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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
854:
one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).
934:
850:
rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of
39:
is a point which the system returns to after a certain number of function iterations or a certain amount of time.
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370:
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494:
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322:
67:
36:
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8:
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between 1 and 3, the value 0 is still periodic but is not attracting, while the value
562:; and if both the stable and unstable manifold have nonzero dimension, it is called a
699:{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),\qquad 0\leq x_{t}\leq 1,\qquad 0\leq r\leq 4}
555:
224:
177:
55:
24:
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between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence
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20:
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Point which a function/system returns to after some time or iterations
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This article incorporates material from hyperbolic fixed point on
550:
of a periodic point or fixed point is zero, the point is called a
185:
708:
exhibits periodicity for various values of the parameter
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360:is defined, then one says that a periodic point is
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765:is an attracting periodic point of period 1. With
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352:
291:
154:
91:
223:(this is not to be confused with the notion of a
1240:
1233:Creative Commons Attribution/Share-Alike License
964:{\displaystyle \Phi :\mathbb {R} \times X\to X}
1221:Periodic points of complex quadratic mappings
110:is called periodic point if there exists an
1146:
1027:
945:
878:
207:is a periodic point with the same period
1185:are periodic with the same prime period.
411:{\displaystyle |f_{n}^{\prime }|\neq 1,}
1125:{\displaystyle \Phi (t,x)=\Phi (t+T,x)}
903:{\displaystyle (\mathbb {R} ,X,\Phi ),}
532:{\displaystyle |f_{n}^{\prime }|>1.}
474:{\displaystyle |f_{n}^{\prime }|<1,}
310:. All periodic points are preperiodic.
1241:
42:
857:
191:satisfying the above is called the
13:
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839:{\displaystyle {\tfrac {r-1}{r}}.}
513:
452:
389:
345:
14:
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758:{\displaystyle {\tfrac {r-1}{r}}}
292:{\displaystyle f_{n}(x)=f_{m}(x)}
680:
654:
578:A period-one point is called a
353:{\displaystyle f_{n}^{\prime }}
1231:, which is licensed under the
1119:
1101:
1092:
1080:
1031:{\displaystyle \Phi (T,x)=x\,}
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797:{\displaystyle 1+{\sqrt {6}},}
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1156:{\displaystyle \mathbb {R} .}
1056:
1045:with this property is called
769:greater than 3 but less than
864:real global dynamical system
155:{\displaystyle \ f_{n}(x)=x}
7:
1189:
573:
10:
1265:
846:As the value of parameter
554:; if the dimension of its
92:{\displaystyle f:X\to X,}
558:is zero, it is called a
230:If there exist distinct
184:. The smallest positive
1171:then all points on the
1167:Given a periodic point
1062:Given a periodic point
323:differentiable manifold
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1041:The smallest positive
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1211:Sharkovsky's theorem
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203:. If every point in
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926:evolution function
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43:Iterated functions
25:iterated functions
23:, in the study of
830:
789:
752:
724:all orbits). For
556:unstable manifold
308:preperiodic point
225:periodic function
126:
29:dynamical systems
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1216:Stationary point
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858:Dynamical system
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548:stable manifold
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325:, so that the
319:diffeomorphism
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168:
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114:>0 so that
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44:
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33:periodic point
15:
9:
6:
4:
3:
2:
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1234:
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1214:
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1199:
1197:
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1174:
1166:
1150:
1116:
1113:
1110:
1107:
1104:
1095:
1089:
1086:
1083:
1061:
1060:
1054:
1049:of the point
1048:
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791:
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430:
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402:
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384:
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363:
340:
336:
328:
324:
320:
311:
309:
283:
275:
271:
267:
261:
253:
249:
241:
240:
239:
228:
226:
218:
199:of the point
198:
194:
187:
179:
149:
146:
140:
132:
128:
117:
116:
115:
86:
83:
77:
74:
71:
64:
63:
62:
61:into itself,
57:
50:
40:
38:
34:
30:
26:
22:
1226:
1225:
1066:with period
1047:prime period
1046:
1040:
987:
983:
973:
861:
707:
590:
587:logistic map
584:
577:
568:saddle point
563:
559:
551:
541:
485:
483:
422:
420:
361:
312:
307:
306:is called a
301:
229:
219:with period
216:
197:least period
196:
193:prime period
192:
164:
101:
46:
32:
18:
1196:Limit cycle
918:phase space
718:0, 0, 0, …,
580:fixed point
421:that it is
21:mathematics
1249:Limit sets
1229:PlanetMath
1206:Stable set
1057:Properties
982:is called
484:and it is
424:attractive
362:hyperbolic
327:derivative
238:such that
215:is called
1201:Limit set
1099:Φ
1078:Φ
1004:Φ
956:→
950:×
939:Φ
892:Φ
821:−
743:−
691:≤
685:≤
672:≤
659:≤
636:−
544:dimension
514:′
486:repelling
453:′
400:≠
390:′
346:′
81:→
1243:Category
1190:See also
1181:through
1132:for all
984:periodic
974:a point
862:Given a
722:attracts
574:Examples
217:periodic
102:a point
47:Given a
37:function
1163:
1138:
1070:, then
910:
867:
804:
771:
546:of the
542:If the
211:, then
186:integer
178:iterate
172:is the
54:from a
49:mapping
1176:γ
988:period
720:which
712:. For
564:saddle
552:source
165:where
125:
1173:orbit
986:with
912:with
321:of a
317:is a
302:then
35:of a
924:the
920:and
916:the
585:The
560:sink
524:>
463:<
234:and
31:, a
27:and
1136:in
993:if
978:in
566:or
488:if
427:if
364:if
313:If
227:).
195:or
180:of
176:th
106:in
56:set
19:In
1245::
1053:.
928:,
582:.
570:.
527:1.
1235:.
1183:x
1178:x
1169:x
1151:.
1147:R
1134:t
1120:)
1117:x
1114:,
1111:T
1108:+
1105:t
1102:(
1096:=
1093:)
1090:x
1087:,
1084:t
1081:(
1068:T
1064:x
1051:x
1043:T
1025:x
1022:=
1019:)
1016:x
1013:,
1010:T
1007:(
991:T
980:X
976:x
959:X
953:X
946:R
942::
922:Φ
914:X
898:,
895:)
889:,
886:X
883:,
879:R
875:(
852:r
848:r
834:.
828:r
824:1
818:r
792:,
787:6
782:+
779:1
767:r
750:r
746:1
740:r
726:r
714:r
710:r
694:4
688:r
682:0
678:,
675:1
667:t
663:x
656:0
652:,
649:)
644:t
640:x
633:1
630:(
625:t
621:x
617:r
614:=
609:1
606:+
603:t
599:x
520:|
509:n
505:f
500:|
469:,
466:1
459:|
448:n
444:f
439:|
406:,
403:1
396:|
385:n
381:f
376:|
341:n
337:f
315:f
304:x
287:)
284:x
281:(
276:m
272:f
268:=
265:)
262:x
259:(
254:n
250:f
236:m
232:n
221:n
213:f
209:n
205:X
201:x
189:n
182:f
174:n
169:n
167:f
150:x
147:=
144:)
141:x
138:(
133:n
129:f
112:n
108:X
104:x
87:,
84:X
78:X
75::
72:f
59:X
52:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.