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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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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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1263:{\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)}}}}
518:
204:. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.
232:, because the path of a star projected onto a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.
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of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding
Poincaré map.
903:
127:
57:
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2133:
1407:{\displaystyle \Phi _{t}(\theta ,r)=\left(\theta +t,{\sqrt {\frac {1}{1+e^{-2t}\left({\frac {1}{r_{0}^{2}}}-1\right)}}}\right)}
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887:{\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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1689:(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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1885:{\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
667:{\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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570:{\displaystyle (\theta ,r)\in \mathbb {S} ^{1}\times \mathbb {R} ^{+}}
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989:{\displaystyle r(t)={\sqrt {\frac {e^{2(t+c)}}{1+e^{2(t+c)}}}}}
27:
Type of map used in mathematics, particularly dynamical systems
29:
2582:
Poincare Map and its application to 'Spinning Magnet' problem
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The behaviour of the orbits of the discrete dynamical system
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of the discrete dynamical system is asymptotically stable.
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The periodic orbit γ of the continuous dynamical system is
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The periodic orbit γ of the continuous dynamical system is
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component we need to separate the variables and integrate:
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Every other point tends monotonically to the equilibrium,
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draws a spiral that tends towards the radius 1 circle.
1067:{\displaystyle r(0)={\sqrt {\frac {e^{2c}}{1+e^{2c}}}}}
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to send the first point to the second, hence the name
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with a certain lower-dimensional subspace, called the
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1613:{\displaystyle \Sigma =\{(\theta ,r)\ :\ \theta =0\}}
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be a local differentiable and transversal section of
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2425:) is a discrete dynamical system with state space
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144:A two-dimensional Poincaré section of the forced
2478:Per definition this system has a fixed point at
2218:{\displaystyle P^{0}:=\operatorname {id} _{U}}
1640:as coordinate on the section. Every point in
1441:increases monotonically and at constant rate.
8:
2493:of the discrete dynamical system is stable.
2468:{\displaystyle P:\mathbb {Z} \times U\to U.}
2340:{\displaystyle P^{-n-1}:=P^{-1}\circ P^{-n}}
1927:{\displaystyle (\Sigma ,\mathbb {Z} ,\Psi )}
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1571:
1417:The behaviour of the flow is the following:
64:. Unsourced material may be challenged and
2178:be the corresponding Poincaré map through
1503:Therefore, the solution with initial data
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1548:{\displaystyle (\theta _{0},r_{0}\neq 1)}
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128:Learn how and when to remove this message
1787:{\displaystyle \Phi _{2\pi }|_{\Sigma }}
738:{\displaystyle \theta (t)=\theta _{0}+t}
239:
139:
207:A Poincaré map can be interpreted as a
2109:Poincaré maps can be interpreted as a
1794:. The Poincaré map is therefore :
2271:{\displaystyle P^{n+1}:=P\circ P^{n}}
7:
2105:Poincaré maps and stability analysis
1660:returns to the section after a time
1273:The flow of the system is therefore
62:adding citations to reliable sources
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228:to study the motion of stars in a
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2064:{\displaystyle \Psi ^{n}(z)\to 1}
2403:{\displaystyle P(n,x):=P^{n}(x)}
2093:{\displaystyle n\to \pm \infty }
897:Inverting last expression gives
34:
2500:if and only if the fixed point
2489:if and only if the fixed point
2134:differentiable dynamical system
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2136:with periodic orbit γ through
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1999:{\displaystyle \Psi ^{n}(1)=1}
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215:A Poincaré map differs from a
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2566:American Mathematical Society
2534:Mironenko reflecting function
1492:{\displaystyle {\bar {r}}=1}
339:Given an open and connected
181:continuous dynamical system
171:, is the intersection of a
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361:{\displaystyle U\subset S}
2111:discrete dynamical system
1464:tends to the equilibrium
697:component we simply have
209:discrete dynamical system
2168:{\displaystyle P:U\to S}
403:{\displaystyle P:U\to S}
244:In the Poincaré section
2429:and evolution function
1722:{\displaystyle \Sigma }
1682:{\displaystyle t=2\pi }
1653:{\displaystyle \Sigma }
1620:: obviously we can use
1434:{\displaystyle \theta }
690:{\displaystyle \theta }
452:) is a neighborhood of
417:for the orbit γ on the
282:global dynamical system
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498:for the first time at
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1745:{\displaystyle 2\pi }
1729:computed at the time
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1702:{\displaystyle \Phi }
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202:first recurrence map
161:first recurrence map
58:improve this article
2579:Shivakumar Jolad,
2514:Poincaré recurrence
1957:{\displaystyle r=1}
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488:positive semi-orbit
248:, the Poincaré map
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424:through the point
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306:evolution function
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155:, particularly in
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2599:Dynamical systems
2539:Invariant measure
2019:{\displaystyle n}
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1633:{\displaystyle r}
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1457:{\displaystyle r}
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758:{\displaystyle r}
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252:projects a point
157:dynamical systems
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16:(Redirected from
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2519:Stroboscopic map
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478:for every point
419:Poincaré section
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330:Poincaré section
312:through a point
185:Poincaré section
146:Duffing equation
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18:Poincaré section
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2529:Recurrence plot
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2604:Henri Poincaré
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2574:External links
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2552:Teschl, Gerald
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1023:
1020:
1017:
1014:
1011:
997:
996:
979:
976:
973:
970:
967:
964:
960:
956:
953:
947:
944:
941:
938:
935:
932:
928:
921:
918:
915:
912:
909:
895:
894:
883:
880:
877:
874:
870:
862:
858:
854:
851:
847:
842:
838:
835:
832:
829:
826:
823:
820:
817:
814:
808:
805:
800:
796:
792:
789:
786:
782:
777:
754:
745:while for the
734:
731:
726:
722:
718:
715:
712:
709:
706:
686:
675:
674:
661:
656:
653:
648:
644:
640:
637:
634:
631:
625:
622:
616:
615:
612:
609:
603:
600:
594:
593:
591:
564:
559:
554:
549:
544:
539:
536:
533:
530:
527:
524:
512:
509:
508:
507:
476:
474:diffeomorphism
443:
411:
410:
399:
396:
393:
390:
387:
357:
354:
351:
310:periodic orbit
237:
234:
173:periodic orbit
169:Henri Poincaré
167:, named after
136:
135:
77:"Poincaré map"
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2616:
2605:
2602:
2600:
2597:
2596:
2594:
2584:
2583:
2578:
2577:
2573:
2567:
2563:
2559:
2558:
2553:
2549:
2548:
2544:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2511:
2507:
2505:
2503:
2499:
2494:
2492:
2488:
2483:
2481:
2462:
2459:
2453:
2450:
2442:
2439:
2432:
2431:
2430:
2428:
2424:
2420:
2416:
2394:
2386:
2382:
2378:
2372:
2369:
2366:
2360:
2353:
2352:
2351:
2332:
2329:
2325:
2321:
2316:
2313:
2309:
2305:
2300:
2297:
2294:
2291:
2287:
2279:
2263:
2259:
2255:
2252:
2249:
2244:
2241:
2238:
2234:
2226:
2210:
2206:
2202:
2197:
2193:
2185:
2184:
2183:
2181:
2162:
2156:
2153:
2150:
2143:
2142:
2141:
2139:
2135:
2131:
2127:
2123:
2118:
2116:
2112:
2104:
2084:
2078:
2058:
2049:
2041:
2028:
2013:
1993:
1990:
1984:
1976:
1964:is fixed, so
1951:
1948:
1945:
1937:
1936:
1935:
1915:
1907:
1892:
1874:
1870:
1867:
1860:
1856:
1852:
1846:
1840:
1837:
1834:
1830:
1826:
1823:
1819:
1813:
1807:
1767:
1764:
1739:
1736:
1676:
1673:
1670:
1667:
1627:
1604:
1601:
1598:
1592:
1583:
1580:
1577:
1568:
1556:
1539:
1536:
1531:
1527:
1523:
1518:
1514:
1486:
1483:
1474:
1451:
1443:
1428:
1420:
1419:
1418:
1400:
1391:
1387:
1384:
1377:
1372:
1368:
1364:
1358:
1352:
1349:
1346:
1342:
1338:
1335:
1331:
1325:
1322:
1319:
1316:
1312:
1308:
1302:
1299:
1296:
1288:
1276:
1275:
1274:
1252:
1248:
1245:
1238:
1233:
1229:
1225:
1219:
1213:
1210:
1207:
1203:
1199:
1196:
1192:
1186:
1176:
1173:
1168:
1165:
1161:
1152:
1147:
1143:
1139:
1136:
1129:
1124:
1120:
1114:
1111:
1107:
1099:
1093:
1087:
1080:
1079:
1078:
1055:
1052:
1048:
1044:
1041:
1035:
1032:
1028:
1021:
1015:
1009:
1002:
1001:
1000:
974:
971:
968:
962:
958:
954:
951:
942:
939:
936:
930:
926:
919:
913:
907:
900:
899:
898:
881:
878:
875:
872:
868:
860:
856:
852:
849:
845:
840:
836:
833:
827:
824:
821:
818:
815:
812:
806:
798:
794:
790:
787:
780:
775:
768:
767:
766:
752:
732:
729:
724:
720:
716:
710:
704:
684:
654:
646:
642:
638:
635:
629:
623:
620:
610:
607:
601:
598:
589:
580:
579:
578:
562:
552:
547:
537:
531:
528:
525:
510:
505:
501:
497:
493:
489:
485:
481:
477:
475:
471:
467:
463:
459:
455:
451:
447:
444:
442:
438:
434:
431:
430:
429:
427:
423:
420:
416:
397:
391:
388:
385:
378:
377:
376:
375:
371:
355:
352:
349:
342:
337:
335:
331:
327:
323:
319:
315:
311:
308:. Let γ be a
307:
303:
299:
295:
291:
287:
283:
279:
275:
271:
263:
259:
255:
251:
247:
242:
235:
233:
231:
227:
222:
218:
213:
210:
205:
203:
199:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
147:
142:
132:
129:
121:
118:December 2020
110:
107:
103:
100:
96:
93:
89:
86:
82:
79: –
78:
74:
73:Find sources:
67:
63:
59:
53:
52:
48:
43:This article
41:
37:
32:
31:
19:
2580:
2556:
2501:
2495:
2490:
2484:
2479:
2477:
2426:
2422:
2418:
2414:
2412:
2349:
2182:. We define
2179:
2177:
2137:
2129:
2125:
2121:
2119:
2108:
1893:
1557:
1502:
1416:
1272:
1076:
998:
896:
676:
514:
503:
499:
495:
491:
483:
479:
469:
465:
461:
457:
453:
449:
445:
440:
436:
432:
425:
421:
418:
415:Poincaré map
414:
412:
369:
341:neighborhood
338:
333:
329:
325:
321:
317:
313:
301:
293:
290:real numbers
285:
277:
273:
269:
267:
261:
257:
253:
249:
245:
226:Michel Hénon
214:
206:
201:
184:
165:Poincaré map
164:
160:
150:
124:
115:
105:
98:
91:
84:
72:
56:Please help
44:
1444:The radius
999:and since
494:intersects
328:, called a
298:phase space
189:transversal
177:state space
153:mathematics
2593:Categories
2562:Providence
2545:References
2006:for every
1938:The point
1421:The angle
413:is called
236:Definition
221:perihelion
88:newspapers
2524:Hénon map
2457:→
2451:×
2330:−
2322:∘
2314:−
2298:−
2292:−
2256:∘
2160:→
2115:stability
2088:∞
2085:±
2082:→
2056:→
2038:Ψ
1973:Ψ
1919:Ψ
1905:Σ
1868:−
1841:π
1835:−
1802:Ψ
1780:Σ
1768:π
1761:Φ
1740:π
1717:Σ
1697:Φ
1677:π
1648:Σ
1599:θ
1578:θ
1566:Σ
1537:≠
1515:θ
1478:¯
1429:θ
1385:−
1347:−
1317:θ
1297:θ
1285:Φ
1246:−
1208:−
1174:−
853:−
837:
831:⟹
822:∫
791:−
776:∫
721:θ
705:θ
685:θ
639:−
624:˙
602:˙
599:θ
553:×
538:∈
526:θ
395:→
353:⊂
45:does not
2585:, (2005)
2508:See also
1077:we find
374:function
332:through
324:through
2132:) be a
511:Example
472:) is a
284:, with
280:) be a
191:to the
175:in the
102:scholar
66:removed
51:sources
2487:stable
2413:then (
2140:. Let
2113:. The
1596:
1590:
486:, the
230:galaxy
104:
97:
90:
83:
75:
2350:and
2120:Let (
268:Let (
179:of a
109:JSTOR
95:books
2071:for
456:and
439:) =
372:, a
316:and
304:the
300:and
296:the
288:the
193:flow
159:, a
81:news
49:any
47:cite
834:log
490:of
482:in
428:if
368:of
198:map
163:or
151:In
60:by
2595::
2564::
2560:.
2554:.
2482:.
2421:,
2417:,
2379::=
2306::=
2250::=
2207:id
2203::=
2128:,
2124:,
1752:,
577::
464:→
336:.
292:,
276:,
272:,
264:).
187:,
2568:.
2502:p
2491:p
2480:p
2463:.
2460:U
2454:U
2447:Z
2443::
2440:P
2427:U
2423:P
2419:U
2415:Z
2398:)
2395:x
2392:(
2387:n
2383:P
2376:)
2373:x
2370:,
2367:n
2364:(
2361:P
2333:n
2326:P
2317:1
2310:P
2301:1
2295:n
2288:P
2264:n
2260:P
2253:P
2245:1
2242:+
2239:n
2235:P
2211:U
2198:0
2194:P
2180:p
2163:S
2157:U
2154::
2151:P
2138:p
2130:φ
2126:M
2122:R
2100:.
2079:n
2059:1
2053:)
2050:z
2047:(
2042:n
2026:.
2014:n
1994:1
1991:=
1988:)
1985:1
1982:(
1977:n
1952:1
1949:=
1946:r
1922:)
1916:,
1912:Z
1908:,
1902:(
1875:)
1871:1
1861:2
1857:r
1853:1
1847:(
1838:4
1831:e
1827:+
1824:1
1820:1
1814:=
1811:)
1808:r
1805:(
1775:|
1765:2
1737:2
1674:2
1671:=
1668:t
1628:r
1608:}
1605:0
1602:=
1593::
1587:)
1584:r
1581:,
1575:(
1572:{
1569:=
1543:)
1540:1
1532:0
1528:r
1524:,
1519:0
1511:(
1487:1
1484:=
1475:r
1452:r
1401:)
1392:)
1388:1
1378:2
1373:0
1369:r
1365:1
1359:(
1353:t
1350:2
1343:e
1339:+
1336:1
1332:1
1326:,
1323:t
1320:+
1313:(
1309:=
1306:)
1303:r
1300:,
1294:(
1289:t
1253:)
1249:1
1239:2
1234:0
1230:r
1226:1
1220:(
1214:t
1211:2
1204:e
1200:+
1197:1
1193:1
1187:=
1180:)
1177:1
1169:t
1166:2
1162:e
1158:(
1153:2
1148:0
1144:r
1140:+
1137:1
1130:2
1125:0
1121:r
1115:t
1112:2
1108:e
1100:=
1097:)
1094:t
1091:(
1088:r
1056:c
1053:2
1049:e
1045:+
1042:1
1036:c
1033:2
1029:e
1022:=
1019:)
1016:0
1013:(
1010:r
978:)
975:c
972:+
969:t
966:(
963:2
959:e
955:+
952:1
946:)
943:c
940:+
937:t
934:(
931:2
927:e
920:=
917:)
914:t
911:(
908:r
882:c
879:+
876:t
873:=
869:)
861:2
857:r
850:1
846:r
841:(
828:t
825:d
819:=
816:r
813:d
807:r
804:)
799:2
795:r
788:1
785:(
781:1
753:r
733:t
730:+
725:0
717:=
714:)
711:t
708:(
655:r
652:)
647:2
643:r
636:1
633:(
630:=
621:r
611:1
608:=
590:{
563:+
558:R
548:1
543:S
535:)
532:r
529:,
523:(
506:)
504:x
502:(
500:P
496:S
492:x
484:U
480:x
470:U
468:(
466:P
462:U
460::
458:P
454:p
450:U
448:(
446:P
441:p
437:p
435:(
433:P
426:p
422:S
398:S
392:U
389::
386:P
370:p
356:S
350:U
334:p
326:p
322:φ
318:S
314:p
302:φ
294:M
286:R
278:φ
274:M
270:R
262:x
260:(
258:P
254:x
250:P
246:S
131:)
125:(
120:)
116:(
106:·
99:·
92:·
85:·
68:.
54:.
20:)
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