3200:
3192:
2701:, has two positive real eigenvalues, the system has an unstable node; if the matrix has two complex eigenvalues with positive real part, the system has an unstable focus (or spiral). Nodes and foci are topologically equivalent but not orbitally equivalent or smoothly equivalent, because their eigenvalues are different (notice that the Jacobians of two locally smoothly equivalent systems must be
2102:
class of two dimensional systems of differential equations that have closed orbits. While the orbits can be transformed to each other to overlap in the spatial sense, the periods of such systems cannot be analogously matched, thus failing to satisfy the topological conjugacy criterion while satisfying the topological equivalence criterion.
2101:
Overall, topological equivalence is a weaker equivalence criterion than topological conjugacy, as it does not require that the time term is mapped along with the orbits and their orientation. An example of a topologically equivalent but not topologically conjugate system would be the non-hyperbolic
1870:
2415:
1696:
2643:
Systems that are smoothly equivalent or orbitally equivalent are also topologically equivalent. However, the reverse is not true. For example, consider linear systems in two dimensions of the form
1995:
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1321:
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83:
3319:
1865:{\displaystyle h({\mathcal {O}}(y,\psi ))=\{h\circ \psi (y,t):t\in \mathbb {R} \}=\{\phi (h(y),t):t\in \mathbb {R} \}={\mathcal {O}}(h(y),\phi )}
3199:
2959:
3152:
2911:
62:, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.
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2004:
1231:, since each class contains all functions which share the same dynamics from the topological viewpoint. For example,
963:
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is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.
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3191:
2410:{\displaystyle f(x)=M^{-1}(x)g(h(x))\quad {\text{where}}\quad M(x)={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}.}
1227:
to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of
131:
3771:
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3172:
3079:
2553:
1278:
3447:
3059:
35:
2420:
In that case, the dynamical systems can be transformed into each other by the coordinate transformation,
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3302:
3064:
3039:
2945:
2724:
Adjoint dynamical systems defined via adjoint functors and natural equivalences in categorical dynamics.
1388:
However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps
3359:
2931:
1060:
531:
2587:
1385:. Speaking informally, topological conjugation is a "change of coordinates" in the topological sense.
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2763:
Arnold V. I. Geometric
Methods in the Theory of Ordinary Differential Equations (Springer, 2020)
2752:
Arnold V. I. Geometric
Methods in the Theory of Ordinary Differential Equations (Springer, 2020)
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50:
that will conjugate the one into the other. Topological conjugacy, and related-but-distinct
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in the space of all continuous surjections of a topological space to itself, by declaring
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into classes of flows sharing the same dynamics, again from the topological viewpoint.
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homeomorphically, and preserving orientation of the orbits. In other words, letting
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3369:
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2717:
There are two reported extensions of the concept of dynamic topological conjugacy:
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641:
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3001:
2839:
2640:. Orbitally equivalent system differ only in the time parametrization.
3622:
2926:
This article incorporates material from topological conjugation on
2941:
2937:
2912:"Analogous systems, Topological Conjugacy and Adjoint Systems"
1183:
Topological conjugation – unlike semiconjugation – defines an
2153:
Two dynamical systems defined by the differential equations,
1833:
1708:
1675:
1901:. In addition, one must line up the flow of time: for each
2458:
Two dynamical systems on the same state space, defined by
2721:
Analogous systems defined as isomorphic dynamical systems
105:
are iterated functions, and there exists a homeomorphism
2110:
More equivalence criteria can be studied if the flows,
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1504:
1484:
1478:, which is requiring more than simply that orbits of
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71:
2775:
Alligood, K. T., Sauer, T., and Yorke, J.A. (1997).
3716:
3533:
3460:
3398:
3268:
3255:
3207:
3138:
2982:
2975:
1518:homeomorphically. This motivates the definition of
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2538:
2494:
2447:
2409:
2273:
2233:
2189:
2142:
2122:
2090:
2064:
1990:{\displaystyle 0<\vert s\vert <t<\delta }
1989:
1945:
1919:
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171:
117:
97:
77:
222:are topologically conjugate. Then one must have
2932:Creative Commons Attribution/Share-Alike License
2713:Generalizations of dynamic topological conjugacy
1522:, which also partitions the set of all flows in
2065:{\displaystyle \phi (h(y),s)=h\circ \psi (y,t)}
1166:For certain values in the parameter space, the
1024:{\displaystyle \phi (h(y),t)=h\circ \psi (y,t)}
283:{\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h,}
1378:{\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h}
2953:
371:{\displaystyle f\colon X\to X,g\colon Y\to Y}
8:
2799:: CS1 maint: multiple names: authors list (
1972:
1966:
1825:
1781:
1775:
1734:
51:
2777:Chaos: An Introduction to Dynamical Systems
297:are topologically conjugate as well. Here,
3265:
2979:
2960:
2946:
2938:
2816:"Shift automorphisms in the HĂ©non mapping"
65:To illustrate this directly: suppose that
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2158:
2135:
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2006:
1958:
1932:
1906:
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1820:
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1707:
1706:
1698:
1674:
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1631:
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1575:
1555:
1527:
1503:
1483:
1463:
1428:
1393:
1363:
1347:
1334:
1328:
1292:
1280:
1260:
1240:
1212:
1192:
1112:
1092:
1071:
1070:
1062:
1036:
965:
941:
921:
897:
863:
837:
813:
793:
773:
753:
725:
705:
681:
657:
621:
593:
573:
533:
509:
489:
465:
443:
423:
383:
333:
302:
265:
249:
236:
230:
207:
187:
145:
133:
110:
90:
70:
1159:The logistic map of unit height and the
54:of flows, are important in the study of
2745:
1458:to be topologically conjugate for each
172:{\displaystyle g=h^{-1}\circ f\circ h,}
2792:
2707:algebraic and geometric multiplicities
2577:{\displaystyle \mu :X\to \mathbb {R} }
1316:{\displaystyle g=h^{-1}\circ f\circ h}
1255:are mapped to homeomorphic orbits of
2894:"Complexity and Categorical Dynamics"
2150:, arise from differential equations.
7:
1131:means, by definition, that they are
612:means, by definition, that they are
3100:Measure-preserving dynamical system
2705:, so their eigenvalues, as well as
2394:
2375:
25:
3668:Oleksandr Mykolayovych Sharkovsky
2814:Devaney, R.; Nitecki, Z. (1979).
2550:if there is a positive function,
1275:through the conjugation. Writing
1078:{\displaystyle t\in \mathbb {R} }
559:{\displaystyle f\circ h=h\circ g}
3198:
3190:
2633:{\displaystyle g(x)=\mu (x)f(x)}
2539:{\displaystyle {\dot {x}}=g(x)}
2495:{\displaystyle {\dot {x}}=f(x)}
2355:
2349:
2234:{\displaystyle {\dot {y}}=g(y)}
2190:{\displaystyle {\dot {x}}=f(x)}
1451:{\displaystyle \psi (\cdot ,t)}
1416:{\displaystyle \phi (\cdot ,t)}
3433:Rabinovich–Fabrikant equations
2930:, which is licensed under the
2870:Elements of Bifurcation Theory
2779:. Springer. pp. 114–124.
2627:
2621:
2615:
2609:
2600:
2594:
2566:
2533:
2527:
2489:
2483:
2442:
2436:
2388:
2382:
2365:
2359:
2346:
2343:
2337:
2331:
2325:
2319:
2300:
2294:
2265:
2228:
2222:
2184:
2178:
2106:Smooth and orbital equivalence
2059:
2047:
2032:
2023:
2017:
2011:
1859:
1850:
1844:
1838:
1808:
1799:
1793:
1787:
1758:
1746:
1728:
1725:
1713:
1703:
1683:{\displaystyle {\mathcal {O}}}
1610:
1594:, if there is a homeomorphism
1445:
1433:
1410:
1398:
1018:
1006:
991:
982:
976:
970:
883:{\displaystyle h\colon Y\to X}
874:
403:{\displaystyle h\colon Y\to X}
394:
362:
344:
52:§ Topological equivalence
1:
2872:(Second ed.). Springer.
2674:{\displaystyle {\dot {x}}=Ax}
1946:{\displaystyle \delta >0}
1163:are topologically conjugate.
1156:are topologically conjugate.
3168:Poincaré recurrence theorem
2868:Kuznetsov, Yuri A. (1998).
1133:topologically semiconjugate
936:means, by definition, that
914:topologically semiconjugate
614:topologically semiconjugate
504:means, by definition, that
482:topologically semiconjugate
3788:
3163:Poincaré–Bendixson theorem
1323:makes this fact evident:
3515:Swinging Atwood's machine
3188:
3158:Krylov–Bogolyubov theorem
3035:
1690:denote an orbit, one has
3423:Lotka–Volterra equations
3247:Synchronization of chaos
3050:axiom A dynamical system
2274:{\displaystyle h:X\to Y}
1619:{\displaystyle h:Y\to X}
1592:topologically equivalent
3408:Double scroll attractor
3173:Stable manifold theorem
3080:False nearest neighbors
1546:Topological equivalence
1520:topological equivalence
1498:be mapped to orbits of
1170:when restricted to its
1129:topologically conjugate
698:topological conjugation
610:topologically conjugate
40:topologically conjugate
18:Topological conjugation
3448:Van der Pol oscillator
3428:Mackey–Glass equations
3060:Box-counting dimension
2695:
2675:
2634:
2578:
2540:
2496:
2449:
2448:{\displaystyle y=h(x)}
2411:
2275:
2235:
2191:
2144:
2124:
2092:
2091:{\displaystyle s>0}
2066:
1991:
1947:
1921:
1920:{\displaystyle y\in Y}
1895:
1894:{\displaystyle y\in Y}
1866:
1684:
1660:
1640:
1620:
1584:
1564:
1550:We say that two flows
1536:
1512:
1492:
1472:
1452:
1417:
1379:
1317:
1269:
1249:
1221:
1201:
1121:
1101:
1079:
1051:
1050:{\displaystyle y\in Y}
1025:
950:
930:
906:
884:
852:
822:
802:
782:
762:
734:
714:
690:
666:
630:
602:
582:
560:
518:
498:
474:
452:
432:
404:
372:
311:
310:{\displaystyle \circ }
284:
216:
196:
173:
119:
99:
79:
3598:Svetlana Jitomirskaya
3505:Multiscroll attractor
3350:Interval exchange map
3303:Dyadic transformation
3288:Complex quadratic map
3130:Topological conjugacy
3065:Correlation dimension
3040:Anosov diffeomorphism
2696:
2676:
2635:
2579:
2541:
2497:
2450:
2412:
2276:
2236:
2192:
2145:
2143:{\displaystyle \psi }
2125:
2123:{\displaystyle \phi }
2093:
2067:
1992:
1948:
1922:
1896:
1867:
1685:
1661:
1659:{\displaystyle \phi }
1641:
1639:{\displaystyle \psi }
1621:
1585:
1583:{\displaystyle \psi }
1565:
1563:{\displaystyle \phi }
1537:
1513:
1511:{\displaystyle \psi }
1493:
1491:{\displaystyle \phi }
1473:
1453:
1418:
1380:
1318:
1270:
1250:
1222:
1202:
1139:is a homeomorphism.
1122:
1120:{\displaystyle \psi }
1102:
1100:{\displaystyle \phi }
1080:
1052:
1026:
951:
931:
929:{\displaystyle \psi }
907:
905:{\displaystyle \phi }
885:
853:
823:
803:
801:{\displaystyle \psi }
783:
763:
761:{\displaystyle \phi }
735:
715:
691:
667:
631:
603:
583:
561:
519:
499:
475:
453:
433:
405:
373:
312:
285:
217:
197:
174:
120:
100:
80:
3767:Topological dynamics
3608:Edward Norton Lorenz
2685:
2647:
2588:
2554:
2548:orbitally equivalent
2506:
2462:
2424:
2288:
2253:
2201:
2157:
2134:
2114:
2076:
2005:
1957:
1931:
1905:
1879:
1697:
1670:
1650:
1630:
1626:, mapping orbits of
1598:
1574:
1554:
1526:
1502:
1482:
1462:
1427:
1392:
1327:
1279:
1259:
1239:
1211:
1191:
1185:equivalence relation
1111:
1091:
1061:
1035:
964:
940:
920:
896:
862:
836:
812:
792:
772:
752:
724:
704:
680:
656:
620:
592:
572:
532:
508:
488:
464:
442:
422:
412:continuous functions
382:
332:
319:function composition
301:
229:
206:
186:
132:
109:
89:
69:
3568:Mitchell Feigenbaum
3510:Population dynamics
3495:Hénon–Heiles system
3355:Irrational rotation
3308:Dynamical billiards
3293:Coupled map lattice
3153:Liouville's theorem
3085:Hausdorff dimension
3070:Conservative system
3055:Bifurcation diagram
2900:on August 19, 2009.
2832:1979CMaPh..67..137D
2735:Commutative diagram
2243:smoothly equivalent
851:{\displaystyle X,Y}
58:and more generally
27:Concept in topology
3746:Santa Fe Institute
3613:Aleksandr Lyapunov
3443:Three-body problem
3330:Gingerbreadman map
3217:Bifurcation theory
3095:Lyapunov stability
2840:10.1007/bf01221362
2709:, must be equal).
2691:
2671:
2630:
2574:
2536:
2492:
2445:
2407:
2271:
2231:
2187:
2140:
2120:
2088:
2062:
1987:
1943:
1917:
1891:
1862:
1680:
1656:
1636:
1616:
1580:
1560:
1532:
1508:
1488:
1468:
1448:
1413:
1375:
1313:
1265:
1245:
1217:
1197:
1117:
1097:
1075:
1047:
1021:
946:
926:
902:
880:
848:
818:
798:
778:
758:
730:
710:
686:
662:
626:
598:
578:
556:
514:
494:
470:
448:
428:
416:topological spaces
400:
368:
307:
280:
212:
192:
169:
115:
95:
75:
56:iterated functions
3754:
3753:
3618:Benoît Mandelbrot
3583:Martin Gutzwiller
3573:Peter Grassberger
3456:
3455:
3438:Rössler attractor
3186:
3185:
3090:Invariant measure
3012:Lyapunov exponent
2694:{\displaystyle A}
2681:. If the matrix,
2659:
2546:, are said to be
2518:
2474:
2402:
2353:
2241:, are said to be
2213:
2169:
1927:, there exists a
1535:{\displaystyle X}
1471:{\displaystyle t}
1268:{\displaystyle f}
1248:{\displaystyle g}
1229:dynamical systems
1220:{\displaystyle g}
1200:{\displaystyle f}
949:{\displaystyle h}
821:{\displaystyle Y}
781:{\displaystyle X}
733:{\displaystyle g}
713:{\displaystyle f}
689:{\displaystyle h}
665:{\displaystyle h}
629:{\displaystyle h}
601:{\displaystyle g}
581:{\displaystyle f}
517:{\displaystyle h}
497:{\displaystyle g}
473:{\displaystyle f}
451:{\displaystyle Y}
431:{\displaystyle X}
215:{\displaystyle g}
195:{\displaystyle f}
118:{\displaystyle h}
98:{\displaystyle g}
78:{\displaystyle f}
60:dynamical systems
16:(Redirected from
3779:
3726:Butterfly effect
3638:Itamar Procaccia
3588:Brosl Hasslacher
3485:Elastic pendulum
3413:Duffing equation
3360:Kaplan–Yorke map
3278:Arnold's cat map
3266:
3242:Stability theory
3227:Dynamical system
3222:Control of chaos
3202:
3194:
3178:Takens's theorem
3110:Poincaré section
2980:
2962:
2955:
2948:
2939:
2920:
2919:
2914:. Archived from
2908:
2902:
2901:
2896:. Archived from
2890:
2884:
2883:
2865:
2859:
2858:
2856:
2854:
2820:Comm. Math. Phys
2811:
2805:
2804:
2798:
2790:
2772:
2766:
2761:
2755:
2750:
2700:
2698:
2697:
2692:
2680:
2678:
2677:
2672:
2661:
2660:
2652:
2639:
2637:
2636:
2631:
2583:
2581:
2580:
2575:
2573:
2545:
2543:
2542:
2537:
2520:
2519:
2511:
2501:
2499:
2498:
2493:
2476:
2475:
2467:
2454:
2452:
2451:
2446:
2416:
2414:
2413:
2408:
2403:
2401:
2397:
2391:
2378:
2372:
2354:
2351:
2318:
2317:
2280:
2278:
2277:
2272:
2240:
2238:
2237:
2232:
2215:
2214:
2206:
2196:
2194:
2193:
2188:
2171:
2170:
2162:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2097:
2095:
2094:
2089:
2071:
2069:
2068:
2063:
2000:
1996:
1994:
1993:
1988:
1952:
1950:
1949:
1944:
1926:
1924:
1923:
1918:
1900:
1898:
1897:
1892:
1871:
1869:
1868:
1863:
1837:
1836:
1824:
1774:
1712:
1711:
1689:
1687:
1686:
1681:
1679:
1678:
1665:
1663:
1662:
1657:
1645:
1643:
1642:
1637:
1625:
1623:
1622:
1617:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1541:
1539:
1538:
1533:
1517:
1515:
1514:
1509:
1497:
1495:
1494:
1489:
1477:
1475:
1474:
1469:
1457:
1455:
1454:
1449:
1422:
1420:
1419:
1414:
1384:
1382:
1381:
1376:
1368:
1367:
1355:
1354:
1339:
1338:
1322:
1320:
1319:
1314:
1300:
1299:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1226:
1224:
1223:
1218:
1206:
1204:
1203:
1198:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1084:
1082:
1081:
1076:
1074:
1056:
1054:
1053:
1048:
1030:
1028:
1027:
1022:
955:
953:
952:
947:
935:
933:
932:
927:
911:
909:
908:
903:
889:
887:
886:
881:
857:
855:
854:
849:
827:
825:
824:
819:
807:
805:
804:
799:
787:
785:
784:
779:
767:
765:
764:
759:
739:
737:
736:
731:
719:
717:
716:
711:
695:
693:
692:
687:
671:
669:
668:
663:
635:
633:
632:
627:
607:
605:
604:
599:
587:
585:
584:
579:
565:
563:
562:
557:
523:
521:
520:
515:
503:
501:
500:
495:
479:
477:
476:
471:
457:
455:
454:
449:
437:
435:
434:
429:
409:
407:
406:
401:
377:
375:
374:
369:
316:
314:
313:
308:
295:iterated systems
289:
287:
286:
281:
270:
269:
257:
256:
241:
240:
221:
219:
218:
213:
201:
199:
198:
193:
178:
176:
175:
170:
153:
152:
124:
122:
121:
116:
104:
102:
101:
96:
84:
82:
81:
76:
21:
3787:
3786:
3782:
3781:
3780:
3778:
3777:
3776:
3757:
3756:
3755:
3750:
3718:
3712:
3658:Caroline Series
3553:Mary Cartwright
3535:
3529:
3480:Double pendulum
3462:
3452:
3401:
3394:
3320:Exponential map
3271:
3257:
3251:
3209:
3203:
3196:
3182:
3148:Ergodic theorem
3141:
3134:
3125:Stable manifold
3115:Recurrence plot
3031:
2985:
2971:
2966:
2923:
2910:
2909:
2905:
2892:
2891:
2887:
2880:
2867:
2866:
2862:
2852:
2850:
2813:
2812:
2808:
2791:
2787:
2774:
2773:
2769:
2762:
2758:
2751:
2747:
2743:
2731:
2715:
2683:
2682:
2645:
2644:
2586:
2585:
2552:
2551:
2504:
2503:
2460:
2459:
2422:
2421:
2392:
2373:
2306:
2286:
2285:
2251:
2250:
2199:
2198:
2155:
2154:
2132:
2131:
2112:
2111:
2108:
2074:
2073:
2003:
2002:
1998:
1955:
1954:
1929:
1928:
1903:
1902:
1877:
1876:
1695:
1694:
1668:
1667:
1648:
1647:
1628:
1627:
1596:
1595:
1572:
1571:
1552:
1551:
1548:
1524:
1523:
1500:
1499:
1480:
1479:
1460:
1459:
1425:
1424:
1390:
1389:
1359:
1343:
1330:
1325:
1324:
1288:
1277:
1276:
1257:
1256:
1237:
1236:
1209:
1208:
1189:
1188:
1181:
1145:
1136:
1109:
1108:
1089:
1088:
1059:
1058:
1033:
1032:
962:
961:
938:
937:
918:
917:
894:
893:
860:
859:
834:
833:
810:
809:
790:
789:
770:
769:
750:
749:
746:
722:
721:
702:
701:
678:
677:
654:
653:
636:is furthermore
618:
617:
590:
589:
570:
569:
530:
529:
506:
505:
486:
485:
462:
461:
440:
439:
420:
419:
380:
379:
330:
329:
327:
299:
298:
261:
245:
232:
227:
226:
204:
203:
184:
183:
141:
130:
129:
107:
106:
87:
86:
67:
66:
38:are said to be
28:
23:
22:
15:
12:
11:
5:
3785:
3783:
3775:
3774:
3772:Homeomorphisms
3769:
3759:
3758:
3752:
3751:
3749:
3748:
3743:
3741:Predictability
3738:
3733:
3728:
3722:
3720:
3714:
3713:
3711:
3710:
3708:Lai-Sang Young
3705:
3703:James A. Yorke
3700:
3698:Amie Wilkinson
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3633:Henri Poincaré
3630:
3625:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3539:
3537:
3531:
3530:
3528:
3527:
3522:
3517:
3512:
3507:
3502:
3500:Kicked rotator
3497:
3492:
3487:
3482:
3477:
3472:
3470:Chua's circuit
3466:
3464:
3458:
3457:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3410:
3404:
3402:
3399:
3396:
3395:
3393:
3392:
3390:Zaslavskii map
3387:
3385:Tinkerbell map
3382:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3316:
3315:
3305:
3300:
3295:
3290:
3285:
3280:
3274:
3272:
3269:
3263:
3253:
3252:
3250:
3249:
3244:
3239:
3234:
3232:Ergodic theory
3229:
3224:
3219:
3213:
3211:
3205:
3204:
3189:
3187:
3184:
3183:
3181:
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3144:
3142:
3139:
3136:
3135:
3133:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3042:
3036:
3033:
3032:
3030:
3029:
3024:
3022:Periodic point
3019:
3014:
3009:
3004:
2999:
2994:
2988:
2986:
2983:
2977:
2973:
2972:
2967:
2965:
2964:
2957:
2950:
2942:
2922:
2921:
2918:on 2015-02-25.
2903:
2885:
2878:
2860:
2826:(2): 137–146.
2806:
2785:
2767:
2756:
2744:
2742:
2739:
2738:
2737:
2730:
2727:
2726:
2725:
2722:
2714:
2711:
2690:
2670:
2667:
2664:
2658:
2655:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2572:
2568:
2565:
2562:
2559:
2535:
2532:
2529:
2526:
2523:
2517:
2514:
2491:
2488:
2485:
2482:
2479:
2473:
2470:
2444:
2441:
2438:
2435:
2432:
2429:
2418:
2417:
2406:
2400:
2396:
2390:
2387:
2384:
2381:
2377:
2370:
2367:
2364:
2361:
2358:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2316:
2313:
2309:
2305:
2302:
2299:
2296:
2293:
2270:
2267:
2264:
2261:
2258:
2247:diffeomorphism
2245:if there is a
2230:
2227:
2224:
2221:
2218:
2212:
2209:
2186:
2183:
2180:
2177:
2174:
2168:
2165:
2139:
2119:
2107:
2104:
2087:
2084:
2081:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1953:such that, if
1942:
1939:
1936:
1916:
1913:
1910:
1890:
1887:
1884:
1873:
1872:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1835:
1830:
1827:
1823:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1773:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1705:
1702:
1677:
1655:
1635:
1615:
1612:
1609:
1606:
1603:
1579:
1559:
1547:
1544:
1531:
1507:
1487:
1467:
1447:
1444:
1441:
1438:
1435:
1432:
1412:
1409:
1406:
1403:
1400:
1397:
1374:
1371:
1366:
1362:
1358:
1353:
1350:
1346:
1342:
1337:
1333:
1312:
1309:
1306:
1303:
1298:
1295:
1291:
1287:
1284:
1264:
1244:
1216:
1196:
1180:
1177:
1176:
1175:
1164:
1157:
1144:
1141:
1116:
1096:
1073:
1069:
1066:
1046:
1043:
1040:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
945:
925:
901:
879:
876:
873:
870:
867:
847:
844:
841:
817:
797:
777:
757:
745:
742:
729:
709:
685:
661:
625:
597:
577:
555:
552:
549:
546:
543:
540:
537:
513:
493:
469:
447:
427:
399:
396:
393:
390:
387:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
326:
323:
306:
291:
290:
279:
276:
273:
268:
264:
260:
255:
252:
248:
244:
239:
235:
211:
191:
180:
179:
168:
165:
162:
159:
156:
151:
148:
144:
140:
137:
114:
94:
74:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3784:
3773:
3770:
3768:
3765:
3764:
3762:
3747:
3744:
3742:
3739:
3737:
3736:Edge of chaos
3734:
3732:
3729:
3727:
3724:
3723:
3721:
3715:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3693:Marcelo Viana
3691:
3689:
3686:
3684:
3683:Audrey Terras
3681:
3679:
3678:Floris Takens
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3578:Celso Grebogi
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3558:Chen Guanrong
3556:
3554:
3551:
3549:
3546:
3544:
3543:Michael Berry
3541:
3540:
3538:
3532:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3459:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3418:Lorenz system
3416:
3414:
3411:
3409:
3406:
3405:
3403:
3397:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3365:Langton's ant
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3340:Horseshoe map
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3314:
3311:
3310:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3275:
3273:
3267:
3264:
3261:
3254:
3248:
3245:
3243:
3240:
3238:
3237:Quantum chaos
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3214:
3212:
3206:
3201:
3197:
3193:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3143:
3137:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3045:Arnold tongue
3043:
3041:
3038:
3037:
3034:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2989:
2987:
2981:
2978:
2974:
2970:
2963:
2958:
2956:
2951:
2949:
2944:
2943:
2940:
2936:
2935:
2933:
2929:
2917:
2913:
2907:
2904:
2899:
2895:
2889:
2886:
2881:
2879:0-387-98382-1
2875:
2871:
2864:
2861:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2810:
2807:
2802:
2796:
2788:
2786:0-387-94677-2
2782:
2778:
2771:
2768:
2765:
2760:
2757:
2754:
2749:
2746:
2740:
2736:
2733:
2732:
2728:
2723:
2720:
2719:
2718:
2712:
2710:
2708:
2704:
2688:
2668:
2665:
2662:
2656:
2653:
2641:
2624:
2618:
2612:
2606:
2603:
2597:
2591:
2563:
2560:
2557:
2549:
2530:
2524:
2521:
2515:
2512:
2486:
2480:
2477:
2471:
2468:
2456:
2439:
2433:
2430:
2427:
2404:
2398:
2385:
2379:
2368:
2362:
2356:
2340:
2334:
2328:
2322:
2314:
2311:
2307:
2303:
2297:
2291:
2284:
2283:
2282:
2268:
2262:
2259:
2256:
2248:
2244:
2225:
2219:
2216:
2210:
2207:
2181:
2175:
2172:
2166:
2163:
2151:
2137:
2117:
2105:
2103:
2099:
2085:
2082:
2079:
2056:
2053:
2050:
2044:
2041:
2038:
2035:
2029:
2026:
2020:
2014:
2008:
2001:is such that
1984:
1981:
1978:
1975:
1969:
1963:
1960:
1940:
1937:
1934:
1914:
1911:
1908:
1888:
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1646:to orbits of
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1169:
1165:
1162:
1161:Bernoulli map
1158:
1155:
1151:
1147:
1146:
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1114:
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839:
831:
815:
795:
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755:
743:
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727:
707:
699:
683:
675:
674:homeomorphism
659:
651:
647:
643:
639:
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611:
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567:
553:
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541:
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467:
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365:
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92:
72:
63:
61:
57:
53:
49:
48:homeomorphism
45:
41:
37:
33:
19:
3688:Mary Tsingou
3653:David Ruelle
3648:Otto Rössler
3593:Michel HĂ©non
3563:Leon O. Chua
3520:Tilt-A-Whirl
3490:FPUT problem
3375:Standard map
3370:Logistic map
3195:
3129:
2969:Chaos theory
2925:
2924:
2916:the original
2906:
2898:the original
2888:
2869:
2863:
2851:. Retrieved
2823:
2819:
2809:
2776:
2770:
2759:
2748:
2716:
2642:
2584:, such that
2547:
2457:
2419:
2281:, such that
2242:
2152:
2109:
2100:
1874:
1591:
1549:
1519:
1387:
1182:
1150:logistic map
1132:
1128:
1087:
913:
892:
747:
697:
696:is termed a
613:
609:
568:
481:
460:
328:
292:
181:
64:
44:there exists
39:
29:
3673:Nina Snaith
3663:Yakov Sinai
3548:Rufus Bowen
3298:Duffing map
3283:Baker's map
3208:Theoretical
3120:SRB measure
3027:Phase space
2997:Bifurcation
2853:2 September
1031:, for each
748:Similarly,
676:; further,
293:and so the
125:such that
32:mathematics
3761:Categories
3731:Complexity
3628:Edward Ott
3475:Convection
3400:Continuous
3075:Ergodicity
2928:PlanetMath
2741:References
1997:, and if
1179:Discussion
960:such that
958:surjection
890:as above.
652:too; i.e.
650:continuous
644:, and its
528:such that
526:surjection
325:Definition
3643:Mary Rees
3603:Bryna Kra
3536:theorists
3345:Ikeda map
3335:HĂ©non map
3325:Gauss map
3007:Limit set
2992:Attractor
2848:121479458
2795:cite book
2657:˙
2607:μ
2567:→
2558:μ
2516:˙
2472:˙
2312:−
2266:→
2211:˙
2167:˙
2138:ψ
2118:ϕ
2045:ψ
2042:∘
2009:ϕ
1985:δ
1935:δ
1912:∈
1886:∈
1875:for each
1857:ϕ
1818:∈
1785:ϕ
1768:∈
1744:ψ
1741:∘
1723:ψ
1654:ϕ
1634:ψ
1611:→
1578:ψ
1558:ϕ
1506:ψ
1486:ϕ
1437:⋅
1431:ψ
1402:⋅
1396:ϕ
1370:∘
1357:∘
1349:−
1308:∘
1302:∘
1294:−
1172:Julia set
1168:HĂ©non map
1115:ψ
1095:ϕ
1068:∈
1042:∈
1004:ψ
1001:∘
968:ϕ
924:ψ
900:ϕ
875:→
869::
796:ψ
756:ϕ
642:bijective
638:injective
551:∘
539:∘
395:→
389::
363:→
357::
345:→
339::
305:∘
272:∘
259:∘
251:−
161:∘
155:∘
147:−
36:functions
3719:articles
3461:Physical
3380:Tent map
3270:Discrete
3210:branches
3140:Theorems
2976:Concepts
2729:See also
1154:tent map
1152:and the
1143:Examples
700:between
317:denotes
182:so that
3717:Related
3525:Weather
3463:systems
3256:Chaotic
3002:Fractal
2828:Bibcode
2703:similar
2072:, then
832:, with
646:inverse
640:, then
3623:Hee Oh
3258:maps (
3105:Mixing
2876:
2846:
2783:
1233:orbits
1127:being
912:being
858:, and
788:, and
608:being
480:being
378:, and
34:, two
3534:Chaos
3313:outer
3017:Orbit
2844:S2CID
2352:where
1423:and
956:is a
830:flows
744:Flows
672:is a
524:is a
3260:list
2984:Core
2874:ISBN
2855:2016
2801:link
2781:ISBN
2502:and
2197:and
2130:and
2083:>
1982:<
1976:<
1964:<
1938:>
1590:are
1570:and
1207:and
1148:The
1135:and
1107:and
828:are
720:and
616:and
588:and
438:and
410:are
202:and
85:and
2836:doi
1235:of
916:to
808:on
768:on
648:is
484:to
414:on
42:if
30:In
3763::
2842:.
2834:.
2824:67
2822:.
2818:.
2797:}}
2793:{{
2455:.
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2098:.
1085:.
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740:.
566:.
458:.
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2961:e
2954:t
2947:v
2934:.
2882:.
2857:.
2838::
2830::
2803:)
2789:.
2689:A
2669:x
2666:A
2663:=
2654:x
2628:)
2625:x
2622:(
2619:f
2616:)
2613:x
2610:(
2604:=
2601:)
2598:x
2595:(
2592:g
2571:R
2564:X
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2522:=
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2428:y
2405:.
2399:x
2395:d
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2376:d
2369:=
2366:)
2363:x
2360:(
2357:M
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2338:(
2335:h
2332:(
2329:g
2326:)
2323:x
2320:(
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2298:x
2295:(
2292:f
2269:Y
2263:X
2260::
2257:h
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2226:y
2223:(
2220:g
2217:=
2208:y
2185:)
2182:x
2179:(
2176:f
2173:=
2164:x
2086:0
2080:s
2060:)
2057:t
2054:,
2051:y
2048:(
2039:h
2036:=
2033:)
2030:s
2027:,
2024:)
2021:y
2018:(
2015:h
2012:(
1999:s
1979:t
1973:|
1970:s
1967:|
1961:0
1941:0
1915:Y
1909:y
1889:Y
1883:y
1860:)
1854:,
1851:)
1848:y
1845:(
1842:h
1839:(
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1829:=
1826:}
1822:R
1815:t
1812::
1809:)
1806:t
1803:,
1800:)
1797:y
1794:(
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1788:(
1782:{
1779:=
1776:}
1772:R
1765:t
1762::
1759:)
1756:t
1753:,
1750:y
1747:(
1738:h
1735:{
1732:=
1729:)
1726:)
1720:,
1717:y
1714:(
1709:O
1704:(
1701:h
1676:O
1614:X
1608:Y
1605::
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1466:t
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1443:t
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1434:(
1411:)
1408:t
1405:,
1399:(
1373:h
1365:n
1361:f
1352:1
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1341:=
1336:n
1332:g
1311:h
1305:f
1297:1
1290:h
1286:=
1283:g
1263:f
1243:g
1215:g
1195:f
1137:h
1072:R
1065:t
1045:Y
1039:y
1019:)
1016:t
1013:,
1010:y
1007:(
998:h
995:=
992:)
989:t
986:,
983:)
980:y
977:(
974:h
971:(
944:h
878:X
872:Y
866:h
846:Y
843:,
840:X
816:Y
776:X
728:g
708:f
684:h
660:h
624:h
596:g
576:f
554:g
548:h
545:=
542:h
536:f
512:h
492:g
468:f
446:Y
426:X
398:X
392:Y
386:h
366:Y
360:Y
354:g
351:,
348:X
342:X
336:f
278:,
275:h
267:n
263:f
254:1
247:h
243:=
238:n
234:g
210:g
190:f
167:,
164:h
158:f
150:1
143:h
139:=
136:g
113:h
93:g
73:f
20:)
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