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with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many properties of periodic and quasiperiodic orbits of the original system and has a lower-dimensional state space, it is often used for analyzing the original system in a simpler
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184:
of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a
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is a recurrence plot; the locus of the Moon when it passes through the plane perpendicular to the Earth's orbit and passing through the Sun and the Earth at perihelion is a
Poincaré map. It was used by
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1252:{\displaystyle r(t)={\sqrt {\frac {e^{2t}r_{0}^{2}}{1+r_{0}^{2}(e^{2t}-1)}}}={\sqrt {\frac {1}{1+e^{-2t}\left({\frac {1}{r_{0}^{2}}}-1\right)}}}}
507:
193:. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.
221:, because the path of a star projected onto a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.
2106:
of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding
Poincaré map.
892:
97:
1396:{\displaystyle \Phi _{t}(\theta ,r)=\left(\theta +t,{\sqrt {\frac {1}{1+e^{-2t}\left({\frac {1}{r_{0}^{2}}}-1\right)}}}\right)}
69:
994:
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76:
876:{\displaystyle \int {\frac {1}{(1-r^{2})r}}dr=\int dt\Longrightarrow \log \left({\frac {r}{\sqrt {1-r^{2}}}}\right)=t+c}
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in, that space, not time, determines when to plot a point. For instance, the locus of the Moon when the Earth is at
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1678:(this can be understood by looking at the evolution of the angle): we can take as Poincaré map the restriction of
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way. In practice this is not always possible as there is no general method to construct a
Poincaré map.
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1874:{\displaystyle \Psi (r)={\sqrt {\frac {1}{1+e^{-4\pi }\left({\frac {1}{r^{2}}}-1\right)}}}}
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We can take as
Poincaré section for this flow the positive horizontal axis, namely
656:{\displaystyle {\begin{cases}{\dot {\theta }}=1\\{\dot {r}}=(1-r^{2})r\end{cases}}}
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Consider the following system of differential equations in polar coordinates,
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The flow of the system can be obtained by integrating the equation: for the
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559:{\displaystyle (\theta ,r)\in \mathbb {S} ^{1}\times \mathbb {R} ^{+}}
24:
978:{\displaystyle r(t)={\sqrt {\frac {e^{2(t+c)}}{1+e^{2(t+c)}}}}}
229:
218:
16:
Type of map used in mathematics, particularly dynamical systems
2571:
Poincare Map and its application to 'Spinning Magnet' problem
1883:
The behaviour of the orbits of the discrete dynamical system
2493:
of the discrete dynamical system is asymptotically stable.
2485:
The periodic orbit γ of the continuous dynamical system is
2474:
The periodic orbit γ of the continuous dynamical system is
754:
component we need to separate the variables and integrate:
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2018:
Every other point tends monotonically to the equilibrium,
2093:
1544:
draws a spiral that tends towards the radius 1 circle.
1056:{\displaystyle r(0)={\sqrt {\frac {e^{2c}}{1+e^{2c}}}}}
189:
to send the first point to the second, hence the name
172:
with a certain lower-dimensional subspace, called the
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1602:{\displaystyle \Sigma =\{(\theta ,r)\ :\ \theta =0\}}
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be a local differentiable and transversal section of
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2414:) is a discrete dynamical system with state space
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133:A two-dimensional Poincaré section of the forced
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2467:Per definition this system has a fixed point at
2207:{\displaystyle P^{0}:=\operatorname {id} _{U}}
1629:as coordinate on the section. Every point in
1430:increases monotonically and at constant rate.
2482:of the discrete dynamical system is stable.
2457:{\displaystyle P:\mathbb {Z} \times U\to U.}
2329:{\displaystyle P^{-n-1}:=P^{-1}\circ P^{-n}}
1916:{\displaystyle (\Sigma ,\mathbb {Z} ,\Psi )}
1596:
1560:
1406:The behaviour of the flow is the following:
53:. Unsourced material may be challenged and
2167:be the corresponding Poincaré map through
1492:Therefore, the solution with initial data
2435:
1900:
1537:{\displaystyle (\theta _{0},r_{0}\neq 1)}
546:
531:
117:Learn how and when to remove this message
1776:{\displaystyle \Phi _{2\pi }|_{\Sigma }}
727:{\displaystyle \theta (t)=\theta _{0}+t}
228:
128:
196:A Poincaré map can be interpreted as a
2580:
2098:Poincaré maps can be interpreted as a
1783:. The Poincaré map is therefore :
2260:{\displaystyle P^{n+1}:=P\circ P^{n}}
2094:Poincaré maps and stability analysis
1649:returns to the section after a time
1262:The flow of the system is therefore
51:adding citations to reliable sources
18:
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217:to study the motion of stars in a
14:
2604:
2562:
2053:{\displaystyle \Psi ^{n}(z)\to 1}
2392:{\displaystyle P(n,x):=P^{n}(x)}
2082:{\displaystyle n\to \pm \infty }
886:Inverting last expression gives
23:
2489:if and only if the fixed point
2478:if and only if the fixed point
2123:differentiable dynamical system
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2125:with periodic orbit γ through
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1988:{\displaystyle \Psi ^{n}(1)=1}
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204:A Poincaré map differs from a
1:
2555:American Mathematical Society
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2523:Mironenko reflecting function
224:
1481:{\displaystyle {\bar {r}}=1}
328:Given an open and connected
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170:continuous dynamical system
160:, is the intersection of a
10:
2609:
499:
350:{\displaystyle U\subset S}
2100:discrete dynamical system
1453:tends to the equilibrium
686:component we simply have
198:discrete dynamical system
2157:{\displaystyle P:U\to S}
392:{\displaystyle P:U\to S}
233:In the Poincaré section
2418:and evolution function
1711:{\displaystyle \Sigma }
1671:{\displaystyle t=2\pi }
1642:{\displaystyle \Sigma }
1609:: obviously we can use
1423:{\displaystyle \theta }
679:{\displaystyle \theta }
441:) is a neighborhood of
406:for the orbit γ on the
271:global dynamical system
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487:for the first time at
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1734:{\displaystyle 2\pi }
1718:computed at the time
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1691:{\displaystyle \Phi }
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191:first recurrence map
150:first recurrence map
47:improve this article
2568:Shivakumar Jolad,
2503:Poincaré recurrence
1946:{\displaystyle r=1}
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477:positive semi-orbit
237:, the Poincaré map
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413:through the point
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295:evolution function
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144:, particularly in
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2588:Dynamical systems
2528:Invariant measure
2008:{\displaystyle n}
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1622:{\displaystyle r}
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1446:{\displaystyle r}
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747:{\displaystyle r}
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241:projects a point
146:dynamical systems
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2508:Stroboscopic map
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467:for every point
408:Poincaré section
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319:Poincaré section
301:through a point
174:Poincaré section
135:Duffing equation
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2518:Recurrence plot
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206:recurrence plot
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2593:Henri Poincaré
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2563:External links
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2541:Teschl, Gerald
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1009:
1006:
1003:
1000:
986:
985:
968:
965:
962:
959:
956:
953:
949:
945:
942:
936:
933:
930:
927:
924:
921:
917:
910:
907:
904:
901:
898:
884:
883:
872:
869:
866:
863:
859:
851:
847:
843:
840:
836:
831:
827:
824:
821:
818:
815:
812:
809:
806:
803:
797:
794:
789:
785:
781:
778:
775:
771:
766:
743:
734:while for the
723:
720:
715:
711:
707:
704:
701:
698:
695:
675:
664:
663:
650:
645:
642:
637:
633:
629:
626:
623:
620:
614:
611:
605:
604:
601:
598:
592:
589:
583:
582:
580:
553:
548:
543:
538:
533:
528:
525:
522:
519:
516:
513:
501:
498:
497:
496:
465:
463:diffeomorphism
432:
400:
399:
388:
385:
382:
379:
376:
346:
343:
340:
299:periodic orbit
226:
223:
162:periodic orbit
158:Henri Poincaré
156:, named after
125:
124:
66:"Poincaré map"
31:
29:
22:
15:
9:
6:
4:
3:
2:
2605:
2594:
2591:
2589:
2586:
2585:
2583:
2573:
2572:
2567:
2566:
2556:
2552:
2548:
2547:
2542:
2538:
2537:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2500:
2494:
2492:
2488:
2483:
2481:
2477:
2472:
2470:
2451:
2448:
2442:
2439:
2431:
2428:
2421:
2420:
2419:
2417:
2413:
2409:
2405:
2383:
2375:
2371:
2367:
2361:
2358:
2355:
2349:
2342:
2341:
2340:
2321:
2318:
2314:
2310:
2305:
2302:
2298:
2294:
2289:
2286:
2283:
2280:
2276:
2268:
2252:
2248:
2244:
2241:
2238:
2233:
2230:
2227:
2223:
2215:
2199:
2195:
2191:
2186:
2182:
2174:
2173:
2172:
2170:
2151:
2145:
2142:
2139:
2132:
2131:
2130:
2128:
2124:
2120:
2116:
2112:
2107:
2105:
2101:
2073:
2067:
2047:
2038:
2030:
2017:
2002:
1982:
1979:
1973:
1965:
1953:is fixed, so
1940:
1937:
1934:
1926:
1925:
1924:
1904:
1896:
1881:
1863:
1859:
1856:
1849:
1845:
1841:
1835:
1829:
1826:
1823:
1819:
1815:
1812:
1808:
1802:
1796:
1756:
1753:
1728:
1725:
1665:
1662:
1659:
1656:
1616:
1593:
1590:
1587:
1581:
1572:
1569:
1566:
1557:
1545:
1528:
1525:
1520:
1516:
1512:
1507:
1503:
1475:
1472:
1463:
1440:
1432:
1417:
1409:
1408:
1407:
1389:
1380:
1376:
1373:
1366:
1361:
1357:
1353:
1347:
1341:
1338:
1335:
1331:
1327:
1324:
1320:
1314:
1311:
1308:
1305:
1301:
1297:
1291:
1288:
1285:
1277:
1265:
1264:
1263:
1241:
1237:
1234:
1227:
1222:
1218:
1214:
1208:
1202:
1199:
1196:
1192:
1188:
1185:
1181:
1175:
1165:
1162:
1157:
1154:
1150:
1141:
1136:
1132:
1128:
1125:
1118:
1113:
1109:
1103:
1100:
1096:
1088:
1082:
1076:
1069:
1068:
1067:
1044:
1041:
1037:
1033:
1030:
1024:
1021:
1017:
1010:
1004:
998:
991:
990:
989:
963:
960:
957:
951:
947:
943:
940:
931:
928:
925:
919:
915:
908:
902:
896:
889:
888:
887:
870:
867:
864:
861:
857:
849:
845:
841:
838:
834:
829:
825:
822:
816:
813:
810:
807:
804:
801:
795:
787:
783:
779:
776:
769:
764:
757:
756:
755:
741:
721:
718:
713:
709:
705:
699:
693:
673:
643:
635:
631:
627:
624:
618:
612:
609:
599:
596:
590:
587:
578:
569:
568:
567:
551:
541:
536:
526:
520:
517:
514:
494:
490:
486:
482:
478:
474:
470:
466:
464:
460:
456:
452:
448:
444:
440:
436:
433:
431:
427:
423:
420:
419:
418:
416:
412:
409:
405:
386:
380:
377:
374:
367:
366:
365:
364:
360:
344:
341:
338:
331:
326:
324:
320:
316:
312:
308:
304:
300:
297:. Let γ be a
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
252:
248:
244:
240:
236:
231:
222:
220:
216:
211:
207:
202:
199:
194:
192:
188:
183:
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
136:
131:
121:
118:
110:
107:December 2020
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
48:
42:
41:
37:
32:This article
30:
26:
21:
20:
2569:
2545:
2490:
2484:
2479:
2473:
2468:
2466:
2415:
2411:
2407:
2403:
2401:
2338:
2171:. We define
2168:
2166:
2126:
2118:
2114:
2110:
2108:
2097:
1882:
1546:
1491:
1405:
1261:
1065:
987:
885:
665:
503:
492:
488:
484:
480:
472:
468:
458:
454:
450:
446:
442:
438:
434:
429:
425:
421:
414:
410:
407:
404:Poincaré map
403:
401:
358:
330:neighborhood
327:
322:
318:
314:
310:
306:
302:
290:
282:
279:real numbers
274:
266:
262:
258:
256:
250:
246:
242:
238:
234:
215:Michel Hénon
203:
195:
190:
173:
154:Poincaré map
153:
149:
139:
113:
104:
94:
87:
80:
73:
61:
45:Please help
33:
1433:The radius
988:and since
483:intersects
317:, called a
287:phase space
178:transversal
166:state space
142:mathematics
2582:Categories
2551:Providence
2534:References
1995:for every
1927:The point
1410:The angle
402:is called
225:Definition
210:perihelion
77:newspapers
2513:Hénon map
2446:→
2440:×
2319:−
2311:∘
2303:−
2287:−
2281:−
2245:∘
2149:→
2104:stability
2077:∞
2074:±
2071:→
2045:→
2027:Ψ
1962:Ψ
1908:Ψ
1894:Σ
1857:−
1830:π
1824:−
1791:Ψ
1769:Σ
1757:π
1750:Φ
1729:π
1706:Σ
1686:Φ
1666:π
1637:Σ
1588:θ
1567:θ
1555:Σ
1526:≠
1504:θ
1467:¯
1418:θ
1374:−
1336:−
1306:θ
1286:θ
1274:Φ
1235:−
1197:−
1163:−
842:−
826:
820:⟹
811:∫
780:−
765:∫
710:θ
694:θ
674:θ
628:−
613:˙
591:˙
588:θ
542:×
527:∈
515:θ
384:→
342:⊂
34:does not
2574:, (2005)
2497:See also
1066:we find
363:function
321:through
313:through
2121:) be a
500:Example
461:) is a
273:, with
269:) be a
180:to the
164:in the
91:scholar
55:removed
40:sources
2476:stable
2402:then (
2129:. Let
2102:. The
1585:
1579:
475:, the
219:galaxy
93:
86:
79:
72:
64:
2339:and
2109:Let (
257:Let (
168:of a
98:JSTOR
84:books
2060:for
445:and
428:) =
361:, a
305:and
293:the
289:and
285:the
277:the
182:flow
148:, a
70:news
38:any
36:cite
823:log
479:of
471:in
417:if
357:of
187:map
152:or
140:In
49:by
2584::
2553::
2549:.
2543:.
2471:.
2410:,
2406:,
2368::=
2295::=
2239::=
2196:id
2192::=
2117:,
2113:,
1741:,
566::
453:→
325:.
281:,
265:,
261:,
253:).
176:,
2557:.
2491:p
2480:p
2469:p
2452:.
2449:U
2443:U
2436:Z
2432::
2429:P
2416:U
2412:P
2408:U
2404:Z
2387:)
2384:x
2381:(
2376:n
2372:P
2365:)
2362:x
2359:,
2356:n
2353:(
2350:P
2322:n
2315:P
2306:1
2299:P
2290:1
2284:n
2277:P
2253:n
2249:P
2242:P
2234:1
2231:+
2228:n
2224:P
2200:U
2187:0
2183:P
2169:p
2152:S
2146:U
2143::
2140:P
2127:p
2119:φ
2115:M
2111:R
2089:.
2068:n
2048:1
2042:)
2039:z
2036:(
2031:n
2015:.
2003:n
1983:1
1980:=
1977:)
1974:1
1971:(
1966:n
1941:1
1938:=
1935:r
1911:)
1905:,
1901:Z
1897:,
1891:(
1864:)
1860:1
1850:2
1846:r
1842:1
1836:(
1827:4
1820:e
1816:+
1813:1
1809:1
1803:=
1800:)
1797:r
1794:(
1764:|
1754:2
1726:2
1663:2
1660:=
1657:t
1617:r
1597:}
1594:0
1591:=
1582::
1576:)
1573:r
1570:,
1564:(
1561:{
1558:=
1532:)
1529:1
1521:0
1517:r
1513:,
1508:0
1500:(
1476:1
1473:=
1464:r
1441:r
1390:)
1381:)
1377:1
1367:2
1362:0
1358:r
1354:1
1348:(
1342:t
1339:2
1332:e
1328:+
1325:1
1321:1
1315:,
1312:t
1309:+
1302:(
1298:=
1295:)
1292:r
1289:,
1283:(
1278:t
1242:)
1238:1
1228:2
1223:0
1219:r
1215:1
1209:(
1203:t
1200:2
1193:e
1189:+
1186:1
1182:1
1176:=
1169:)
1166:1
1158:t
1155:2
1151:e
1147:(
1142:2
1137:0
1133:r
1129:+
1126:1
1119:2
1114:0
1110:r
1104:t
1101:2
1097:e
1089:=
1086:)
1083:t
1080:(
1077:r
1045:c
1042:2
1038:e
1034:+
1031:1
1025:c
1022:2
1018:e
1011:=
1008:)
1005:0
1002:(
999:r
967:)
964:c
961:+
958:t
955:(
952:2
948:e
944:+
941:1
935:)
932:c
929:+
926:t
923:(
920:2
916:e
909:=
906:)
903:t
900:(
897:r
871:c
868:+
865:t
862:=
858:)
850:2
846:r
839:1
835:r
830:(
817:t
814:d
808:=
805:r
802:d
796:r
793:)
788:2
784:r
777:1
774:(
770:1
742:r
722:t
719:+
714:0
706:=
703:)
700:t
697:(
644:r
641:)
636:2
632:r
625:1
622:(
619:=
610:r
600:1
597:=
579:{
552:+
547:R
537:1
532:S
524:)
521:r
518:,
512:(
495:)
493:x
491:(
489:P
485:S
481:x
473:U
469:x
459:U
457:(
455:P
451:U
449::
447:P
443:p
439:U
437:(
435:P
430:p
426:p
424:(
422:P
415:p
411:S
387:S
381:U
378::
375:P
359:p
345:S
339:U
323:p
315:p
311:φ
307:S
303:p
291:φ
283:M
275:R
267:φ
263:M
259:R
251:x
249:(
247:P
243:x
239:P
235:S
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
57:.
43:.
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