3189:
3181:
2690:, 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
2091:
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.
2090:
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
1859:
2404:
1685:
2632:
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
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3308:
1854:{\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 )}
3188:
2948:
3141:
2900:
51:, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.
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1993:
1220:, since each class contains all functions which share the same dynamics from the topological viewpoint. For example,
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217:
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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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3180:
2399:{\displaystyle f(x)=M^{-1}(x)g(h(x))\quad {\text{where}}\quad M(x)={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}.}
1216:
to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of
120:
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3161:
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24:
2409:
In that case, the dynamical systems can be transformed into each other by the coordinate transformation,
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3053:
3028:
2934:
2713:
Adjoint dynamical systems defined via adjoint functors and natural equivalences in categorical dynamics.
1377:
However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps
3348:
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1049:
520:
2576:
1374:. Speaking informally, topological conjugation is a "change of coordinates" in the topological sense.
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2752:
Arnold V. I. Geometric
Methods in the Theory of Ordinary Differential Equations (Springer, 2020)
2741:
Arnold V. I. Geometric
Methods in the Theory of Ordinary Differential Equations (Springer, 2020)
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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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2706:
There are two reported extensions of the concept of dynamic topological conjugacy:
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20:
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2629:. Orbitally equivalent system differ only in the time parametrization.
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2915:
This article incorporates material from topological conjugation on
2930:
2926:
2901:"Analogous systems, Topological Conjugacy and Adjoint Systems"
1172:
Topological conjugation – unlike semiconjugation – defines an
2142:
Two dynamical systems defined by the differential equations,
1822:
1697:
1664:
1890:. In addition, one must line up the flow of time: for each
2447:
Two dynamical systems on the same state space, defined by
2710:
Analogous systems defined as isomorphic dynamical systems
94:
are iterated functions, and there exists a homeomorphism
2099:
More equivalence criteria can be studied if the flows,
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1467:, which is requiring more than simply that orbits of
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2764:
Alligood, K. T., Sauer, T., and Yorke, J.A. (1997).
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3257:
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1507:homeomorphically. This motivates the definition of
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1979:{\displaystyle 0<\vert s\vert <t<\delta }
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106:
86:
66:
211:are topologically conjugate. Then one must have
2921:Creative Commons Attribution/Share-Alike License
2702:Generalizations of dynamic topological conjugacy
1511:, which also partitions the set of all flows in
2054:{\displaystyle \phi (h(y),s)=h\circ \psi (y,t)}
1155:For certain values in the parameter space, the
1013:{\displaystyle \phi (h(y),t)=h\circ \psi (y,t)}
272:{\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h,}
1367:{\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h}
2942:
360:{\displaystyle f\colon X\to X,g\colon Y\to Y}
8:
2788:: CS1 maint: multiple names: authors list (
1961:
1955:
1814:
1770:
1764:
1723:
40:
2766:Chaos: An Introduction to Dynamical Systems
286:are topologically conjugate as well. Here,
3254:
2968:
2949:
2935:
2927:
2805:"Shift automorphisms in the HĂ©non mapping"
54:To illustrate this directly: suppose that
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1947:
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1317:
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1229:
1201:
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852:
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714:
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522:
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372:
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238:
225:
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196:
176:
134:
122:
99:
79:
59:
1148:The logistic map of unit height and the
43:of flows, are important in the study of
2734:
1447:to be topologically conjugate for each
161:{\displaystyle g=h^{-1}\circ f\circ h,}
2781:
2696:algebraic and geometric multiplicities
2566:{\displaystyle \mu :X\to \mathbb {R} }
1305:{\displaystyle g=h^{-1}\circ f\circ h}
1244:are mapped to homeomorphic orbits of
2883:"Complexity and Categorical Dynamics"
2139:, arise from differential equations.
7:
1120:means, by definition, that they are
601:means, by definition, that they are
3089:Measure-preserving dynamical system
2694:, so their eigenvalues, as well as
2383:
2364:
14:
3657:Oleksandr Mykolayovych Sharkovsky
2803:Devaney, R.; Nitecki, Z. (1979).
2539:if there is a positive function,
1264:through the conjugation. Writing
1067:{\displaystyle t\in \mathbb {R} }
548:{\displaystyle f\circ h=h\circ g}
3187:
3179:
2622:{\displaystyle g(x)=\mu (x)f(x)}
2528:{\displaystyle {\dot {x}}=g(x)}
2484:{\displaystyle {\dot {x}}=f(x)}
2344:
2338:
2223:{\displaystyle {\dot {y}}=g(y)}
2179:{\displaystyle {\dot {x}}=f(x)}
1440:{\displaystyle \psi (\cdot ,t)}
1405:{\displaystyle \phi (\cdot ,t)}
3422:Rabinovich–Fabrikant equations
2919:, which is licensed under the
2859:Elements of Bifurcation Theory
2768:. Springer. pp. 114–124.
2616:
2610:
2604:
2598:
2589:
2583:
2555:
2522:
2516:
2478:
2472:
2431:
2425:
2377:
2371:
2354:
2348:
2335:
2332:
2326:
2320:
2314:
2308:
2289:
2283:
2254:
2217:
2211:
2173:
2167:
2095:Smooth and orbital equivalence
2048:
2036:
2021:
2012:
2006:
2000:
1848:
1839:
1833:
1827:
1797:
1788:
1782:
1776:
1747:
1735:
1717:
1714:
1702:
1692:
1672:{\displaystyle {\mathcal {O}}}
1599:
1583:, if there is a homeomorphism
1434:
1422:
1399:
1387:
1007:
995:
980:
971:
965:
959:
872:{\displaystyle h\colon Y\to X}
863:
392:{\displaystyle h\colon Y\to X}
383:
351:
333:
41:§ Topological equivalence
1:
2861:(Second ed.). Springer.
2663:{\displaystyle {\dot {x}}=Ax}
1935:{\displaystyle \delta >0}
1152:are topologically conjugate.
1145:are topologically conjugate.
3157:Poincaré recurrence theorem
2857:Kuznetsov, Yuri A. (1998).
1122:topologically semiconjugate
925:means, by definition, that
903:topologically semiconjugate
603:topologically semiconjugate
493:means, by definition, that
471:topologically semiconjugate
3777:
3152:Poincaré–Bendixson theorem
1312:makes this fact evident:
3504:Swinging Atwood's machine
3177:
3147:Krylov–Bogolyubov theorem
3024:
1679:denote an orbit, one has
3412:Lotka–Volterra equations
3236:Synchronization of chaos
3039:axiom A dynamical system
2263:{\displaystyle h:X\to Y}
1608:{\displaystyle h:Y\to X}
1581:topologically equivalent
3397:Double scroll attractor
3162:Stable manifold theorem
3069:False nearest neighbors
1535:Topological equivalence
1509:topological equivalence
1487:be mapped to orbits of
1159:when restricted to its
1118:topologically conjugate
687:topological conjugation
599:topologically conjugate
29:topologically conjugate
3437:Van der Pol oscillator
3417:Mackey–Glass equations
3049:Box-counting dimension
2684:
2664:
2623:
2567:
2529:
2485:
2438:
2437:{\displaystyle y=h(x)}
2400:
2264:
2224:
2180:
2133:
2113:
2081:
2080:{\displaystyle s>0}
2055:
1980:
1936:
1910:
1909:{\displaystyle y\in Y}
1884:
1883:{\displaystyle y\in Y}
1855:
1673:
1649:
1629:
1609:
1573:
1553:
1539:We say that two flows
1525:
1501:
1481:
1461:
1441:
1406:
1368:
1306:
1258:
1238:
1210:
1190:
1110:
1090:
1068:
1040:
1039:{\displaystyle y\in Y}
1014:
939:
919:
895:
873:
841:
811:
791:
771:
751:
723:
703:
679:
655:
619:
591:
571:
549:
507:
487:
463:
441:
421:
393:
361:
300:
299:{\displaystyle \circ }
273:
205:
185:
162:
108:
88:
68:
3587:Svetlana Jitomirskaya
3494:Multiscroll attractor
3339:Interval exchange map
3292:Dyadic transformation
3277:Complex quadratic map
3119:Topological conjugacy
3054:Correlation dimension
3029:Anosov diffeomorphism
2685:
2665:
2624:
2568:
2530:
2486:
2439:
2401:
2265:
2225:
2181:
2134:
2132:{\displaystyle \psi }
2114:
2112:{\displaystyle \phi }
2082:
2056:
1981:
1937:
1911:
1885:
1856:
1674:
1650:
1648:{\displaystyle \phi }
1630:
1628:{\displaystyle \psi }
1610:
1574:
1572:{\displaystyle \psi }
1554:
1552:{\displaystyle \phi }
1526:
1502:
1500:{\displaystyle \psi }
1482:
1480:{\displaystyle \phi }
1462:
1442:
1407:
1369:
1307:
1259:
1239:
1211:
1191:
1128:is a homeomorphism.
1111:
1109:{\displaystyle \psi }
1091:
1089:{\displaystyle \phi }
1069:
1041:
1015:
940:
920:
918:{\displaystyle \psi }
896:
894:{\displaystyle \phi }
874:
842:
812:
792:
790:{\displaystyle \psi }
772:
752:
750:{\displaystyle \phi }
724:
704:
680:
656:
620:
592:
572:
550:
508:
488:
464:
442:
422:
394:
362:
301:
274:
206:
186:
163:
109:
89:
69:
3756:Topological dynamics
3597:Edward Norton Lorenz
2674:
2636:
2577:
2543:
2537:orbitally equivalent
2495:
2451:
2413:
2277:
2242:
2190:
2146:
2123:
2103:
2065:
1994:
1946:
1920:
1894:
1868:
1686:
1659:
1639:
1619:
1615:, mapping orbits of
1587:
1563:
1543:
1515:
1491:
1471:
1451:
1416:
1381:
1316:
1268:
1248:
1228:
1200:
1180:
1174:equivalence relation
1100:
1080:
1050:
1024:
953:
929:
909:
885:
851:
825:
801:
781:
761:
741:
713:
693:
669:
645:
609:
581:
561:
521:
497:
477:
453:
431:
411:
401:continuous functions
371:
321:
308:function composition
290:
218:
195:
175:
121:
98:
78:
58:
3557:Mitchell Feigenbaum
3499:Population dynamics
3484:Hénon–Heiles system
3344:Irrational rotation
3297:Dynamical billiards
3282:Coupled map lattice
3142:Liouville's theorem
3074:Hausdorff dimension
3059:Conservative system
3044:Bifurcation diagram
2889:on August 19, 2009.
2821:1979CMaPh..67..137D
2724:Commutative diagram
2232:smoothly equivalent
840:{\displaystyle X,Y}
47:and more generally
16:Concept in topology
3735:Santa Fe Institute
3602:Aleksandr Lyapunov
3432:Three-body problem
3319:Gingerbreadman map
3206:Bifurcation theory
3084:Lyapunov stability
2829:10.1007/bf01221362
2698:, must be equal).
2680:
2660:
2619:
2563:
2525:
2481:
2434:
2396:
2260:
2220:
2176:
2129:
2109:
2077:
2051:
1976:
1932:
1906:
1880:
1851:
1669:
1645:
1625:
1605:
1569:
1549:
1521:
1497:
1477:
1457:
1437:
1402:
1364:
1302:
1254:
1234:
1206:
1186:
1106:
1086:
1064:
1036:
1010:
935:
915:
891:
869:
837:
807:
787:
767:
747:
719:
699:
675:
651:
615:
587:
567:
545:
503:
483:
459:
437:
417:
405:topological spaces
389:
357:
296:
269:
201:
181:
158:
104:
84:
64:
45:iterated functions
3743:
3742:
3607:Benoît Mandelbrot
3572:Martin Gutzwiller
3562:Peter Grassberger
3445:
3444:
3427:Rössler attractor
3175:
3174:
3079:Invariant measure
3001:Lyapunov exponent
2683:{\displaystyle A}
2670:. If the matrix,
2648:
2535:, are said to be
2507:
2463:
2391:
2342:
2230:, are said to be
2202:
2158:
1916:, there exists a
1524:{\displaystyle X}
1460:{\displaystyle t}
1257:{\displaystyle f}
1237:{\displaystyle g}
1218:dynamical systems
1209:{\displaystyle g}
1189:{\displaystyle f}
938:{\displaystyle h}
810:{\displaystyle Y}
770:{\displaystyle X}
722:{\displaystyle g}
702:{\displaystyle f}
678:{\displaystyle h}
654:{\displaystyle h}
618:{\displaystyle h}
590:{\displaystyle g}
570:{\displaystyle f}
506:{\displaystyle h}
486:{\displaystyle g}
462:{\displaystyle f}
440:{\displaystyle Y}
420:{\displaystyle X}
204:{\displaystyle g}
184:{\displaystyle f}
107:{\displaystyle h}
87:{\displaystyle g}
67:{\displaystyle f}
49:dynamical systems
3768:
3715:Butterfly effect
3627:Itamar Procaccia
3577:Brosl Hasslacher
3474:Elastic pendulum
3402:Duffing equation
3349:Kaplan–Yorke map
3267:Arnold's cat map
3255:
3231:Stability theory
3216:Dynamical system
3211:Control of chaos
3191:
3183:
3167:Takens's theorem
3099:Poincaré section
2969:
2951:
2944:
2937:
2928:
2909:
2908:
2903:. Archived from
2897:
2891:
2890:
2885:. Archived from
2879:
2873:
2872:
2854:
2848:
2847:
2845:
2843:
2809:Comm. Math. Phys
2800:
2794:
2793:
2787:
2779:
2761:
2755:
2750:
2744:
2739:
2689:
2687:
2686:
2681:
2669:
2667:
2666:
2661:
2650:
2649:
2641:
2628:
2626:
2625:
2620:
2572:
2570:
2569:
2564:
2562:
2534:
2532:
2531:
2526:
2509:
2508:
2500:
2490:
2488:
2487:
2482:
2465:
2464:
2456:
2443:
2441:
2440:
2435:
2405:
2403:
2402:
2397:
2392:
2390:
2386:
2380:
2367:
2361:
2343:
2340:
2307:
2306:
2269:
2267:
2266:
2261:
2229:
2227:
2226:
2221:
2204:
2203:
2195:
2185:
2183:
2182:
2177:
2160:
2159:
2151:
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2086:
2084:
2083:
2078:
2060:
2058:
2057:
2052:
1989:
1985:
1983:
1982:
1977:
1941:
1939:
1938:
1933:
1915:
1913:
1912:
1907:
1889:
1887:
1886:
1881:
1860:
1858:
1857:
1852:
1826:
1825:
1813:
1763:
1701:
1700:
1678:
1676:
1675:
1670:
1668:
1667:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1578:
1576:
1575:
1570:
1558:
1556:
1555:
1550:
1530:
1528:
1527:
1522:
1506:
1504:
1503:
1498:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1411:
1409:
1408:
1403:
1373:
1371:
1370:
1365:
1357:
1356:
1344:
1343:
1328:
1327:
1311:
1309:
1308:
1303:
1289:
1288:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1215:
1213:
1212:
1207:
1195:
1193:
1192:
1187:
1127:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1073:
1071:
1070:
1065:
1063:
1045:
1043:
1042:
1037:
1019:
1017:
1016:
1011:
944:
942:
941:
936:
924:
922:
921:
916:
900:
898:
897:
892:
878:
876:
875:
870:
846:
844:
843:
838:
816:
814:
813:
808:
796:
794:
793:
788:
776:
774:
773:
768:
756:
754:
753:
748:
728:
726:
725:
720:
708:
706:
705:
700:
684:
682:
681:
676:
660:
658:
657:
652:
624:
622:
621:
616:
596:
594:
593:
588:
576:
574:
573:
568:
554:
552:
551:
546:
512:
510:
509:
504:
492:
490:
489:
484:
468:
466:
465:
460:
446:
444:
443:
438:
426:
424:
423:
418:
398:
396:
395:
390:
366:
364:
363:
358:
305:
303:
302:
297:
284:iterated systems
278:
276:
275:
270:
259:
258:
246:
245:
230:
229:
210:
208:
207:
202:
190:
188:
187:
182:
167:
165:
164:
159:
142:
141:
113:
111:
110:
105:
93:
91:
90:
85:
73:
71:
70:
65:
3776:
3775:
3771:
3770:
3769:
3767:
3766:
3765:
3746:
3745:
3744:
3739:
3707:
3701:
3647:Caroline Series
3542:Mary Cartwright
3524:
3518:
3469:Double pendulum
3451:
3441:
3390:
3383:
3309:Exponential map
3260:
3246:
3240:
3198:
3192:
3185:
3171:
3137:Ergodic theorem
3130:
3123:
3114:Stable manifold
3104:Recurrence plot
3020:
2974:
2960:
2955:
2912:
2899:
2898:
2894:
2881:
2880:
2876:
2869:
2856:
2855:
2851:
2841:
2839:
2802:
2801:
2797:
2780:
2776:
2763:
2762:
2758:
2751:
2747:
2740:
2736:
2732:
2720:
2704:
2672:
2671:
2634:
2633:
2575:
2574:
2541:
2540:
2493:
2492:
2449:
2448:
2411:
2410:
2381:
2362:
2295:
2275:
2274:
2240:
2239:
2188:
2187:
2144:
2143:
2121:
2120:
2101:
2100:
2097:
2063:
2062:
1992:
1991:
1987:
1944:
1943:
1918:
1917:
1892:
1891:
1866:
1865:
1684:
1683:
1657:
1656:
1637:
1636:
1617:
1616:
1585:
1584:
1561:
1560:
1541:
1540:
1537:
1513:
1512:
1489:
1488:
1469:
1468:
1449:
1448:
1414:
1413:
1379:
1378:
1348:
1332:
1319:
1314:
1313:
1277:
1266:
1265:
1246:
1245:
1226:
1225:
1198:
1197:
1178:
1177:
1170:
1134:
1125:
1098:
1097:
1078:
1077:
1048:
1047:
1022:
1021:
951:
950:
927:
926:
907:
906:
883:
882:
849:
848:
823:
822:
799:
798:
779:
778:
759:
758:
739:
738:
735:
711:
710:
691:
690:
667:
666:
643:
642:
625:is furthermore
607:
606:
579:
578:
559:
558:
519:
518:
495:
494:
475:
474:
451:
450:
429:
428:
409:
408:
369:
368:
319:
318:
316:
288:
287:
250:
234:
221:
216:
215:
193:
192:
173:
172:
130:
119:
118:
96:
95:
76:
75:
56:
55:
27:are said to be
17:
12:
11:
5:
3774:
3772:
3764:
3763:
3761:Homeomorphisms
3758:
3748:
3747:
3741:
3740:
3738:
3737:
3732:
3730:Predictability
3727:
3722:
3717:
3711:
3709:
3703:
3702:
3700:
3699:
3697:Lai-Sang Young
3694:
3692:James A. Yorke
3689:
3687:Amie Wilkinson
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3622:Henri Poincaré
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3528:
3526:
3520:
3519:
3517:
3516:
3511:
3506:
3501:
3496:
3491:
3489:Kicked rotator
3486:
3481:
3476:
3471:
3466:
3461:
3459:Chua's circuit
3455:
3453:
3447:
3446:
3443:
3442:
3440:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3393:
3391:
3388:
3385:
3384:
3382:
3381:
3379:Zaslavskii map
3376:
3374:Tinkerbell map
3371:
3366:
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3305:
3304:
3294:
3289:
3284:
3279:
3274:
3269:
3263:
3261:
3258:
3252:
3242:
3241:
3239:
3238:
3233:
3228:
3223:
3221:Ergodic theory
3218:
3213:
3208:
3202:
3200:
3194:
3193:
3178:
3176:
3173:
3172:
3170:
3169:
3164:
3159:
3154:
3149:
3144:
3139:
3133:
3131:
3128:
3125:
3124:
3122:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3086:
3081:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3025:
3022:
3021:
3019:
3018:
3013:
3011:Periodic point
3008:
3003:
2998:
2993:
2988:
2983:
2977:
2975:
2972:
2966:
2962:
2961:
2956:
2954:
2953:
2946:
2939:
2931:
2911:
2910:
2907:on 2015-02-25.
2892:
2874:
2867:
2849:
2815:(2): 137–146.
2795:
2774:
2756:
2745:
2733:
2731:
2728:
2727:
2726:
2719:
2716:
2715:
2714:
2711:
2703:
2700:
2679:
2659:
2656:
2653:
2647:
2644:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2561:
2557:
2554:
2551:
2548:
2524:
2521:
2518:
2515:
2512:
2506:
2503:
2480:
2477:
2474:
2471:
2468:
2462:
2459:
2433:
2430:
2427:
2424:
2421:
2418:
2407:
2406:
2395:
2389:
2385:
2379:
2376:
2373:
2370:
2366:
2359:
2356:
2353:
2350:
2347:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2305:
2302:
2298:
2294:
2291:
2288:
2285:
2282:
2259:
2256:
2253:
2250:
2247:
2236:diffeomorphism
2234:if there is a
2219:
2216:
2213:
2210:
2207:
2201:
2198:
2175:
2172:
2169:
2166:
2163:
2157:
2154:
2128:
2108:
2096:
2093:
2076:
2073:
2070:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1942:such that, if
1931:
1928:
1925:
1905:
1902:
1899:
1879:
1876:
1873:
1862:
1861:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1824:
1819:
1816:
1812:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1762:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1699:
1694:
1691:
1666:
1644:
1624:
1604:
1601:
1598:
1595:
1592:
1568:
1548:
1536:
1533:
1520:
1496:
1476:
1456:
1436:
1433:
1430:
1427:
1424:
1421:
1401:
1398:
1395:
1392:
1389:
1386:
1363:
1360:
1355:
1351:
1347:
1342:
1339:
1335:
1331:
1326:
1322:
1301:
1298:
1295:
1292:
1287:
1284:
1280:
1276:
1273:
1253:
1233:
1205:
1185:
1169:
1166:
1165:
1164:
1153:
1146:
1133:
1130:
1105:
1085:
1062:
1058:
1055:
1035:
1032:
1029:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
934:
914:
890:
868:
865:
862:
859:
856:
836:
833:
830:
806:
786:
766:
746:
734:
731:
718:
698:
674:
650:
614:
586:
566:
544:
541:
538:
535:
532:
529:
526:
502:
482:
458:
436:
416:
388:
385:
382:
379:
376:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
315:
312:
295:
280:
279:
268:
265:
262:
257:
253:
249:
244:
241:
237:
233:
228:
224:
200:
180:
169:
168:
157:
154:
151:
148:
145:
140:
137:
133:
129:
126:
103:
83:
63:
15:
13:
10:
9:
6:
4:
3:
2:
3773:
3762:
3759:
3757:
3754:
3753:
3751:
3736:
3733:
3731:
3728:
3726:
3725:Edge of chaos
3723:
3721:
3718:
3716:
3713:
3712:
3710:
3704:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3682:Marcelo Viana
3680:
3678:
3675:
3673:
3672:Audrey Terras
3670:
3668:
3667:Floris Takens
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3567:Celso Grebogi
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3547:Chen Guanrong
3545:
3543:
3540:
3538:
3535:
3533:
3532:Michael Berry
3530:
3529:
3527:
3521:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3456:
3454:
3448:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3407:Lorenz system
3405:
3403:
3400:
3398:
3395:
3394:
3392:
3386:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3354:Langton's ant
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3329:Horseshoe map
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3303:
3300:
3299:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3264:
3262:
3256:
3253:
3250:
3243:
3237:
3234:
3232:
3229:
3227:
3226:Quantum chaos
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3203:
3201:
3195:
3190:
3186:
3182:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3134:
3132:
3126:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3034:Arnold tongue
3032:
3030:
3027:
3026:
3023:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2978:
2976:
2970:
2967:
2963:
2959:
2952:
2947:
2945:
2940:
2938:
2933:
2932:
2929:
2925:
2924:
2922:
2918:
2906:
2902:
2896:
2893:
2888:
2884:
2878:
2875:
2870:
2868:0-387-98382-1
2864:
2860:
2853:
2850:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2799:
2796:
2791:
2785:
2777:
2775:0-387-94677-2
2771:
2767:
2760:
2757:
2754:
2749:
2746:
2743:
2738:
2735:
2729:
2725:
2722:
2721:
2717:
2712:
2709:
2708:
2707:
2701:
2699:
2697:
2693:
2677:
2657:
2654:
2651:
2645:
2642:
2630:
2613:
2607:
2601:
2595:
2592:
2586:
2580:
2552:
2549:
2546:
2538:
2519:
2513:
2510:
2504:
2501:
2475:
2469:
2466:
2460:
2457:
2445:
2428:
2422:
2419:
2416:
2393:
2387:
2374:
2368:
2357:
2351:
2345:
2329:
2323:
2317:
2311:
2303:
2300:
2296:
2292:
2286:
2280:
2273:
2272:
2271:
2257:
2251:
2248:
2245:
2237:
2233:
2214:
2208:
2205:
2199:
2196:
2170:
2164:
2161:
2155:
2152:
2140:
2126:
2106:
2094:
2092:
2088:
2074:
2071:
2068:
2045:
2042:
2039:
2033:
2030:
2027:
2024:
2018:
2015:
2009:
2003:
1997:
1990:is such that
1973:
1970:
1967:
1964:
1958:
1952:
1949:
1929:
1926:
1923:
1903:
1900:
1897:
1877:
1874:
1871:
1845:
1842:
1836:
1830:
1817:
1806:
1803:
1800:
1794:
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1773:
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1726:
1720:
1711:
1708:
1705:
1689:
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1681:
1680:
1642:
1635:to orbits of
1622:
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1596:
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1566:
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1183:
1175:
1167:
1162:
1158:
1154:
1151:
1150:Bernoulli map
1147:
1144:
1140:
1136:
1135:
1131:
1129:
1123:
1119:
1103:
1083:
1075:
1056:
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1033:
1030:
1027:
1004:
1001:
998:
992:
989:
986:
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977:
974:
968:
962:
956:
948:
932:
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834:
831:
828:
820:
804:
784:
764:
744:
732:
730:
716:
696:
688:
672:
664:
663:homeomorphism
648:
640:
636:
632:
628:
612:
604:
600:
584:
564:
556:
542:
539:
536:
533:
530:
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516:
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480:
472:
456:
448:
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406:
402:
386:
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354:
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313:
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198:
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155:
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131:
127:
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117:
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115:
101:
81:
61:
52:
50:
46:
42:
38:
37:homeomorphism
34:
30:
26:
22:
3677:Mary Tsingou
3642:David Ruelle
3637:Otto Rössler
3582:Michel HĂ©non
3552:Leon O. Chua
3509:Tilt-A-Whirl
3479:FPUT problem
3364:Standard map
3359:Logistic map
3184:
3118:
2958:Chaos theory
2914:
2913:
2905:the original
2895:
2887:the original
2877:
2858:
2852:
2840:. Retrieved
2812:
2808:
2798:
2765:
2759:
2748:
2737:
2705:
2631:
2573:, such that
2536:
2446:
2408:
2270:, such that
2231:
2141:
2098:
2089:
1863:
1580:
1538:
1508:
1376:
1171:
1139:logistic map
1121:
1117:
1076:
902:
881:
736:
686:
685:is termed a
602:
598:
557:
470:
449:
317:
281:
170:
53:
33:there exists
28:
18:
3662:Nina Snaith
3652:Yakov Sinai
3537:Rufus Bowen
3287:Duffing map
3272:Baker's map
3197:Theoretical
3109:SRB measure
3016:Phase space
2986:Bifurcation
2842:2 September
1020:, for each
737:Similarly,
665:; further,
282:and so the
114:such that
21:mathematics
3750:Categories
3720:Complexity
3617:Edward Ott
3464:Convection
3389:Continuous
3064:Ergodicity
2917:PlanetMath
2730:References
1986:, and if
1168:Discussion
949:such that
947:surjection
879:as above.
641:too; i.e.
639:continuous
633:, and its
517:such that
515:surjection
314:Definition
3632:Mary Rees
3592:Bryna Kra
3525:theorists
3334:Ikeda map
3324:HĂ©non map
3314:Gauss map
2996:Limit set
2981:Attractor
2837:121479458
2784:cite book
2646:˙
2596:μ
2556:→
2547:μ
2505:˙
2461:˙
2301:−
2255:→
2200:˙
2156:˙
2127:ψ
2107:ϕ
2034:ψ
2031:∘
1998:ϕ
1974:δ
1924:δ
1901:∈
1875:∈
1864:for each
1846:ϕ
1807:∈
1774:ϕ
1757:∈
1733:ψ
1730:∘
1712:ψ
1643:ϕ
1623:ψ
1600:→
1567:ψ
1547:ϕ
1495:ψ
1475:ϕ
1426:⋅
1420:ψ
1391:⋅
1385:ϕ
1359:∘
1346:∘
1338:−
1297:∘
1291:∘
1283:−
1161:Julia set
1157:HĂ©non map
1104:ψ
1084:ϕ
1057:∈
1031:∈
993:ψ
990:∘
957:ϕ
913:ψ
889:ϕ
864:→
858::
785:ψ
745:ϕ
631:bijective
627:injective
540:∘
528:∘
384:→
378::
352:→
346::
334:→
328::
294:∘
261:∘
248:∘
240:−
150:∘
144:∘
136:−
25:functions
3708:articles
3450:Physical
3369:Tent map
3259:Discrete
3199:branches
3129:Theorems
2965:Concepts
2718:See also
1143:tent map
1141:and the
1132:Examples
689:between
306:denotes
171:so that
3706:Related
3514:Weather
3452:systems
3245:Chaotic
2991:Fractal
2817:Bibcode
2692:similar
2061:, then
821:, with
635:inverse
629:, then
3612:Hee Oh
3247:maps (
3094:Mixing
2865:
2835:
2772:
1222:orbits
1116:being
901:being
847:, and
777:, and
597:being
469:being
367:, and
23:, two
3523:Chaos
3302:outer
3006:Orbit
2833:S2CID
2341:where
1412:and
945:is a
819:flows
733:Flows
661:is a
513:is a
3249:list
2973:Core
2863:ISBN
2844:2016
2790:link
2770:ISBN
2491:and
2186:and
2119:and
2072:>
1971:<
1965:<
1953:<
1927:>
1579:are
1559:and
1196:and
1137:The
1124:and
1096:and
817:are
709:and
605:and
577:and
427:and
399:are
191:and
74:and
2825:doi
1224:of
905:to
797:on
757:on
637:is
473:to
403:on
31:if
19:In
3752::
2831:.
2823:.
2813:67
2811:.
2807:.
2786:}}
2782:{{
2444:.
2238:,
2087:.
1074:.
1046:,
729:.
555:.
447:.
407:,
310:.
35:a
3251:)
2950:e
2943:t
2936:v
2923:.
2871:.
2846:.
2827::
2819::
2792:)
2778:.
2678:A
2658:x
2655:A
2652:=
2643:x
2617:)
2614:x
2611:(
2608:f
2605:)
2602:x
2599:(
2593:=
2590:)
2587:x
2584:(
2581:g
2560:R
2553:X
2550::
2523:)
2520:x
2517:(
2514:g
2511:=
2502:x
2479:)
2476:x
2473:(
2470:f
2467:=
2458:x
2432:)
2429:x
2426:(
2423:h
2420:=
2417:y
2394:.
2388:x
2384:d
2378:)
2375:x
2372:(
2369:h
2365:d
2358:=
2355:)
2352:x
2349:(
2346:M
2336:)
2333:)
2330:x
2327:(
2324:h
2321:(
2318:g
2315:)
2312:x
2309:(
2304:1
2297:M
2293:=
2290:)
2287:x
2284:(
2281:f
2258:Y
2252:X
2249::
2246:h
2218:)
2215:y
2212:(
2209:g
2206:=
2197:y
2174:)
2171:x
2168:(
2165:f
2162:=
2153:x
2075:0
2069:s
2049:)
2046:t
2043:,
2040:y
2037:(
2028:h
2025:=
2022:)
2019:s
2016:,
2013:)
2010:y
2007:(
2004:h
2001:(
1988:s
1968:t
1962:|
1959:s
1956:|
1950:0
1930:0
1904:Y
1898:y
1878:Y
1872:y
1849:)
1843:,
1840:)
1837:y
1834:(
1831:h
1828:(
1823:O
1818:=
1815:}
1811:R
1804:t
1801::
1798:)
1795:t
1792:,
1789:)
1786:y
1783:(
1780:h
1777:(
1771:{
1768:=
1765:}
1761:R
1754:t
1751::
1748:)
1745:t
1742:,
1739:y
1736:(
1727:h
1724:{
1721:=
1718:)
1715:)
1709:,
1706:y
1703:(
1698:O
1693:(
1690:h
1665:O
1603:X
1597:Y
1594::
1591:h
1519:X
1455:t
1435:)
1432:t
1429:,
1423:(
1400:)
1397:t
1394:,
1388:(
1362:h
1354:n
1350:f
1341:1
1334:h
1330:=
1325:n
1321:g
1300:h
1294:f
1286:1
1279:h
1275:=
1272:g
1252:f
1232:g
1204:g
1184:f
1126:h
1061:R
1054:t
1034:Y
1028:y
1008:)
1005:t
1002:,
999:y
996:(
987:h
984:=
981:)
978:t
975:,
972:)
969:y
966:(
963:h
960:(
933:h
867:X
861:Y
855:h
835:Y
832:,
829:X
805:Y
765:X
717:g
697:f
673:h
649:h
613:h
585:g
565:f
543:g
537:h
534:=
531:h
525:f
501:h
481:g
457:f
435:Y
415:X
387:X
381:Y
375:h
355:Y
349:Y
343:g
340:,
337:X
331:X
325:f
267:,
264:h
256:n
252:f
243:1
236:h
232:=
227:n
223:g
199:g
179:f
156:,
153:h
147:f
139:1
132:h
128:=
125:g
102:h
82:g
62:f
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