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1110:
25:
767:
1105:{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial x}}(0,r_{0})&=0,&{\frac {\partial ^{2}f}{\partial x^{2}}}(0,r_{0})&=0,&{\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{0})&\neq 0,\\{\frac {\partial f}{\partial r}}(0,r_{0})&=0,&{\frac {\partial ^{2}f}{\partial x\partial r}}(0,r_{0})&\neq 0.\end{aligned}}}
1303:
1308:
Note that subcritical and supercritical describe the stability of the outer lines of the pitchfork (dashed or solid, respectively) and are not dependent on which direction the pitchfork faces. For example, the negative of the first ODE above,
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772:
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1298:{\displaystyle {\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{0}){\begin{cases}<0,&{\text{supercritical}}\\>0,&{\text{subcritical}}\end{cases}}\,\,}
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where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like
1122:
1247:
120:
Supercritical case: solid lines represent stable points, while dotted line represents unstable one.
345:
Subcritical case: solid line represents stable point, while dotted lines represent unstable ones.
105:
35:
116:
1388:
Non-linear
Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering
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424:
251:
199:
8:
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538:
450:
277:
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1369:
90:
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104:—i.e. flows—pitchfork bifurcations occur generically in systems with
1404:
1360:, faces the same direction as the first picture but reverses the stability.
757:
1176:. The form of the pitchfork is given by the sign of the third derivative:
82:
49: in this section. Unsourced material may be challenged and removed.
16:
Bifurcation from a system having one fixed point to three fixed points
24:
1395:
Introduction to
Applied Nonlinear Dynamical Systems and Chaos
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is stable, and there are two unstable equilibria at
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128:of the supercritical pitchfork bifurcation is
100:In continuous dynamical systems described by
411:{\displaystyle {\frac {dx}{dt}}=rx+x^{3}.}
186:{\displaystyle {\frac {dx}{dt}}=rx-x^{3}.}
1294:
1293:
745:
744:
675:
612:
65:Learn how and when to remove this message
340:
115:
1403:
626:described by a one parameter function
222:, there is one stable equilibrium at
111:
564:
274:there is an unstable equilibrium at
47:adding citations to reliable sources
18:
1353:{\displaystyle {\dot {x}}=x^{3}-rx}
749:{\displaystyle -f(x,r)=f(-x,r)\,\,}
616:{\displaystyle {\dot {x}}=f(x,r)\,}
336:
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502:{\displaystyle x=\pm {\sqrt {-r}}}
14:
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682:{\displaystyle r\in \mathbb {R} }
326:{\displaystyle x=\pm {\sqrt {r}}}
23:
1169:{\displaystyle (x,r)=(0,r_{0})}
300:, and two stable equilibria at
34:needs additional citations for
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89:is a particular type of local
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353:for the subcritical case is
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1397:, Springer-Verlag, 1990.
1390:, Perseus Books, 2000.
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654:{\displaystyle f(x,r)}
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528:{\displaystyle r>0}
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440:{\displaystyle r<0}
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267:{\displaystyle r>0}
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215:{\displaystyle r<0}
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1117:pitchfork bifurcation
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87:pitchfork bifurcation
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43:improve this article
1375:Bifurcation diagram
554:{\displaystyle x=0}
535:the equilibrium at
466:{\displaystyle x=0}
447:the equilibrium at
293:{\displaystyle x=0}
241:{\displaystyle x=0}
1411:Bifurcation theory
1370:Bifurcation theory
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112:Supercritical case
79:bifurcation theory
1386:Steven Strogatz,
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565:Formal definition
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95:Hopf bifurcations
81:, a field within
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1263:supercritical
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845:
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827:
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645:
642:
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633:
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591:
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582:
572:
571:
570:
562:
561:is unstable.
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32:This section
30:
26:
21:
20:
1394:
1393:S. Wiggins,
1387:
1307:
1116:
1114:
758:odd function
689:satisfying:
625:
568:
420:
348:
195:
123:
99:
86:
76:
61:
55:October 2017
52:
41:Please help
36:verification
33:
1283:subcritical
351:normal form
126:normal form
91:bifurcation
83:mathematics
1381:References
1342:−
1323:˙
1205:∂
1191:∂
1093:≠
1058:∂
1052:∂
1038:∂
988:∂
980:∂
964:≠
922:∂
908:∂
851:∂
837:∂
787:∂
779:∂
730:−
700:−
672:∈
586:˙
492:−
487:±
314:±
168:−
1405:Category
1364:See also
106:symmetry
569:An ODE
1115:has a
509:. For
248:. For
661:with
1271:>
1251:<
520:>
432:<
349:The
259:>
207:<
196:For
124:The
102:ODEs
85:, a
1119:at
77:In
45:by
1407::
1096:0.
760:),
333:.
108:.
1348:x
1345:r
1337:3
1333:x
1329:=
1320:x
1277:,
1274:0
1257:,
1254:0
1245:{
1240:)
1235:0
1231:r
1227:,
1224:0
1221:(
1213:3
1209:x
1200:f
1195:3
1164:)
1159:0
1155:r
1151:,
1148:0
1145:(
1142:=
1139:)
1136:r
1133:,
1130:x
1127:(
1086:)
1081:0
1077:r
1073:,
1070:0
1067:(
1061:r
1055:x
1047:f
1042:2
1029:,
1026:0
1023:=
1016:)
1011:0
1007:r
1003:,
1000:0
997:(
991:r
983:f
970:,
967:0
957:)
952:0
948:r
944:,
941:0
938:(
930:3
926:x
917:f
912:3
899:,
896:0
893:=
886:)
881:0
877:r
873:,
870:0
867:(
859:2
855:x
846:f
841:2
828:,
825:0
822:=
815:)
810:0
806:r
802:,
799:0
796:(
790:x
782:f
742:)
739:r
736:,
733:x
727:(
724:f
721:=
718:)
715:r
712:,
709:x
706:(
703:f
676:R
669:r
649:)
646:r
643:,
640:x
637:(
634:f
610:)
607:r
604:,
601:x
598:(
595:f
592:=
583:x
549:0
546:=
543:x
523:0
517:r
495:r
484:=
481:x
461:0
458:=
455:x
435:0
429:r
406:.
401:3
397:x
393:+
390:x
387:r
384:=
378:t
375:d
370:x
367:d
319:r
311:=
308:x
288:0
285:=
282:x
262:0
256:r
236:0
233:=
230:x
210:0
204:r
181:.
176:3
172:x
165:x
162:r
159:=
153:t
150:d
145:x
142:d
68:)
62:(
57:)
53:(
39:.
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