.
1194:x
1189:x
1180:x
1162:.
1158:R
1145:t
1131:)
1128:x
1125:,
1122:T
1119:+
1116:t
1113:(
1107:=
1104:)
1101:x
1098:,
1095:t
1092:(
1079:T
1075:x
1062:x
1054:T
1036:x
1033:=
1030:)
1027:x
1024:,
1021:T
1018:(
1002:T
991:X
987:x
970:X
964:X
957:R
953::
933:Φ
925:X
909:,
906:)
900:,
897:X
894:,
890:R
886:(
863:r
859:r
845:.
839:r
835:1
829:r
803:,
798:6
793:+
790:1
778:r
761:r
757:1
751:r
737:r
725:r
721:r
705:4
699:r
693:0
689:,
686:1
678:t
674:x
667:0
663:,
660:)
655:t
651:x
644:1
641:(
636:t
632:x
628:r
625:=
620:1
617:+
614:t
610:x
531:|
520:n
516:f
511:|
480:,
477:1
470:|
459:n
455:f
450:|
417:,
414:1
407:|
396:n
392:f
387:|
352:n
348:f
326:f
315:x
298:)
295:x
292:(
287:m
283:f
279:=
276:)
273:x
270:(
265:n
261:f
247:m
243:n
232:n
224:f
220:n
216:X
212:x
200:n
193:f
185:n
180:n
178:f
161:x
158:=
155:)
152:x
149:(
144:n
140:f
123:n
119:X
115:x
98:,
95:X
89:X
86::
83:f
70:X
63:f
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.