878:
682:
866:
1944:
1001:
1936:
877:
127:
681:
621:. Homoclinic bifurcations can occur supercritically or subcritically. The variant above is the "small" or "type I" homoclinic bifurcation. In 2D there is also the "big" or "type II" homoclinic bifurcation in which the homoclinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. In three or more dimensions, higher codimension bifurcations can occur, producing complicated, possibly
31:
980:
and a number of theoretical examples which are difficult to access experimentally such as the kicked top and coupled quantum wells. The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits
522:
has an eigenvalue with modulus equal to one. If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to â1, it is a period-doubling (or flip) bifurcation, and otherwise,
134:
A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (described by maps), this corresponds to a fixed
956:
of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the codimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Hopf bifurcations are the only generic local
653:
of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period bifurcation occurs at this critical value. Beyond the critical value, the two fixed points emerge continuously from each other on the limit cycle to
602:
Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations. In fact, the changes in
635:. Heteroclinic bifurcations are of two types: resonance bifurcations and transverse bifurcations. Both types of bifurcation will result in the change of stability of the heteroclinic cycle. At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the
989:. Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations.
143:
at the bifurcation point. The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local').
240:
957:
bifurcations which are really codimension-one (the others all having higher codimension). However, transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter.
573:
A phase portrait before, at, and after a homoclinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a
643:. A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. This will also cause a change in stability of the heteroclinic cycle.
77:
occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by
116:
Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed
150:
520:
344:
420:
805:
742:
570:
557:
466:
286:
857:
350:
with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely imaginary, this is a
936:
831:
1445:
Stamatiou, G. & Ghikas, D. P. K. (2007). "Quantum entanglement dependence on bifurcations and scars in non-autonomous systems. The case of quantum kicked top".
901:
1498:
Galan, J.; Freire, E. (1999). "Chaos in a Mean Field Model of
Coupled Quantum Wells; Bifurcations of Periodic Orbits in a Symmetric Hamiltonian System".
1332:
Founargiotakis, M.; Farantos, S. C.; Skokos, Ch.; Contopoulos, G. (1997). "Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2".
972:
Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems, molecular systems, and
594:: As the bifurcation parameter increases, the limit cycle grows until it exactly intersects the saddle point, yielding an orbit of infinite duration.
977:
903:
increases from zero, a stable limit cycle emerges out of the origin via Hopf bifurcation. Here we plot the limit cycle parametrically, up to order
1410:
Wieczorek, S.; Krauskopf, B.; Simpson, T. B. & Lenstra, D. (2005). "The dynamical complexity of optically injected semiconductor lasers".
2233:
2063:
865:
1226:
Peters, A. D.; Jaffé, C.; Delos, J. B. (1994). "Quantum
Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model".
1669:
1664:
360:
1367:
Monteiro, T. S. & Saraga, D. S. (2001). "Quantum Wells in Tilted Fields:Semiclassical
Amplitudes and Phase Coherence Times".
1943:
1703:
1675:
347:
1648:
1617:
1585:
1076:
1896:
1843:
471:
295:
2176:
2411:
1150:
1125:
17:
2218:
1906:
1181:"Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electric fields"
2093:
1278:
289:
147:
More technically, consider the continuous dynamical system described by the ordinary differential equation (ODE)
747:
130:
Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.
79:
1911:
1277:
Courtney, Michael; Jiao, Hong; Spellmeyer, Neal; Kleppner, Daniel; Gao, J.; Delos, J. B.; et al. (1995).
690:
235:{\displaystyle {\dot {x}}=f(x,\lambda )\quad f\colon \mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}.}
1901:
87:
1533:
Kleppner, D.; Delos, J. B. (2001). "Beyond quantum mechanics: Insights from the work of Martin
Gutzwiller".
2258:
2166:
2031:
636:
425:
245:
2510:
961:
546:
83:
109:
Local bifurcations, which can be analysed entirely through changes in the local stability properties of
2515:
2171:
1696:
669:
2238:
1627:
2286:
836:
536:
639:
of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a
1990:
973:
1935:
2151:
1916:
1823:
530:
1891:
906:
2191:
1803:
113:, periodic orbits or other invariant sets as parameters cross through critical thresholds; and
2341:
2248:
2046:
1873:
1808:
1783:
1689:
1631:
1044:
939:
541:
62:
810:
668:
Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g.
2351:
2103:
2003:
1848:
1542:
1507:
1464:
1419:
1376:
1341:
1290:
1235:
1192:
1034:
1029:
661:
2181:
1630:. "Journal of differential equations", Febrer 2011, vol. 250, nĂșm. 4, pp. 1967â2023.
886:
598:: When the bifurcation parameter increases further, the limit cycle disappears completely.
8:
2311:
2268:
2253:
2098:
2051:
2036:
2021:
1921:
1828:
1813:
1798:
1014:
650:
1853:
1546:
1511:
1468:
1423:
1380:
1345:
1294:
1239:
1196:
2489:
2356:
2186:
2073:
2068:
1838:
1558:
1480:
1454:
1392:
1314:
1259:
1208:
1024:
1019:
1006:
632:
136:
1519:
1353:
649:
in which a stable node and saddle point simultaneously occur on a limit cycle. As the
2361:
2326:
2316:
2213:
1833:
1755:
1644:
1613:
1581:
1562:
1396:
1306:
1251:
1212:
1146:
1121:
1072:
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1000:
982:
110:
70:
2376:
1484:
1318:
94:
2469:
2381:
2331:
2228:
2156:
2108:
1985:
1970:
1965:
1760:
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1515:
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1427:
1384:
1349:
1298:
1263:
1243:
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575:
552:
351:
50:
1431:
2401:
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2223:
2056:
1868:
1858:
1638:
1605:
1109:
2391:
2336:
1302:
1247:
1165:
James P. Keener, "Infinite Period
Bifurcation and Global Bifurcation Branches",
2484:
2451:
2446:
2441:
2243:
2133:
2128:
2026:
1975:
1765:
1117:
1039:
640:
631:
in which a limit cycle collides with two or more saddle points; they involve a
54:
1554:
1388:
27:
Study of sudden qualitative behavior changes caused by small parameter changes
2504:
2479:
2436:
2426:
2421:
2321:
2301:
2161:
2083:
1980:
1788:
986:
1204:
2431:
2396:
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2118:
2113:
1712:
1310:
1255:
655:
622:
618:
583:
58:
2078:
1180:
2416:
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2041:
1863:
1770:
1459:
953:
614:
587:
66:
46:
42:
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1818:
1331:
603:
topology extend out to an arbitrarily large distance (hence 'global').
2386:
2346:
2088:
1750:
1735:
967:
2123:
97:
in 1885 in the first paper in mathematics showing such a behavior.
1628:
Generic bifurcations of low codimension of planar
Filippov Systems
126:
90:
differential equations) and discrete systems (described by maps).
1793:
1745:
569:
1409:
1093:
L'Ăquilibre d'une masse fluide animĂ©e d'un mouvement de rotation
960:
An example of a well-studied codimension-two bifurcation is the
105:
It is useful to divide bifurcations into two principal classes:
2366:
1681:
1276:
578:. After the bifurcation there is no longer a periodic orbit.
139:
with modulus equal to one. In both cases, the equilibrium is
30:
976:. Bifurcation theory has also been applied to the study of
833:, around the origin. A homoclinic bifurcation occurs around
664:
in which a limit cycle collides with a nonhyperbolic cycle.
1626:
Guardia, M.; Martinez-Seara, M.; Teixeira, M. A. (2011).
1279:"Closed Orbit Bifurcations in Continuum Stark Spectra"
909:
889:
839:
813:
750:
693:
474:
428:
363:
298:
248:
153:
1062:
996:
357:
For discrete dynamical systems, consider the system
1169:, Vol. 41, No. 1 (August, 1981), pp. 127–144.
1143:
Bifurcation Theory and
Methods of Dynamical Systems
515:{\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}
339:{\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}
930:
895:
851:
825:
799:
736:
514:
460:
414:
338:
280:
234:
1640:Global bifurcations and Chaos: Analytical Methods
1603:
968:Applications in semiclassical and quantum physics
2502:
1225:
1444:
947:
93:The name "bifurcation" was first introduced by
1366:
871:A detailed view of the homoclinic bifurcation.
34:Phase portrait showing saddle-node bifurcation
1697:
1532:
938:. The exact computation is explained on the
415:{\displaystyle x_{n+1}=f(x_{n},\lambda )\,.}
1497:
1704:
1690:
1575:
800:{\displaystyle {\dot {y}}=-x+\mu y+2x^{2}}
1610:Bifurcation Theory and Catastrophe Theory
1458:
606:Examples of global bifurcations include:
582:: For small parameter values, there is a
408:
219:
210:
196:
1578:Chaos in Classical and Quantum Mechanics
1178:
1108:
737:{\displaystyle {\dot {x}}=\mu x+y-x^{2}}
687:A Hopf bifurcation occurs in the system
568:
526:Examples of local bifurcations include:
125:
29:
1636:
45:study of changes in the qualitative or
14:
2503:
1670:Bifurcations and Two Dimensional Flows
564:
1685:
1071:. London: Thompson. pp. 96â111.
654:disrupt the oscillation and form two
121:
461:{\displaystyle (x_{0},\lambda _{0})}
281:{\displaystyle (x_{0},\lambda _{0})}
100:
1844:Measure-preserving dynamical system
1726:
1167:SIAM Journal on Applied Mathematics
1140:
422:Then a local bifurcation occurs at
61:, and the solutions of a family of
24:
1676:Introduction to Bifurcation theory
985:points out in his classic work on
25:
2527:
2412:Oleksandr Mykolayovych Sharkovsky
1658:
1942:
1934:
1711:
1145:. World Scientific. p. 26.
1099:, vol.7, pp. 259-380, Sept 1885.
999:
876:
864:
680:
1569:
1526:
1500:Reports on Mathematical Physics
1491:
1438:
1403:
1360:
852:{\displaystyle \mu =0.06605695}
187:
65:. Most commonly applied to the
2177:RabinovichâFabrikant equations
1576:Gutzwiller, Martin C. (1990).
1477:10.1016/j.physleta.2007.04.003
1325:
1270:
1219:
1179:Gao, J.; Delos, J. B. (1997).
1172:
1159:
1134:
1102:
1085:
1056:
455:
429:
405:
386:
275:
249:
242:A local bifurcation occurs at
214:
184:
172:
13:
1:
1632:DOI:10.1016/j.jde.2010.11.016
1597:
1580:. New York: Springer-Verlag.
1520:10.1016/S0034-4877(99)80148-7
1432:10.1016/j.physrep.2005.06.003
1354:10.1016/S0009-2614(97)00931-7
1114:Nonlinear Dynamics and Chaos
948:Codimension of a bifurcation
560:(secondary Hopf) bifurcation
7:
1912:Poincaré recurrence theorem
1303:10.1103/PhysRevLett.74.1538
1248:10.1103/PhysRevLett.73.2825
992:
962:BogdanovâTakens bifurcation
647:Infinite-period bifurcation
10:
2532:
1907:PoincarĂ©âBendixson theorem
931:{\displaystyle \mu ^{3/2}}
523:it is a Hopf bifurcation.
2460:
2277:
2259:Swinging Atwood's machine
2204:
2142:
2012:
1999:
1951:
1932:
1902:KrylovâBogolyubov theorem
1882:
1779:
1719:
1637:Wiggins, Stephen (1988).
974:resonant tunneling diodes
537:Transcritical bifurcation
2167:LotkaâVolterra equations
1991:Synchronization of chaos
1794:axiom A dynamical system
1334:Chemical Physics Letters
1050:
676:Examples of bifurcations
629:Heteroclinic bifurcation
2152:Double scroll attractor
1917:Stable manifold theorem
1824:False nearest neighbors
1555:10.1023/A:1017512925106
1389:10.1023/A:1017546721313
1205:10.1103/PhysRevA.56.356
590:in the first quadrant.
2192:Van der Pol oscillator
2172:MackeyâGlass equations
1804:Box-counting dimension
1678:by John David Crawford
1643:. New York: Springer.
1608:; et al. (1994).
1535:Foundations of Physics
1369:Foundations of Physics
1069:Differential Equations
1067:; Hall, G. R. (2006).
932:
897:
853:
827:
826:{\displaystyle \mu =0}
801:
738:
611:Homoclinic bifurcation
599:
516:
462:
416:
340:
282:
236:
131:
63:differential equations
35:
2342:Svetlana Jitomirskaya
2249:Multiscroll attractor
2094:Interval exchange map
2047:Dyadic transformation
2032:Complex quadratic map
1874:Topological conjugacy
1809:Correlation dimension
1784:Anosov diffeomorphism
1141:Luo, Dingjun (1997).
1045:Tennis racket theorem
933:
898:
854:
828:
802:
739:
572:
542:Pitchfork bifurcation
517:
463:
417:
341:
283:
237:
129:
49:structure of a given
33:
2352:Edward Norton Lorenz
1604:Afrajmovich, V. S.;
1035:Geomagnetic reversal
1030:Feigenbaum constants
907:
896:{\displaystyle \mu }
887:
837:
811:
748:
691:
662:Blue sky catastrophe
586:at the origin and a
472:
426:
361:
296:
246:
151:
2312:Mitchell Feigenbaum
2254:Population dynamics
2239:HĂ©nonâHeiles system
2099:Irrational rotation
2052:Dynamical billiards
2037:Coupled map lattice
1897:Liouville's theorem
1829:Hausdorff dimension
1814:Conservative system
1799:Bifurcation diagram
1547:2001FoPh...31..593K
1512:1999RpMP...44...87G
1469:2007PhLA..368..206S
1424:2005PhR...416....1W
1381:2001FoPh...31..355M
1346:1997CPL...277..456F
1295:1995PhRvL..74.1538C
1240:1994PhRvL..73.2825P
1197:1997PhRvA..56..356G
1110:Strogatz, Steven H.
1015:Bifurcation diagram
565:Global bifurcations
2511:Bifurcation theory
2490:Santa Fe Institute
2357:Aleksandr Lyapunov
2187:Three-body problem
2074:Gingerbreadman map
1961:Bifurcation theory
1839:Lyapunov stability
1665:Nonlinear dynamics
1025:Catastrophe theory
1020:Bifurcation memory
1007:Mathematics portal
981:becomes large, as
928:
893:
849:
823:
797:
734:
633:heteroclinic cycle
600:
549:(flip) bifurcation
533:(fold) bifurcation
512:
458:
412:
336:
278:
232:
137:Floquet multiplier
132:
122:Local bifurcations
39:Bifurcation theory
36:
2516:Nonlinear systems
2498:
2497:
2362:BenoĂźt Mandelbrot
2327:Martin Gutzwiller
2317:Peter Grassberger
2200:
2199:
2182:Rössler attractor
1930:
1929:
1834:Invariant measure
1756:Lyapunov exponent
1672:by Elmer G. Wiens
1650:978-0-387-96775-2
1619:978-3-540-65379-0
1587:978-0-387-97173-5
1447:Physics Letters A
1234:(21): 2825â2828.
1091:Henri Poincaré. "
1078:978-0-495-01265-8
983:Martin Gutzwiller
760:
703:
479:
303:
163:
101:Bifurcation types
71:dynamical systems
18:Bifurcation point
16:(Redirected from
2523:
2470:Butterfly effect
2382:Itamar Procaccia
2332:Brosl Hasslacher
2229:Elastic pendulum
2157:Duffing equation
2104:KaplanâYorke map
2022:Arnold's cat map
2010:
2009:
1986:Stability theory
1971:Dynamical system
1966:Control of chaos
1946:
1938:
1922:Takens's theorem
1854:Poincaré section
1724:
1723:
1706:
1699:
1692:
1683:
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1523:
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1462:
1460:quant-ph/0702172
1453:(3â4): 206â214.
1442:
1436:
1435:
1407:
1401:
1400:
1364:
1358:
1357:
1340:(5â6): 456â464.
1329:
1323:
1322:
1289:(9): 1538â1541.
1274:
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1267:
1223:
1217:
1216:
1176:
1170:
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1138:
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1106:
1100:
1097:Acta Mathematica
1089:
1083:
1082:
1060:
1009:
1004:
1003:
940:Hopf bifurcation
937:
935:
934:
929:
927:
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684:
617:collides with a
576:homoclinic orbit
553:Hopf bifurcation
521:
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352:Hopf bifurcation
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51:family of curves
21:
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2520:
2501:
2500:
2499:
2494:
2462:
2456:
2402:Caroline Series
2297:Mary Cartwright
2279:
2273:
2224:Double pendulum
2206:
2196:
2145:
2138:
2064:Exponential map
2015:
2001:
1995:
1953:
1947:
1940:
1926:
1892:Ergodic theorem
1885:
1878:
1869:Stable manifold
1859:Recurrence plot
1775:
1729:
1715:
1710:
1661:
1651:
1620:
1600:
1595:
1588:
1574:
1570:
1531:
1527:
1496:
1492:
1443:
1439:
1412:Physics Reports
1408:
1404:
1365:
1361:
1330:
1326:
1283:Phys. Rev. Lett
1275:
1271:
1228:Phys. Rev. Lett
1224:
1220:
1177:
1173:
1164:
1160:
1153:
1139:
1135:
1128:
1120:. p. 262.
1107:
1103:
1090:
1086:
1079:
1063:Blanchard, P.;
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1005:
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724:
695:
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689:
688:
685:
567:
547:Period-doubling
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135:point having a
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57:of a family of
55:integral curves
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2485:Predictability
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2452:Lai-Sang Young
2449:
2447:James A. Yorke
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2244:Kicked rotator
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2214:Chua's circuit
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2134:Zaslavskii map
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2129:Tinkerbell map
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1976:Ergodic theory
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1659:External links
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1541:(4): 593â612.
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1506:(1â2): 87â94.
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1418:(1â2): 1â128.
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1375:(2): 355â370.
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1191:(1): 356â364.
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1118:Addison-Wesley
1101:
1084:
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1065:Devaney, R. L.
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1047:
1042:
1040:Phase portrait
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994:
991:
978:laser dynamics
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641:periodic orbit
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95:Henri Poincaré
53:, such as the
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2480:Edge of chaos
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2427:Audrey Terras
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2302:Chen Guanrong
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1981:Quantum chaos
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1152:981-02-2094-4
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987:quantum chaos
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656:saddle points
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59:vector fields
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48:
44:
40:
32:
19:
2432:Mary Tsingou
2397:David Ruelle
2392:Otto Rössler
2337:Michel HĂ©non
2307:Leon O. Chua
2264:Tilt-A-Whirl
2234:FPUT problem
2119:Standard map
2114:Logistic map
1960:
1939:
1740:
1713:Chaos theory
1639:
1612:. Springer.
1609:
1577:
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1185:Phys. Rev. A
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619:saddle point
610:
605:
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592:Middle panel
591:
584:saddle point
579:
525:
356:
146:
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133:
104:
92:
74:
67:mathematical
43:mathematical
38:
37:
2417:Nina Snaith
2407:Yakov Sinai
2292:Rufus Bowen
2042:Duffing map
2027:Baker's map
1952:Theoretical
1864:SRB measure
1771:Phase space
1741:Bifurcation
954:codimension
637:eigenvalues
615:limit cycle
613:in which a
596:Right panel
588:limit cycle
531:Saddle-node
75:bifurcation
47:topological
2505:Categories
2475:Complexity
2372:Edward Ott
2219:Convection
2144:Continuous
1819:Ergodicity
1598:References
847:0.06605695
580:Left panel
348:eigenvalue
111:equilibria
2387:Mary Rees
2347:Bryna Kra
2280:theorists
2089:Ikeda map
2079:HĂ©non map
2069:Gauss map
1751:Limit set
1736:Attractor
1563:116944147
1397:120968155
1213:120255640
912:μ
891:μ
841:μ
815:μ
776:μ
767:−
758:˙
722:−
710:μ
701:˙
625:dynamics.
502:λ
447:λ
403:λ
326:λ
267:λ
215:→
207:×
192::
182:λ
161:˙
69:study of
2463:articles
2205:Physical
2124:Tent map
2014:Discrete
1954:branches
1884:Theorems
1720:Concepts
1485:15562617
1319:21573702
1311:10059054
1256:10057205
1112:(1994).
993:See also
290:Jacobian
117:points).
80:ordinary
2461:Related
2269:Weather
2207:systems
2000:Chaotic
1746:Fractal
1543:Bibcode
1508:Bibcode
1465:Bibcode
1420:Bibcode
1377:Bibcode
1342:Bibcode
1291:Bibcode
1264:1641622
1236:Bibcode
1193:Bibcode
807:, when
623:chaotic
346:has an
292:matrix
288:if the
88:partial
41:is the
2367:Hee Oh
2002:maps (
1849:Mixing
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1124:
1075:
670:crises
2278:Chaos
2057:outer
1761:Orbit
1559:S2CID
1481:S2CID
1455:arXiv
1393:S2CID
1315:S2CID
1260:S2CID
1209:S2CID
1051:Notes
942:page.
651:limit
84:delay
2004:list
1728:Core
1645:ISBN
1614:ISBN
1582:ISBN
1307:PMID
1252:PMID
1147:ISBN
1122:ISBN
1073:ISBN
952:The
744:and
73:, a
1551:doi
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