1609:
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617:
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25:
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122:
609:
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1353:
notation: It shifts the binary point one place to the right; then, if what appears to the left of the binary point is a "one" it changes all ones to zeroes and vice versa (with the exception of the final bit "one" in the case of a finite binary expansion); starting from an irrational number, this
1086:
1915:
892:
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of the map – that is, it is the smallest invariant subset of the real line under this map. If μ is greater than the square root of 2, these intervals merge, and the Julia set is the whole interval from μ − μ/2 to μ/2 (see bifurcation
1267:{\displaystyle {\frac {\mu }{\mu ^{4}+1}}\to {\frac {\mu ^{2}}{\mu ^{4}+1}}\to {\frac {\mu ^{3}}{\mu ^{4}+1}}\to {\frac {\mu ^{4}}{\mu ^{4}+1}}\to {\frac {\mu }{\mu ^{4}+1}}{\mbox{ appears at }}\mu \approx 1.8393}
743:
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1711:
776:
1069:{\displaystyle {\frac {\mu }{\mu ^{3}+1}}\to {\frac {\mu ^{2}}{\mu ^{3}+1}}\to {\frac {\mu ^{3}}{\mu ^{3}+1}}\to {\frac {\mu }{\mu ^{3}+1}}{\mbox{ appears at }}\mu ={\frac {1+{\sqrt {5}}}{2}}}
237:
1280:
If μ equals 2 the system maps the interval onto itself. There are now periodic points with every orbit length within this interval, as well as non-periodic points. The periodic points are
2130:
522:{\displaystyle x_{n+1}=f_{\mu }(x_{n})={\begin{cases}\mu x_{n}&\mathrm {for} ~~x_{n}<{\frac {1}{2}}\\\mu (1-x_{n})&\mathrm {for} ~~{\frac {1}{2}}\leq x_{n}\end{cases}}}
2017:
1649:
A closer look at the orbit diagram shows that there are 4 separated regions at μ ≈ 1. For further magnification, 2 reference lines (red) are drawn from the tip to suitable
1982:
1703:
1452:
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2044:
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1414:
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1316:
318:
2163:. Thus data from an asymmetric tent map cannot be distinguished, using the autocorrelation function, from data generated by a first-order autoregressive process.
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1666:
With distance measured from the corresponding reference lines, further detail appears in the upper and lower part of the map. (total 8 separated regions at some μ)
2171:
The tent map has found applications in social cognitive optimization, chaos in economics, image encryption, on risk and market sentiments for pricing, etc.
1591:
within the interval . The Julia set still contains an infinite number of both non-periodic and periodic points (including orbits for any orbit length) but
680:
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of
767:
are unstable (i.e. nearby points move away from the orbits rather than towards them). Orbits with longer lengths appear as μ increases. For example:
3008:
2838:
701:
1910:{\displaystyle v_{n+1}={\begin{cases}v_{n}/a&\mathrm {for} ~~v_{n}\in \\(1-v_{n})/(1-a)&\mathrm {for} ~~v_{n}\in \end{cases}}}
1494:
1599:(obtained by successively deleting middle thirds from subsets of the unit line) is the Julia set of the tent map for μ = 3.
887:{\displaystyle {\frac {\mu }{\mu ^{2}+1}}\to {\frac {\mu ^{2}}{\mu ^{2}+1}}\to {\frac {\mu }{\mu ^{2}+1}}{\mbox{ appears at }}\mu =1}
1608:
684:
close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at
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2478:
2671:
643:
Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic.
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172:
89:
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108:
61:
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the system maps a set of intervals between μ − μ/2 and μ/2 to themselves. This set of intervals is the
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may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting
68:
46:
2686:
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3033:
2941:
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532:
where μ is a positive real constant. Choosing for instance the parameter μ = 2, the effect of the function
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If μ is between 1 and 2 the interval contains both periodic and non-periodic points, although all of the
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variable with respect to number of iterations) stops fluctuating and no values are observed after
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Bifurcation diagram for the tent map. Higher density indicates increased probability of the
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2623:
2256:"Social cognitive optimization with tent map for combined heat and power economic dispatch"
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of the interval assumes new subsequent positions as described above, generating a sequence
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8:
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640:, and thus the behaviours of the two maps are in this sense identical under iteration.
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If μ is greater than 2 the map's Julia set becomes disconnected, and breaks up into a
82:
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2240:
Brock, W. A., "Distinguishing random and deterministic systems: Abridged version,"
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2046:} will have the same autocorrelation function as will data from the first-order
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2299:"Chapter 40 Non-linear dynamical systems: Instability and chaos in economics"
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1354:
process goes on forever without repeating itself. The invariant measure for
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point within will now eventually diverge towards infinity. The canonical
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variable acquiring that value for the given value of the μ parameter.
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2816:
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2545:
2180:
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1417:
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2399:"On risk and market sentiments driving financial share price dynamics"
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2510:
2397:
Lampart, Marek; Lampartová, Alžběta; Orlando, Giuseppe (2023-09-01).
2185:
1619:= 2.0 which shows numerical error: "the plot of time series (plot of
1281:
756:
738:{\displaystyle 0.61\to 0.585\to 0.6225\to 0.56625\to 0.650625\ldots }
652:
1640:
24:
2272:
2568:
2520:
1657:
1574:{\displaystyle x_{n}={\tfrac {2}{\pi }}\sin ^{-1}(y_{n}^{1/2}).}
588:
case of the tent map is a non-linear transformation of both the
543:
to get again the interval . Iterating the procedure, any point
3141:
2456:
2358:
Li, Chunhu; Luo, Guangchun; Qin, Ke; Li, Chunbao (2017-01-01).
1458:
to each other: Denoting the logistically evolving variable as
1675:
The asymmetric tent map is essentially a distorted, but still
1389:} will show zero autocorrelation at all non-zero lags. Thus
121:
688:= 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at
1903:
608:
515:
243:
129:
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less than or equal to 1/2 are fixed points of the system.
140: = 0.4 over the tent map with μ = 1.9.
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2396:
2260:
2360:"An image encryption scheme based on chaotic tent map"
1998:
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1322:. This can be seen by noting what the map does when
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is the uniform density over the unit interval. The
257:. For the values of the parameter μ within 0 and 2,
2297:Brock, William A.; Dechert, W. Davis (1991-01-01),
1420:using the autocorrelation function. Note that the
49:. Unsourced material may be challenged and removed.
2226:Iterated Maps on the Interval as Dynamical Systems
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232:{\displaystyle f_{\mu }(x):=\mu \min\{x,\,1-x\},}
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2335:10.1007/978-3-030-70982-2#editorsandaffiliations
1661:Further magnification shows 8 separated regions.
201:
1635:
1644:Magnification near the tip shows more details.
1288:. In fact, the dynamics will be non-periodic
2472:
2296:
1984:case of the tent map is the present case of
223:
204:
2305:, vol. 4, Elsevier, pp. 2209–2235
2125:{\displaystyle w_{n+1}=(2a-1)w_{n}+u_{n+1}}
133:Example of iterating the initial condition
2479:
2465:
2357:
628:The tent map with parameter μ = 2 and the
2424:
2414:
2342:
2281:
2271:
2161:independently and identically distributed
213:
109:Learn how and when to remove this message
1705:case of the tent map. It is defined by
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1639:
1607:
658:of the system for all initial values of
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128:
120:
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2234:
3278:
2205:Conjugating the Tent and Logistic Maps
1670:
662:i.e. the system will converge towards
2460:
2254:Sun, Jiaze; Li, Yang (January 2019).
2253:
2231:
47:adding citations to reliable sources
18:
2619:Measure-preserving dynamical system
2501:
1653:at certain μ (e.g., 1.10) as shown.
1603:
13:
2303:Handbook of Mathematical Economics
1862:
1859:
1856:
1769:
1766:
1763:
1615:of the Tent map for the parameter
1362:for a sufficiently long sequence {
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14:
3297:
3187:Oleksandr Mykolayovych Sharkovsky
2445:
2012:{\displaystyle a={\tfrac {1}{2}}}
2717:
2709:
2486:
23:
2166:
1631:= 2.0, initial point is random.
34:needs additional citations for
2952:Rabinovich–Fabrikant equations
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2214:
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708:
666:= 0 from any initial value of
647:If μ is less than 1 the point
612:Orbits of unit-height tent map
467:
448:
372:
359:
192:
186:
1:
2323:"Nonlinearities in Economics"
2191:
1416:cannot be distinguished from
293:in gives rise to a sequence
271:into itself, thus defining a
1636:Magnifying the orbit diagram
603:
7:
2687:Poincaré recurrence theorem
2228:, Boston: Birkhauser, 1980.
2174:
1284:in , so the map has become
10:
3302:
2682:Poincaré–Bendixson theorem
2416:10.1007/s11071-023-08702-5
2244:40, October 1986, 168-195.
2242:Journal of Economic Theory
751:If μ is between 1 and the
242:the name being due to the
125:Graph of tent map function
3235:
3052:
3034:Swinging Atwood's machine
2979:
2917:
2787:
2774:
2726:
2707:
2677:Krylov–Bogolyubov theorem
2657:
2554:
2494:
2376:10.1007/s11071-016-3030-8
1454:case of the tent map are
2942:Lotka–Volterra equations
2766:Synchronization of chaos
2569:axiom A dynamical system
2211:, University of Michigan
1360:autocorrelation function
673:If μ is 1 all values of
152:with parameter μ is the
2927:Double scroll attractor
2692:Stable manifold theorem
2599:False nearest neighbors
638:topologically conjugate
278:on it (equivalently, a
2967:Van der Pol oscillator
2947:Mackey–Glass equations
2579:Box-counting dimension
2153:
2126:
2048:autoregressive process
2040:
2013:
1978:
1977:{\displaystyle \mu =2}
1952:
1911:
1699:
1698:{\displaystyle \mu =2}
1662:
1645:
1632:
1575:
1485:, the homeomorphism is
1479:
1448:
1447:{\displaystyle \mu =2}
1410:
1383:
1343:
1312:
1268:
1251: appears at
1070:
1034: appears at
888:
871: appears at
739:
625:
613:
582:
581:{\displaystyle \mu =2}
523:
314:
233:
141:
126:
3117:Svetlana Jitomirskaya
3024:Multiscroll attractor
2869:Interval exchange map
2822:Dyadic transformation
2807:Complex quadratic map
2649:Topological conjugacy
2584:Correlation dimension
2559:Anosov diffeomorphism
2220:Collett, Pierre, and
2154:
2152:{\displaystyle u_{n}}
2127:
2041:
2039:{\displaystyle v_{n}}
2014:
1979:
1953:
1951:{\displaystyle a\in }
1912:
1700:
1660:
1643:
1611:
1576:
1480:
1478:{\displaystyle y_{n}}
1449:
1411:
1409:{\displaystyle x_{n}}
1384:
1382:{\displaystyle x_{n}}
1344:
1342:{\displaystyle x_{n}}
1313:
1311:{\displaystyle x_{0}}
1269:
1071:
889:
740:
619:
611:
583:
524:
315:
313:{\displaystyle x_{n}}
234:
132:
124:
3127:Edward Norton Lorenz
2222:Eckmann, Jean-Pierre
2136:
2053:
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1962:
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777:
702:
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327:
297:
173:
43:improve this article
3087:Mitchell Feigenbaum
3029:Population dynamics
3014:Hénon–Heiles system
2874:Irrational rotation
2827:Dynamical billiards
2812:Coupled map lattice
2672:Liouville's theorem
2604:Hausdorff dimension
2589:Conservative system
2574:Bifurcation diagram
2409:(17): 16585–16604.
1671:Asymmetric tent map
1564:
280:recurrence relation
246:-like shape of the
3265:Santa Fe Institute
3132:Aleksandr Lyapunov
2962:Three-body problem
2849:Gingerbreadman map
2736:Bifurcation theory
2614:Lyapunov stability
2403:Nonlinear Dynamics
2364:Nonlinear Dynamics
2149:
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282:). In particular,
229:
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127:
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3272:
3137:Benoît Mandelbrot
3102:Martin Gutzwiller
3092:Peter Grassberger
2975:
2974:
2957:Rössler attractor
2705:
2704:
2609:Invariant measure
2531:Lyapunov exponent
2283:10.1002/etep.2660
2006:
1871:
1868:
1778:
1775:
1679:, version of the
1627:= 50". Parameter
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119:
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3293:
3245:Butterfly effect
3157:Itamar Procaccia
3107:Brosl Hasslacher
3004:Elastic pendulum
2932:Duffing equation
2879:Kaplan–Yorke map
2797:Arnold's cat map
2785:
2784:
2761:Stability theory
2746:Dynamical system
2741:Control of chaos
2721:
2713:
2697:Takens's theorem
2629:Poincaré section
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1677:piecewise linear
1604:Numerical errors
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1424:= 4 case of the
1415:
1413:
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1405:
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1349:is expressed in
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753:square root of 2
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596:= 4 case of the
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276:dynamical system
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19:
16:Mathematical map
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3177:Caroline Series
3072:Mary Cartwright
3054:
3048:
2999:Double pendulum
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2920:
2913:
2839:Exponential map
2790:
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2715:
2701:
2667:Ergodic theorem
2660:
2653:
2644:Stable manifold
2634:Recurrence plot
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2019:. A sequence {
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1779:
1762:
1760:
1752:
1746:
1742:
1735:
1734:
1719:
1715:
1713:
1710:
1709:
1684:
1681:
1680:
1673:
1638:
1606:
1555:
1551:
1546:
1527:
1523:
1511:
1502:
1498:
1496:
1493:
1492:
1469:
1465:
1463:
1460:
1459:
1433:
1430:
1429:
1400:
1396:
1394:
1391:
1390:
1373:
1369:
1367:
1364:
1363:
1333:
1329:
1327:
1324:
1323:
1302:
1298:
1296:
1293:
1292:
1248:
1233:
1229:
1228:
1223:
1205:
1201:
1200:
1194:
1190:
1188:
1170:
1166:
1165:
1159:
1155:
1153:
1135:
1131:
1130:
1124:
1120:
1118:
1100:
1096:
1095:
1090:
1088:
1085:
1084:
1053:
1046:
1044:
1031:
1016:
1012:
1011:
1006:
988:
984:
983:
977:
973:
971:
953:
949:
948:
942:
938:
936:
918:
914:
913:
908:
906:
903:
902:
868:
853:
849:
848:
843:
825:
821:
820:
814:
810:
808:
790:
786:
785:
780:
778:
775:
774:
703:
700:
699:
632:with parameter
606:
567:
564:
563:
558:
549:
538:
513:
512:
506:
502:
489:
472:
470:
461:
457:
442:
441:
431:
422:
418:
401:
399:
393:
389:
379:
378:
366:
362:
353:
349:
334:
330:
328:
325:
324:
304:
300:
298:
295:
294:
292:
263:
256:
180:
176:
174:
171:
170:
165:
139:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3299:
3289:
3288:
3271:
3270:
3268:
3267:
3262:
3260:Predictability
3257:
3252:
3247:
3241:
3239:
3233:
3232:
3230:
3229:
3227:Lai-Sang Young
3224:
3222:James A. Yorke
3219:
3217:Amie Wilkinson
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3164:
3159:
3154:
3152:Henri Poincaré
3149:
3144:
3139:
3134:
3129:
3124:
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3058:
3056:
3050:
3049:
3047:
3046:
3041:
3036:
3031:
3026:
3021:
3019:Kicked rotator
3016:
3011:
3006:
3001:
2996:
2991:
2989:Chua's circuit
2985:
2983:
2977:
2976:
2973:
2972:
2970:
2969:
2964:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2923:
2921:
2918:
2915:
2914:
2912:
2911:
2909:Zaslavskii map
2906:
2904:Tinkerbell map
2901:
2896:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2836:
2835:
2834:
2824:
2819:
2814:
2809:
2804:
2799:
2793:
2791:
2788:
2782:
2772:
2771:
2769:
2768:
2763:
2758:
2753:
2751:Ergodic theory
2748:
2743:
2738:
2732:
2730:
2724:
2723:
2708:
2706:
2703:
2702:
2700:
2699:
2694:
2689:
2684:
2679:
2674:
2669:
2663:
2661:
2658:
2655:
2654:
2652:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2571:
2566:
2561:
2555:
2552:
2551:
2549:
2548:
2543:
2541:Periodic point
2538:
2533:
2528:
2523:
2518:
2513:
2507:
2505:
2502:
2496:
2492:
2491:
2484:
2483:
2476:
2469:
2461:
2455:
2454:
2447:
2446:External links
2444:
2441:
2440:
2389:
2370:(1): 127–133.
2350:
2314:
2289:
2246:
2230:
2213:
2196:
2195:
2193:
2190:
2189:
2188:
2183:
2176:
2173:
2168:
2165:
2146:
2142:
2119:
2116:
2113:
2109:
2105:
2100:
2096:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2069:
2066:
2063:
2059:
2033:
2029:
2005:
2002:
1996:
1993:
1973:
1970:
1967:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1920:for parameter
1918:
1917:
1904:
1899:
1896:
1893:
1890:
1887:
1884:
1879:
1875:
1864:
1861:
1858:
1854:
1852:
1849:
1846:
1843:
1840:
1836:
1832:
1827:
1823:
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1810:
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1637:
1634:
1605:
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1584:
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1570:
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1562:
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1526:
1519:
1516:
1510:
1505:
1501:
1487:
1486:
1472:
1468:
1443:
1440:
1437:
1403:
1399:
1376:
1372:
1336:
1332:
1305:
1301:
1290:if and only if
1277:
1276:
1275:
1274:
1263:
1260:
1257:
1244:
1241:
1236:
1232:
1227:
1222:
1216:
1213:
1208:
1204:
1197:
1193:
1187:
1181:
1178:
1173:
1169:
1162:
1158:
1152:
1146:
1143:
1138:
1134:
1127:
1123:
1117:
1111:
1108:
1103:
1099:
1094:
1079:
1078:
1077:
1076:
1063:
1057:
1052:
1049:
1043:
1040:
1027:
1024:
1019:
1015:
1010:
1005:
999:
996:
991:
987:
980:
976:
970:
964:
961:
956:
952:
945:
941:
935:
929:
926:
921:
917:
912:
897:
896:
895:
894:
883:
880:
877:
864:
861:
856:
852:
847:
842:
836:
833:
828:
824:
817:
813:
807:
801:
798:
793:
789:
784:
769:
768:
761:
748:
747:
746:
745:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
694:
693:
678:
671:
605:
602:
577:
574:
571:
554:
547:
536:
530:
529:
516:
509:
505:
501:
496:
493:
481:
478:
475:
471:
469:
464:
460:
456:
453:
450:
447:
444:
443:
438:
435:
430:
425:
421:
410:
407:
404:
400:
396:
392:
388:
385:
384:
382:
377:
374:
369:
365:
361:
356:
352:
348:
343:
340:
337:
333:
307:
303:
290:
261:
254:
240:
239:
228:
225:
222:
219:
216:
212:
209:
206:
203:
200:
197:
194:
191:
188:
183:
179:
163:
137:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3298:
3287:
3284:
3283:
3281:
3266:
3263:
3261:
3258:
3256:
3255:Edge of chaos
3253:
3251:
3248:
3246:
3243:
3242:
3240:
3234:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3212:Marcelo Viana
3210:
3208:
3205:
3203:
3202:Audrey Terras
3200:
3198:
3197:Floris Takens
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3097:Celso Grebogi
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3077:Chen Guanrong
3075:
3073:
3070:
3068:
3065:
3063:
3062:Michael Berry
3060:
3059:
3057:
3051:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2986:
2984:
2978:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2937:Lorenz system
2935:
2933:
2930:
2928:
2925:
2924:
2922:
2916:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2884:Langton's ant
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2859:Horseshoe map
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2833:
2830:
2829:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2794:
2792:
2786:
2783:
2780:
2773:
2767:
2764:
2762:
2759:
2757:
2756:Quantum chaos
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2733:
2731:
2725:
2720:
2716:
2712:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2664:
2662:
2656:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2564:Arnold tongue
2562:
2560:
2557:
2556:
2553:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2508:
2506:
2500:
2497:
2493:
2489:
2482:
2477:
2475:
2470:
2468:
2463:
2462:
2459:
2453:
2452:ChaosBook.org
2450:
2449:
2436:
2432:
2427:
2422:
2417:
2412:
2408:
2404:
2400:
2393:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2354:
2345:
2340:
2336:
2332:
2328:
2324:
2318:
2304:
2300:
2293:
2284:
2279:
2274:
2269:
2265:
2261:
2257:
2250:
2243:
2237:
2235:
2227:
2223:
2217:
2210:
2209:Jeffrey Rauch
2206:
2201:
2197:
2187:
2184:
2182:
2179:
2178:
2172:
2164:
2162:
2144:
2140:
2117:
2114:
2111:
2107:
2103:
2098:
2094:
2087:
2084:
2081:
2078:
2072:
2067:
2064:
2061:
2057:
2049:
2031:
2027:
2003:
2000:
1994:
1991:
1971:
1968:
1965:
1942:
1939:
1936:
1930:
1927:
1894:
1891:
1888:
1882:
1877:
1873:
1847:
1844:
1841:
1834:
1825:
1821:
1817:
1814:
1801:
1798:
1795:
1789:
1784:
1780:
1757:
1753:
1747:
1743:
1736:
1731:
1726:
1723:
1720:
1716:
1708:
1707:
1706:
1692:
1689:
1686:
1678:
1665:
1664:
1659:
1652:
1648:
1647:
1642:
1630:
1626:
1622:
1618:
1614:
1610:
1598:
1594:
1590:
1586:
1585:
1568:
1560:
1556:
1552:
1547:
1543:
1536:
1531:
1528:
1524:
1517:
1514:
1508:
1503:
1499:
1491:
1490:
1489:
1488:
1470:
1466:
1457:
1441:
1438:
1435:
1427:
1423:
1419:
1401:
1397:
1374:
1370:
1361:
1357:
1352:
1334:
1330:
1321:
1303:
1299:
1291:
1287:
1283:
1279:
1278:
1261:
1258:
1255:
1242:
1239:
1234:
1230:
1225:
1214:
1211:
1206:
1202:
1195:
1191:
1179:
1176:
1171:
1167:
1160:
1156:
1144:
1141:
1136:
1132:
1125:
1121:
1109:
1106:
1101:
1097:
1092:
1083:
1082:
1081:
1080:
1061:
1055:
1050:
1047:
1041:
1038:
1025:
1022:
1017:
1013:
1008:
997:
994:
989:
985:
978:
974:
962:
959:
954:
950:
943:
939:
927:
924:
919:
915:
910:
901:
900:
899:
898:
881:
878:
875:
862:
859:
854:
850:
845:
834:
831:
826:
822:
815:
811:
799:
796:
791:
787:
782:
773:
772:
771:
770:
766:
762:
758:
754:
750:
749:
732:
729:
723:
717:
711:
705:
698:
697:
696:
695:
692:= 0.61 we get
691:
687:
683:
679:
676:
672:
669:
665:
661:
657:
654:
650:
646:
645:
644:
641:
639:
635:
631:
623:
618:
610:
601:
599:
595:
591:
590:bit shift map
575:
572:
569:
560:
557:
553:
546:
542:
535:
507:
503:
499:
494:
491:
462:
458:
454:
451:
445:
436:
433:
428:
423:
419:
394:
390:
386:
380:
375:
367:
363:
354:
350:
346:
341:
338:
335:
331:
323:
322:
321:
305:
301:
289:
285:
281:
277:
274:
273:discrete-time
270:
269:unit interval
266:
260:
253:
249:
245:
226:
220:
217:
214:
210:
207:
198:
195:
189:
181:
177:
169:
168:
167:
162:
159:
155:
151:
147:
136:
131:
123:
113:
110:
102:
99:December 2009
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3286:Chaotic maps
3207:Mary Tsingou
3172:David Ruelle
3167:Otto Rössler
3112:Michel Hénon
3082:Leon O. Chua
3039:Tilt-A-Whirl
3009:FPUT problem
2898:
2894:Standard map
2889:Logistic map
2714:
2488:Chaos theory
2426:10084/152214
2406:
2402:
2392:
2367:
2363:
2353:
2344:11581/480148
2327:SpringerLink
2326:
2317:
2307:, retrieved
2302:
2292:
2266:(1): e2660.
2263:
2259:
2249:
2241:
2225:
2216:
2200:
2170:
2167:Applications
1919:
1674:
1650:
1628:
1624:
1620:
1616:
1593:almost every
1456:homeomorphic
1426:logistic map
1421:
1355:
689:
685:
681:
674:
667:
663:
659:
648:
642:
633:
630:logistic map
627:
621:
598:logistic map
593:
561:
555:
551:
544:
533:
531:
287:
258:
251:
241:
160:
149:
143:
134:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3192:Nina Snaith
3182:Yakov Sinai
3067:Rufus Bowen
2817:Duffing map
2802:Baker's map
2727:Theoretical
2639:SRB measure
2546:Phase space
2516:Bifurcation
2181:Shift space
1613:Time series
1418:white noise
656:fixed point
166:defined by
146:mathematics
3250:Complexity
3147:Edward Ott
2994:Convection
2919:Continuous
2594:Ergodicity
2309:2023-09-29
2273:1809.03616
2192:References
1597:Cantor set
1589:Cantor set
1320:irrational
653:attractive
651:= 0 is an
69:newspapers
58:"Tent map"
3162:Mary Rees
3122:Bryna Kra
3055:theorists
2864:Ikeda map
2854:Hénon map
2844:Gauss map
2526:Limit set
2511:Attractor
2435:1573-269X
2384:1573-269X
2186:Gray code
2085:−
1966:μ
1931:∈
1883:∈
1845:−
1818:−
1790:∈
1687:μ
1537:
1529:−
1518:π
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760:diagram).
757:Julia set
733:…
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604:Behaviour
570:μ
500:≤
455:−
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387:μ
355:μ
284:iterating
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3280:Category
3238:articles
2980:Physical
2899:Tent map
2789:Discrete
2729:branches
2659:Theorems
2495:Concepts
2175:See also
1428:and the
730:0.650625
636:= 4 are
592:and the
541:interval
286:a point
158:function
156:-valued
150:tent map
3236:Related
3044:Weather
2982:systems
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1958:. The
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