4139:
3883:
4011:
4562:
1091:
1630:
1083:
5280:
4435:
1037:
9737:
4273:
25:
4374:
1182:
3217:
is not also a one-bit repeating sequence): 001, 010, 100, 110, 101, or 011. The iterate 001001001... maps into 010010010..., which maps into 100100100..., which in turn maps into the original 001001001...; so this is a 3-cycle of the bit shift map. And the other three binary-expansion repeating sequences give the 3-cycle 110110110... → 101101101... → 011011011... → 110110110.... Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as
5722:
4334:
9729:
5214:
4489:
1289:
5515:
1164:
5726:
5724:
5729:
5728:
5723:
439:
5730:
5218:
5216:
5221:
5220:
5215:
4493:
4491:
4496:
4490:
4495:
5222:
4497:
1621:
Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states arbitrarily far into the
3216:
is written in binary notation, the map moves the binary point one place to the right (and if the bit to the left of the binary point has become a "1", this "1" is changed to a "0"). A cycle of length 3, for example, occurs if an iterate has a 3-bit repeating sequence in its binary expansion (which
637:
increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive bifurcation
5727:
8045:
The logistic map exhibits numerous characteristics of both periodic and chaotic solutions, whereas the logistic ordinary differential equation (ODE) exhibits regular solutions, commonly referred to as the S-shaped sigmoid function. The logistic map can be seen as the discrete counterpart of the
5519:
1312:
of the chaotic logistic map. In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. Hence, predictions about
4138:
3882:
5517:
5522:
5516:
4010:
388:
5521:
5523:
5219:
1232:
The relative simplicity of the logistic map makes it a widely used point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions—a property of the logistic map for most values of
3126:
4494:
672:) is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
2242:
758:, characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices. There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of
7886:
5520:
7042:
8035:
3307:
logistic map gives the corresponding logistic cycle 0.611260467... → 0.950484434... → 0.188255099... → 0.611260467.... We could similarly translate the other bit-shift 3-cycle into its corresponding logistic cycle. Likewise, cycles of any length
1070:
and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between
7756:
1237:
between about 3.57 and 4 (as noted above). A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the
1277:
the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map. Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of
4968:
4744:
6400:
In general, each period-multiplying route to chaos has its own pair of
Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.
6628:
By universality, we can use another family of functions that also undergoes repeated period-doubling on its route to chaos, and even though it is not exactly the logistic map, it would still yield the same
Feigenbaum constants.
2813:
2438:
60:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
610:
2697:
equal to 1. Just as the squared sine function in the trigonometric solution leads to neither shrinkage nor expansion of the set of points visited, in the latter solution this effect is accomplished by the unit modulus of
7287:
619:
between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial (sequence
2688:
2587:
5725:
2975:
293:(density-dependent mortality), where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
2352:
7400:
2907:
case, from almost all initial conditions the iterate sequence is chaotic. Nevertheless, there exist an infinite number of initial conditions that lead to cycles, and indeed there exist cycles of length
6911:
5217:
6395:
6157:
6046:
3207:
4492:
2094:
7931:
6315:
5387:
5081:
1288:
7639:
7077:, the period-2 stable orbit undergoes period-doubling bifurcation again, yielding a period-4 stable orbit. In order to find out what the stable orbit is like, we "zoom in" around the region of
1616:
5966:
5708:
276:
is a number between zero and one, which represents the ratio of existing population to the maximum possible population. This nonlinear difference equation is intended to capture two effects:
3485:, and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length
6351:
6159:
As another example, period-4-pling has a pair of
Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define
1094:
Stable regions within the chaotic region, where a tangent bifurcation occurs at the boundary between the chaotic and periodic attractor, giving intermittent trajectories as described in the
5518:
7761:
6002:
5171:
1317:) worse when there are even very small errors in our knowledge of the initial state. This quality of unpredictability and apparent randomness led the logistic map equation to be used as a
375:
6493:
8108:
986:
7936:
6441:
745:
5208:
251:
7577:
6794:
6703:
6906:
892:
5469:
5416:
5273:
6556:
6530:
5804:
5768:
5317:
4823:
4784:
4599:
4555:
4464:
3868:
6203:
5854:
5612:
7522:
7477:
3483:
6078:
5638:
5557:
5509:
3412:= 630 is the number of cycles of length 13. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable.
1420:
5181:). Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant
70:
6611:
7644:
7150:
3561:
6836:
3613:
3535:
1443:
7075:
5440:
5244:
5011:
4991:
4175:
4047:
3919:
3718:
3651:
1360:
7435:
4403:
4258:
4130:
4002:
3842:
6257:
6230:
5908:
5881:
4363:
4302:
4228:
4202:
4074:
3972:
3946:
3793:
3767:
3680:
4828:
4604:
2064:. There are two aspects of the behavior of the logistic map that should be captured by an upper bound in this regime: the asymptotic geometric decay with constant
7101:
4100:
3305:
1386:
7155:
4426:
4325:
8148:
8128:
6723:
6579:
5586:
5101:
4519:
3816:
3741:
3581:
3503:
1463:
9158:"The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models"
2056:
Although exact solutions to the recurrence relation are only available in a small number of cases, a closed-form upper bound on the logistic map is known when
2469:
never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor
2718:
1528:
that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter
2367:
514:
80:
Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
3319:
are irrational, almost all initial conditions of the bit-shift map lead to the non-periodicity of chaos. This is one way to see that the logistic
37:
4746:, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.
8795:
2602:
1252:
The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional
8179:
6083:
2504:
1468:
Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a
370:
The image below shows the amplitude and frequency content of some logistic map iterates for parameter values ranging from 2 to 4.
10026:
9856:
5614:, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants
5016:
3121:{\displaystyle y_{n+1}={\begin{cases}2y_{n}&0\leq y_{n}<{\tfrac {1}{2}}\\2y_{n}-1&{\tfrac {1}{2}}\leq y_{n}<1,\end{cases}}}
1465:
value within each interval where the dynamical system has a stable cycle. This can be seen in the
Lyapunov exponent plot as sharp dips.
1300:
This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see
2474:
3361:
669:
627:
5442:
can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is
1010:
there is at most one stable cycle. If a stable cycle exists, it is globally stable, attracting almost all points. Some values of
2285:
7292:
3371:
has 2 fixed points, 1 cycle of length 2, 2 cycles of length 3 and so on. This sequence takes a particularly simple form for prime
998:, almost all initial values eventually leave the interval and diverge. The set of initial conditions which remain within form a
9205:
Okulov, A Yu; Oraevskiĭ, A N (1986). "Space–temporal behavior of a light pulse propagating in a nonlinear nondispersive medium".
9736:
9496:
2237:{\displaystyle \forall n\in \{0,1,\ldots \}\quad {\text{and}}\quad x_{0},r\in ,\quad x_{n}\leq {\frac {x_{0}}{r^{-n}+x_{0}n}}.}
9378:
9355:
9332:
9057:
Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos
Theoretical Division Annual Report 1975-1976
8771:
8741:
8713:
8680:
6356:
6007:
3140:
65:
8169:
6635:
9689:
7894:
6262:
5322:
3844:, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain
9636:
9400:
4970:, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.
7582:
1567:
9969:
5913:
88:
9121:
Okulov, A Yu; Oraevskiĭ, A N (1984). "Regular and stochastic self-modulation in a ring laser with nonlinear element".
7881:{\displaystyle \lim _{n}{\frac {r_{\infty }-r_{n}}{r_{\infty }-r_{n+1}}}\approx S'(r_{\infty })\approx 1+{\sqrt {17}}}
5643:
101:
Content in this edit is translated from the existing
Japanese Knowledge article at ]; see its history for attribution.
10204:
6320:
10011:
9699:
8884:
7037:{\displaystyle {\begin{cases}p={\frac {1}{2}}(r+{\sqrt {r(r+4)}})\\q={\frac {1}{2}}(r-{\sqrt {r(r+4)}})\end{cases}}}
507:≈ 3.44949 the population will approach permanent oscillations between two values. These two values are dependent on
167:
9886:
8611:
8030:{\displaystyle \alpha \approx r_{\infty }^{2}+4r_{\infty }-3{\sqrt {r_{\infty }^{2}+4r_{\infty }}}\approx -2.24...}
5971:
5106:
9444:
6446:
8057:
901:
163:
162:
in the 1960s to showcase irregular solutions (e.g., Eq. 3 of ), was popularized in a 1976 paper by the biologist
9463:
9453:
8163:
5511:, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
1507:(onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024.
1318:
1095:
755:
708:
9704:
6407:
5559:, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
5184:
182:
9694:
8185:
7527:
6728:
4405:, there are three intersection points, with the middle one unstable, and the two others having slope exactly
1651:
1029:
visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that
448:
with bias (the parameter k from the figure corresponds to the parameter r from the definition in the article)
284:
9416:
782:
for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period
10051:
9959:
9824:
6841:
859:
9459:
8856:
Campbell, Trevor; Broderick, Tamara (2017). "Automated scalable
Bayesian inference via Hilbert coresets".
5445:
5392:
5249:
6535:
6509:
5773:
5737:
5286:
4789:
4753:
4568:
4524:
4467:
4443:
3871:
3847:
649:
6162:
5813:
5591:
171:
9964:
9489:
7485:
7440:
10031:
3428:
3364:): 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161.... This tells us that the logistic map with
10079:
8763:
Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering
6051:
5617:
5530:
5482:
1391:
9090:
8191:
6920:
3003:
9783:
9068:
Feigenbaum, Mitchell (1978). "Quantitative universality for a class of nonlinear transformations".
7751:{\displaystyle r_{\infty }=\lim _{n}r_{n}\approx \lim _{n}S^{-n}(0)={\frac {1}{2}}({\sqrt {17}}-3)}
395:(the parameter k from the figure corresponds to the parameter r from the definition in the article)
96:
9728:
1090:
9944:
9709:
9616:
8174:
6584:
9684:
7106:
3795:, there are three intersection points, with the middle one unstable, and the two others stable.
3540:
9984:
9596:
9410:
9085:
8669:
6802:
4973:
This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by
4365:, there are three intersection points, with the middle one unstable, and the two others stable.
3586:
3508:
2694:
1629:
9449:
4561:
3312:
can be found in the bit-shift map and then translated into the corresponding logistic cycles.
1425:
1143:, the population will approach permanent oscillations between two values, as with the case of
1082:
10303:
10134:
10041:
9839:
9666:
9601:
9576:
9482:
9428:
8811:"Absolutely continuous invariant measures for one-parameter families of one-dimensional maps"
7047:
5425:
5229:
4996:
4976:
4963:{\displaystyle f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots }
4739:{\displaystyle f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots }
4147:
4019:
3891:
3685:
3618:
2924:
1493:
1469:
1327:
117:
7405:
4382:
4233:
4105:
3977:
3821:
1014:
with a stable cycle of some period have infinitely many unstable cycles of various periods.
10144:
9896:
9796:
9641:
9439:
9295:
9263:
9214:
9169:
9130:
9077:
8987:
8822:
8546:
8500:
8411:
8373:
8276:
8228:
6504:
6235:
6208:
5886:
5859:
4342:
4281:
4207:
4180:
4052:
3951:
3924:
3772:
3746:
3659:
1512:
1388:
precisely at the maximum point, so convergence to the equilibrium point is on the order of
1239:
1036:
404:
between 0 and 1, the population will eventually die, independent of the initial population.
159:
9974:
8487:"Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator"
6625:
The
Feigenbaum constants can be estimated by a renormalization argument. (Section 10.7,).
8:
10104:
10061:
10046:
9891:
9844:
9829:
9814:
9714:
9621:
9606:
9591:
9423:
8697:
7080:
6614:
5711:
5279:
5174:
4483:
4079:
3284:
1647:
1525:
1485:
1365:
1242:
1072:
1040:
1021:
at right summarizes this. The horizontal axis shows the possible values of the parameter
1018:
679:
beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of
639:
479:
143:
9646:
9299:
9267:
9218:
9173:
9134:
9081:
8991:
8826:
8550:
8504:
8377:
8280:
8232:
8216:
4434:
4408:
4307:
3974:. Before the period doubling bifurcation occurs. The orbit converges to the fixed point
3212:
The reason that the dyadic transformation is also called the bit-shift map is that when
1422:. Consequently, the equilibrium point is called "superstable". Its Lyapunov exponent is
10282:
10149:
9979:
9866:
9861:
9753:
9631:
9533:
9311:
9230:
9103:
8956:
8938:
8904:
8857:
8838:
8789:
8702:
8580:
8516:
8428:
8391:
8310:
8240:
8133:
8113:
6838:. After the period-doubling bifurcation, we can solve for the period-2 stable orbit by
6708:
6564:
5571:
5086:
4504:
3801:
3726:
3566:
3488:
1448:
1314:
490:
147:
8923:
8332:
3769:, the intersection point splits to two, which is a period doubling. For example, when
1245:
describing it may be thought of as a stretching-and-folding operation on the interval
337:, this leads to negative population sizes. (This problem does not appear in the older
10154:
10119:
10109:
10006:
9626:
9548:
9374:
9351:
9344:
9328:
9315:
9283:
9275:
9251:
9234:
9187:
9107:
9039:
8960:
8908:
8842:
8777:
8767:
8737:
8709:
8676:
8640:
8572:
8520:
8491:
8467:
8432:
8395:
8329:
8302:
8244:
8159:
2272:; however, the general case can only be predicted statistically. The solution when
2088:
term in the recurrence relation. The following bound captures both of these effects:
1634:
1544:
1497:
1489:
1481:
1301:
1185:
895:
139:
110:
92:
10169:
9142:
8265:
May, Robert M. (1976). "Simple mathematical models with very complicated dynamics".
10262:
10174:
10124:
10021:
9949:
9901:
9778:
9763:
9758:
9553:
9303:
9271:
9222:
9177:
9138:
9095:
9031:
9019:
8995:
8948:
8896:
8830:
8620:
8584:
8562:
8554:
8508:
8463:
8459:
8420:
8381:
8314:
8292:
8284:
8267:
8236:
5177:, which appears in most period-doubling routes to chaos (thus it is an instance of
1517:
1309:
754:
varies from approximately 3.56995 to approximately 3.82843 is sometimes called the
2808:{\displaystyle x_{n}={\tfrac {1}{2}}-{\tfrac {1}{2}}\left(1-2x_{0}\right)^{2^{n}}}
1293:
1253:
750:
The development of the chaotic behavior of the logistic sequence as the parameter
10194:
10089:
10016:
9849:
9661:
9651:
9433:
9404:
9367:
9056:
8952:
8810:
8447:
8046:
logistic ODE, and their correlation has been extensively discussed in literature
4272:
3870:, the period doublings become infinite, and the map becomes chaotic. This is the
2433:{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}\left({\sqrt {x_{0}}}\right).}
1623:
1510:
It is often possible, however, to make precise and accurate statements about the
10184:
10129:
9020:"Dependence of universal constants upon multiplication period in nonlinear maps"
8537:
May, R. M. (1976). "Simple mathematical models with very complicated dynamics".
605:{\displaystyle x_{\pm }={\frac {1}{2r}}\left(r+1\pm {\sqrt {(r-3)(r+1)}}\right)}
10277:
10244:
10239:
10234:
10036:
9926:
9921:
9819:
9768:
9558:
9000:
8975:
8598:
8486:
8196:
8040:
2495:
1168:
8781:
8567:
8297:
1181:
10297:
10272:
10229:
10219:
10214:
10114:
10094:
9954:
9876:
9773:
9581:
9191:
9043:
8664:
8602:
8512:
8471:
8424:
8248:
5810:
We can also consider period-tripling route to chaos by picking a sequence of
4373:
3132:
455:
between 2 and 3, the population will also eventually approach the same value
316:
9035:
698:
that show oscillation among three values, and for slightly higher values of
438:
10224:
10189:
10099:
10056:
9911:
9505:
9254:; Procaccia, I. (1983). "Measuring the strangeness of strange attractors".
9226:
8761:
8386:
8361:
8355:
8353:
8351:
8349:
7282:{\displaystyle (T^{-1}\circ f_{r}^{2}\circ T)(x)=-(1+S(r))x+x^{2}+O(x^{3})}
1060:
338:
155:
151:
9871:
9440:"A very brief history of universality in period doubling" by P. Cvitanović
9182:
9157:
6613:
changes dynamics from regular to chaotic one with qualitatively the same
10209:
10199:
10084:
9834:
9656:
9563:
9469:
8576:
8306:
2683:{\displaystyle \alpha =1-2x_{0}\pm {\sqrt {\left(1-2x_{0}\right)^{2}-1}}}
1257:
374:
341:, which also exhibits chaotic dynamics.) One can also consider values of
8346:
5806:, since all period-doubling routes to chaos are the same (universality).
4333:
10267:
10164:
9611:
9307:
9099:
8900:
8834:
8625:
8606:
8182:, of which the logistic map is a special case confined to the real line
6443:, and the relation becomes exact as both numbers increase to infinity:
1305:
1114:
between -2 and -1 the logistic sequence also features chaotic behavior.
999:
136:
99:
to the source of your translation. A model attribution edit summary is
2582:{\displaystyle x_{n}={\frac {-\alpha ^{2^{n}}-\alpha ^{-2^{n}}+2}{4}}}
1163:
894:, two chaotic bands of the bifurcation diagram intersect in the first
822:
value at the end of the infinite sequence of sub-ranges is called the
10179:
10139:
9881:
9543:
9528:
8943:
8558:
8337:
8288:
1521:
1274:
1044:
4825:
looks like a fractal. Furthermore, as we repeat the period-doublings
1267:, and clearly shows the quadratic curve of the difference equation (
9916:
8862:
3344:
327:
315:
case of the logistic map is a nonlinear transformation of both the
9462:
by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr,
9394:
8976:"The problem of deducing the climate from the governing equations"
8217:"The problem of deducing the climate from the governing equations"
4466:, there are infinitely many intersections, and we have arrived at
1171:
of the logistic map, showing chaotic behaviour for most values of
387:
9586:
9538:
5734:
Logistic map approaching the period-doubling chaos scaling limit
4750:
Looking at the images, one can notice that at the point of chaos
4428:, indicating that it is about to undergo another period-doubling.
1076:
9445:"A not so short history of Universal Function" by P. Cvitanović
6503:
Universality of one-dimensional maps with parabolic maxima and
1516:
of a future state in a chaotic system. If a (possibly chaotic)
478:, but first will fluctuate around that value for some time. The
411:
between 1 and 2, the population will quickly approach the value
166:, in part as a discrete-time demographic model analogous to the
57:
10159:
9474:
2923:. We can exploit the relationship of the logistic map to the
9365:
Tufillaro, Nicholas; Abbott, Tyler; Reilly, Jeremiah (1992).
8188:, which illustrates the inverse problem for the logistic map.
8162:, solution of the logistic map's continuous counterpart: the
5770:
from below. At the limit, this has the same shape as that of
3281:. Using the above translation from the bit-shift map to the
2473:
shows the exponential growth of stretching, which results in
2347:{\displaystyle x_{n}=\sin ^{2}\left(2^{n}\theta \pi \right),}
1296:
show the stretching-and-folding structure of the logistic map
1222:. For example, for any initial value on the horizontal axis,
150:, often referred to as an archetypal example of how complex,
9286:(1981). "On the Hausdorff dimension of fractal attractors".
8041:
The
Logistic Map and Logistic Ordinary Differential Equation
7395:{\displaystyle S(r)=r^{2}+4r-2,c=r^{2}+4r-3{\sqrt {r(r+4)}}}
287:
to the current population when the population size is small,
8760:
Strogatz, Steven (2019). "10.1: Fixed Points and
Cobwebs".
8607:"Chaos: Significance, Mechanism, and Economic Applications"
8360:
Tsuchiya, Takashi; Yamagishi, Daisuke (February 11, 1997).
7030:
4327:, indicating that it is about to undergo a period-doubling.
4260:
becomes unstable, splitting into a periodic-2 stable cycle.
3356:
3114:
683:
that show non-chaotic behavior; these are sometimes called
664:
622:
442:
Evolution of different initial conditions as a function of
391:
Evolution of different initial conditions as a function of
9156:
Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin (2023-08-12).
8327:
2889:
1650:
for the logistic map can be visualized with the following
1002:
and the dynamics restricted to this Cantor set is chaotic.
830:
rises there is a succession of new windows with different
69:
to this template: there are already 1,153 articles in the
4132:. is exactly 1, and a period doubling bifurcation occurs.
9470:
Using SAGE to investigate the discrete logistic equation
9369:
An Experimental Approach to Nonlinear Dynamics and Chaos
5910:
window of the bifurcation diagram. For example, we have
5471:, it converges. This is the second Feigenbaum constant.
9419:
by Roger White. Chapter 5 covers the Logistic Equation.
8924:"Chaotic root-finding for a small class of polynomials"
8362:"The Complete Bifurcation Diagram for the Logistic Map"
8170:
Lyapunov stability#Definition for discrete-time systems
6725:
increases, it undergoes period-doubling bifurcation at
6390:{\displaystyle \delta =981.6\dots ,\alpha =38.82\dots }
6152:{\displaystyle f(x)\mapsto -\alpha f(f(f(-x/\alpha )))}
6041:{\displaystyle \delta =55.26\dots ,\alpha =9.277\dots }
3682:, we have a single intersection, with slope bounded in
3202:{\displaystyle x_{n}=\sin ^{2}\left(2\pi y_{n}\right).}
1445:. A similar argument shows that there is a superstable
694:(approximately 3.82843) there is a range of parameters
6498:
6410:
5646:
3078:
3041:
2751:
2736:
2378:
774:-values consisting of a succession of subranges. The
489:, when it is dramatically slow, less than linear (see
9364:
8671:
Ordinary Differential Equations and Dynamical Systems
8136:
8116:
8060:
7939:
7926:{\displaystyle \delta \approx 1+{\sqrt {17}}=5.12...}
7897:
7764:
7647:
7585:
7530:
7488:
7443:
7408:
7295:
7158:
7109:
7083:
7050:
6914:
6844:
6805:
6731:
6711:
6638:
6587:
6567:
6538:
6512:
6449:
6359:
6323:
6310:{\displaystyle r_{1}=3.960102,r_{2}=3.9615554,\dots }
6265:
6238:
6211:
6165:
6086:
6054:
6010:
5974:
5916:
5889:
5862:
5816:
5776:
5740:
5620:
5594:
5574:
5533:
5485:
5448:
5428:
5395:
5325:
5289:
5252:
5232:
5187:
5109:
5089:
5019:
4999:
4979:
4831:
4792:
4756:
4607:
4571:
4527:
4507:
4446:
4411:
4385:
4345:
4310:
4284:
4236:
4210:
4183:
4150:
4108:
4082:
4055:
4022:
3980:
3954:
3927:
3894:
3850:
3824:
3804:
3775:
3749:
3729:
3720:, indicating that it is a stable single fixed point.
3688:
3662:
3621:
3589:
3569:
3543:
3511:
3491:
3431:
3287:
3143:
2978:
2721:
2605:
2507:
2370:
2288:
2097:
1570:
1451:
1428:
1394:
1368:
1330:
1228:
gives the value of the iterate four iterations later.
904:
862:
711:
517:
185:
9250:
9018:
Delbourgo, R.; Hart, W.; Kenny, B. G. (1985-01-01).
8409:
Zhang, Cheng (October 2010). "Period three begins".
8260:
8258:
6705:
The family has an equilibrium point at zero, and as
6353:. This has a different pair of Feigenbaum constants
6004:. This has a different pair of Feigenbaum constants
5382:{\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
5076:{\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
4304:, we have a single intersection, with slope exactly
2265:
can in fact be solved exactly, as can the case with
158:
dynamical equations. The map, initially utilized by
53:
5083:then at the limit, we would end up with a function
1285:corresponding to the steeper sections of the plot.
1025:while the vertical axis shows the set of values of
9436:. Champaign, IL: Wolfram Media, p. 918, 2002.
9366:
9343:
9017:
8704:Iterated Maps on the Interval as Dynamical Systems
8701:
8668:
8142:
8122:
8102:
8029:
7925:
7880:
7750:
7634:{\displaystyle r_{n+1}\approx {\sqrt {r_{n}+6}}-2}
7633:
7571:
7516:
7471:
7429:
7394:
7281:
7144:
7095:
7069:
7036:
6900:
6830:
6788:
6717:
6697:
6605:
6573:
6550:
6524:
6487:
6435:
6389:
6345:
6309:
6251:
6224:
6197:
6151:
6072:
6040:
5996:
5960:
5902:
5875:
5848:
5798:
5762:
5702:
5632:
5606:
5580:
5551:
5503:
5463:
5434:
5410:
5381:
5311:
5267:
5238:
5202:
5165:
5095:
5075:
5005:
4985:
4962:
4817:
4778:
4738:
4593:
4549:
4513:
4458:
4420:
4397:
4357:
4319:
4296:
4252:
4222:
4196:
4169:
4124:
4094:
4068:
4041:
3996:
3966:
3940:
3913:
3862:
3836:
3810:
3787:
3761:
3735:
3712:
3674:
3645:
3607:
3575:
3555:
3529:
3497:
3477:
3326:map is chaotic for almost all initial conditions.
3299:
3201:
3120:
2807:
2682:
2581:
2432:
2346:
2236:
1611:{\displaystyle {\frac {1}{\pi {\sqrt {x(1-x)}}}}.}
1610:
1457:
1437:
1414:
1380:
1354:
980:
886:
739:
604:
245:
9198:
9114:
8855:
8359:
8255:
5961:{\displaystyle r_{1}=3.8284,r_{2}=3.85361,\dots }
5703:{\textstyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}
5319:, as we repeat the functional equation iteration
5246:, the map does not converge to a limit, but when
3420:
898:for the logistic map. It satisfies the equations
16:Simple polynomial map exhibiting chaotic behavior
10295:
9401:An interactive visualization of the logistic map
8931:Journal of Difference Equations and Applications
7766:
7685:
7662:
6450:
6259:window of the bifurcation diagram. Then we have
5418:, we find that the map does converge to a limit.
2848:other than the unstable fixed point 0, the term
9397:. An introductory primer on chaos and fractals.
9204:
9120:
7482:By self-similarity, the third bifurcation when
6346:{\displaystyle r_{\infty }=3.96155658717\dots }
2454:maps into a periodic sequence. But almost all
1543:and the probability measure corresponds to the
1086:Magnification of the chaotic region of the map.
793:. In a sub-range with a stable cycle of period
297:The usual values of interest for the parameter
283:, where the population will increase at a rate
9155:
8597:
174:. Mathematically, the logistic map is written
95:accompanying your translation by providing an
44:Click for important translation instructions.
36:expand this article with text translated from
9490:
8695:
8484:
8180:Periodic points of complex quadratic mappings
8054:In a toy model for discrete laser dynamics:
5997:{\displaystyle r_{\infty }=3.854077963\dots }
5166:{\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}
2044:
1158:
1063:: if we zoom in on the above-mentioned value
355:
9452:by Marek Bodnar after work by Phil Ramsden,
8921:
6488:{\displaystyle \lim \delta /\alpha ^{2}=2/3}
2125:
2107:
1313:future states become progressively (indeed,
9282:
8103:{\displaystyle x\rightarrow Gx(1-\tanh(x))}
8037:. These are within 10% of the true values.
6620:
2039:
1641:
981:{\displaystyle r^{3}-2r^{2}-4r-8=0,x=1-1/r}
789:. This sequence of sub-ranges is called a
687:. For instance, beginning at 1 +
9497:
9483:
9460:Multiplicative coupling of 2 logistic maps
9067:
8794:: CS1 maint: location missing publisher (
8445:
2475:sensitive dependence on initial conditions
360:
9181:
9089:
8999:
8942:
8861:
8727:
8725:
8624:
8566:
8385:
8296:
6436:{\textstyle 3\delta \approx 2\alpha ^{2}}
3329:The number of cycles of (minimal) length
1622:future, and use this knowledge to inform
1561:. Specifically, the invariant measure is
740:{\displaystyle r=1+{\sqrt {8}}=3.8284...}
9341:
8883:
8808:
8759:
8150:is laser gain as bifurcation parameter.
5720:
5513:
5278:
5212:
5203:{\displaystyle \delta =4.6692016\cdots }
4560:
4487:
3425:In the logistic map, we have a function
3354:) is a known integer sequence (sequence
2477:, while the squared sine function keeps
1628:
1180:
1162:
1102:We can also consider negative values of
1089:
1081:
1035:
702:oscillation among 6 values, then 12 etc.
437:
434:, independent of the initial population.
386:
246:{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}
8879:
8877:
8875:
8873:
8641:"Misiurewicz Point of the Logistic Map"
8478:
7572:{\displaystyle r_{n}\approx S(r_{n+1})}
6789:{\displaystyle r=r_{0},r_{1},r_{2},...}
5563:
5226:For the wrong values of scaling factor
4102:. The tangent slope at the fixed point
841:; all subsequent windows involving odd
10296:
9322:
8973:
8815:Communications in Mathematical Physics
8731:
8722:
8663:
8485:Jeffries, Carson; Pérez, José (1982).
8214:
7437:, the second bifurcation occurs, thus
6698:{\displaystyle f_{r}(x)=-(1+r)x+x^{2}}
3583:points, and the slope of the graph of
2498:instead of trigonometric functions is
2447:, after a finite number of iterations
2357:where the initial condition parameter
2247:
1051:is shown on the vertical line at that
800:, there are unstable cycles of period
384:, the following behavior is observed:
9478:
9123:Soviet Journal of Quantum Electronics
9013:
9011:
8887:(1870). "Ueber iterirte Functionen".
8755:
8753:
8532:
8530:
8408:
8328:
8130:stands for electric field amplitude,
6901:{\displaystyle f_{r}(p)=q,f_{r}(q)=p}
3315:However, since almost all numbers in
1539:, the attractor is also the interval
887:{\displaystyle r=3.678...,x=0.728...}
154:behaviour can arise from very simple
8870:
5464:{\displaystyle \alpha =2.5029\dots }
5411:{\displaystyle \alpha =2.5029\dots }
5268:{\displaystyle \alpha =2.5029\dots }
2931:) to find cycles of any length. If
2458:are irrational, and, for irrational
747:, the stable period-3 cycle emerges.
301:are those in the interval , so that
176:
18:
9637:Measure-preserving dynamical system
9519:
8536:
8264:
6551:{\displaystyle \alpha =2.502907...}
6525:{\displaystyle \delta =4.669201...}
6499:Feigenbaum universality of 1-D maps
5799:{\displaystyle r^{*}=3.5699\cdots }
5763:{\displaystyle r^{*}=3.84943\dots }
5312:{\displaystyle r^{*}=3.5699\cdots }
4818:{\displaystyle f_{r^{*}}^{\infty }}
4779:{\displaystyle r^{*}=3.5699\cdots }
4601:, as we repeat the period-doublings
4594:{\displaystyle r^{*}=3.5699\cdots }
4550:{\displaystyle r^{*}=3.5699\cdots }
4468:chaos via the period-doubling route
4459:{\displaystyle r\approx 3.56994567}
3863:{\displaystyle r\approx 3.56994567}
2494:an equivalent solution in terms of
778:th subrange contains the values of
648:. This behavior is an example of a
13:
9417:Complexity & Chaos (audiobook)
9008:
8750:
8527:
8448:"The Birth of Period 3, Revisited"
8241:10.1111/j.2153-3490.1964.tb00136.x
8011:
7990:
7972:
7951:
7854:
7808:
7783:
7653:
7152:. Now, by routine algebra, we have
6597:
6329:
6232:is the lowest value in the period-
6198:{\displaystyle r_{1},r_{2},\dots }
6065:
5980:
5883:is the lowest value in the period-
5849:{\displaystyle r_{1},r_{2},\dots }
5607:{\displaystyle r\approx 3.8494344}
5544:
5496:
4810:
3505:, we would find that the graph of
2890:Finding cycles of any length when
2098:
2068:, and the fast initial decay when
1626:based on the state of the system.
1432:
1287:
14:
10315:
10205:Oleksandr Mykolayovych Sharkovsky
9388:
7517:{\displaystyle S(r)\approx r_{1}}
7472:{\displaystyle S(r_{1})\approx 0}
5474:
4501:Approach to the scaling limit as
2596:is either of the complex numbers
1191:(blue) and its iterated versions
826:of the cascade of harmonics. As
9735:
9727:
9504:
8612:Journal of Economic Perspectives
8215:Lorenz, Edward N. (1964-02-01).
6799:The first bifurcation occurs at
6080:converges to the fixed point to
4477:
4433:
4372:
4332:
4271:
4137:
4009:
3881:
3478:{\displaystyle f_{r}(x)=rx(1-x)}
1154:, and given by the same formula.
849:starting with arbitrarily large
373:
23:
9323:Sprott, Julien Clinton (2003).
9149:
9143:10.1070/QE1984v014n09ABEH006171
9061:
9050:
8967:
8922:Little, M.; Heesch, D. (2004).
8915:
8849:
8802:
8689:
8657:
8633:
8446:Bechhoefer, John (1996-04-01).
6073:{\displaystyle f_{r}^{\infty }}
5633:{\displaystyle \delta ,\alpha }
5552:{\displaystyle f_{r}^{\infty }}
5504:{\displaystyle f_{r}^{\infty }}
3415:
2870:goes to the stable fixed point
2705:By contrast, the solution when
2172:
2134:
2128:
1415:{\displaystyle \delta ^{2^{n}}}
1047:for any value of the parameter
9970:Rabinovich–Fabrikant equations
9464:Wolfram Demonstrations Project
9454:Wolfram Demonstrations Project
9325:Chaos and Time-Series Analysis
9288:Journal of Statistical Physics
9070:Journal of Statistical Physics
8591:
8464:10.1080/0025570X.1996.11996402
8439:
8402:
8321:
8208:
8164:Logistic differential equation
8097:
8094:
8088:
8073:
8064:
8049:
7859:
7846:
7745:
7729:
7713:
7707:
7641:. Iterating this map, we find
7566:
7547:
7498:
7492:
7460:
7447:
7418:
7412:
7387:
7375:
7305:
7299:
7276:
7263:
7238:
7235:
7229:
7217:
7208:
7202:
7199:
7159:
7119:
7113:
7024:
7019:
7007:
6993:
6970:
6965:
6953:
6939:
6889:
6883:
6861:
6855:
6676:
6664:
6655:
6649:
6600:
6588:
6146:
6143:
6140:
6123:
6117:
6111:
6099:
6096:
6090:
5697:
5694:
5677:
5671:
5659:
5656:
5650:
5376:
5373:
5356:
5350:
5338:
5335:
5329:
5160:
5157:
5140:
5134:
5119:
5113:
5070:
5067:
5050:
5044:
5032:
5029:
5023:
3872:period-doubling route to chaos
3707:
3689:
3640:
3622:
3547:
3472:
3460:
3448:
3442:
3421:Period-doubling route to chaos
3131:then the two are related by a
2166:
2154:
1597:
1585:
1349:
1337:
1319:pseudo-random number generator
834:values. The first one is for
592:
580:
577:
565:
237:
218:
105:You may also add the template
1:
9244:
8186:Radial basis function network
7103:, using the affine transform
3396:. For example: 2 ⋅
2077:is close to 1, driven by the
1121:between -1 and 1 -
1059:The bifurcation diagram is a
845:occur in decreasing order of
9346:Nonlinear Dynamics and Chaos
9276:10.1016/0167-2789(83)90298-1
8953:10.1080/10236190412331285351
8766:(2nd ed.). Boca Raton.
7891:Thus, we have the estimates
1496:of approximately 0.5170976 (
1147:between 3 and 1 +
500:between 3 and 1 +
7:
9705:Poincaré recurrence theorem
9373:. Addison-Wesley New York.
9327:. Oxford University Press.
8734:Chaos: Making a New Science
8153:
6617:as those for logistic map.
6606:{\displaystyle [0,\infty )}
1292:Two- and three-dimensional
1269:
259:
107:{{Translated|ja|ロジスティック写像}}
77:will aid in categorization.
10:
10320:
9700:Poincaré–Bendixson theorem
9450:Discrete Logistic Equation
9411:The Logistic Map and Chaos
9001:10.3402/tellusa.v16i1.8893
7524:, and so on. Thus we have
7145:{\displaystyle T(x)=x/c+p}
4481:
3556:{\displaystyle x\mapsto x}
3336:for the logistic map with
2484:folded within the range .
1159:Chaos and the logistic map
1096:Pomeau–Manneville scenario
1043:for the logistic map. The
756:Pomeau–Manneville scenario
356:Characteristics of the map
345:in the interval , so that
52:Machine translation, like
10253:
10070:
10052:Swinging Atwood's machine
9997:
9935:
9805:
9792:
9744:
9725:
9695:Krylov–Bogolyubov theorem
9675:
9572:
9512:
9342:Strogatz, Steven (2000).
8736:. London: Penguin Books.
6831:{\displaystyle r=r_{0}=0}
3653:at those intersections.
3608:{\displaystyle f_{r}^{n}}
3530:{\displaystyle f_{r}^{n}}
2935:follows the logistic map
1304:), evidenced also by the
638:intervals approaches the
380:By varying the parameter
308:remains bounded on . The
38:the corresponding article
9960:Lotka–Volterra equations
9784:Synchronization of chaos
9587:axiom A dynamical system
9424:History of iterated maps
8513:10.1103/PhysRevA.26.2117
8425:10.4169/002557010x521859
8202:
6621:Renormalization estimate
6561:The gradual increase of
5714:. This is an example of
2040:Special cases of the map
2019:"logistic map"
1656:
1642:Graphical representation
1535:and an initial state in
1438:{\displaystyle -\infty }
172:Pierre François Verhulst
9945:Double scroll attractor
9710:Stable manifold theorem
9617:False nearest neighbors
9395:The Chaos Hypertextbook
9036:10.1103/PhysRevA.31.514
8974:Lorenz, Edward (1964).
8645:sprott.physics.wisc.edu
8175:Malthusian growth model
7070:{\displaystyle r=r_{1}}
5527:In the chaotic regime,
5479:In the chaotic regime,
5435:{\displaystyle \alpha }
5239:{\displaystyle \alpha }
5006:{\displaystyle \alpha }
4993:for a certain constant
4986:{\displaystyle \alpha }
4170:{\displaystyle x_{n+2}}
4042:{\displaystyle x_{n+2}}
3914:{\displaystyle x_{n+2}}
3713:{\displaystyle (-1,+1)}
3646:{\displaystyle (-1,+1)}
1725:# number of repetitions
1355:{\displaystyle rx(1-x)}
1324:At r = 2, the function
650:period-doubling cascade
116:For more guidance, see
9985:Van der Pol oscillator
9965:Mackey–Glass equations
9597:Box-counting dimension
9350:. Perseus Publishing.
9227:10.1364/JOSAB.3.000741
8732:Gleick, James (1987).
8387:10.1515/zna-1997-6-708
8144:
8124:
8104:
8031:
7927:
7882:
7752:
7635:
7573:
7518:
7473:
7431:
7430:{\displaystyle S(r)=0}
7396:
7283:
7146:
7097:
7071:
7038:
6902:
6832:
6790:
6719:
6699:
6607:
6575:
6552:
6526:
6489:
6437:
6391:
6347:
6311:
6253:
6226:
6199:
6153:
6074:
6042:
5998:
5962:
5904:
5877:
5850:
5807:
5800:
5764:
5704:
5634:
5608:
5582:
5560:
5553:
5505:
5465:
5436:
5419:
5412:
5383:
5313:
5283:At the point of chaos
5276:
5269:
5240:
5204:
5167:
5097:
5077:
5007:
4987:
4964:
4819:
4780:
4747:
4740:
4595:
4565:At the point of chaos
4558:
4551:
4515:
4460:
4422:
4399:
4398:{\displaystyle r=3.45}
4359:
4321:
4298:
4254:
4253:{\displaystyle x_{f2}}
4224:
4198:
4171:
4126:
4125:{\displaystyle x_{f2}}
4096:
4070:
4043:
3998:
3997:{\displaystyle x_{f2}}
3968:
3942:
3915:
3864:
3838:
3837:{\displaystyle r=3.45}
3812:
3789:
3763:
3737:
3714:
3676:
3647:
3609:
3577:
3557:
3531:
3499:
3479:
3301:
3203:
3122:
2809:
2684:
2583:
2434:
2348:
2238:
1638:
1612:
1524:, then there exists a
1459:
1439:
1416:
1382:
1356:
1297:
1256:of the logistic map's
1229:
1178:
1099:
1087:
1056:
982:
888:
764:period-doubling window
741:
606:
482:is linear, except for
449:
396:
361:Behavior dependent on
247:
10135:Svetlana Jitomirskaya
10042:Multiscroll attractor
9887:Interval exchange map
9840:Dyadic transformation
9825:Complex quadratic map
9667:Topological conjugacy
9602:Correlation dimension
9577:Anosov diffeomorphism
9429:A New Kind of Science
9183:10.3390/atmos14081279
8889:Mathematische Annalen
8809:Jakobson, M. (1981).
8145:
8125:
8105:
8032:
7928:
7883:
7753:
7636:
7574:
7519:
7474:
7432:
7397:
7284:
7147:
7098:
7072:
7039:
6903:
6833:
6791:
6720:
6700:
6608:
6576:
6553:
6527:
6490:
6438:
6392:
6348:
6312:
6254:
6252:{\displaystyle 4^{n}}
6227:
6225:{\displaystyle r_{n}}
6200:
6154:
6075:
6043:
5999:
5963:
5905:
5903:{\displaystyle 3^{n}}
5878:
5876:{\displaystyle r_{n}}
5851:
5801:
5765:
5733:
5705:
5635:
5609:
5583:
5554:
5526:
5506:
5466:
5437:
5413:
5384:
5314:
5282:
5270:
5241:
5225:
5205:
5168:
5098:
5078:
5008:
4988:
4965:
4820:
4781:
4741:
4596:
4564:
4552:
4516:
4500:
4461:
4423:
4400:
4360:
4358:{\displaystyle r=3.4}
4322:
4299:
4297:{\displaystyle r=3.0}
4255:
4225:
4223:{\displaystyle a=3.3}
4199:
4197:{\displaystyle x_{n}}
4172:
4144:Relationship between
4127:
4097:
4071:
4069:{\displaystyle x_{n}}
4044:
4016:Relationship between
3999:
3969:
3967:{\displaystyle a=2.7}
3943:
3941:{\displaystyle x_{n}}
3916:
3888:Relationship between
3865:
3839:
3813:
3790:
3788:{\displaystyle r=3.4}
3764:
3762:{\displaystyle r=3.0}
3738:
3715:
3677:
3675:{\displaystyle r=3.0}
3648:
3610:
3578:
3558:
3532:
3500:
3480:
3302:
3204:
3123:
2968:dyadic transformation
2925:dyadic transformation
2863:goes to infinity, so
2810:
2685:
2584:
2435:
2349:
2239:
1632:
1613:
1494:information dimension
1470:correlation dimension
1460:
1440:
1417:
1383:
1357:
1291:
1184:
1166:
1093:
1085:
1039:
983:
889:
824:point of accumulation
742:
607:
441:
390:
352:remains bounded on .
248:
118:Knowledge:Translation
89:copyright attribution
10145:Edward Norton Lorenz
8698:Eckmann, Jean-Pierre
8452:Mathematics Magazine
8412:Mathematics Magazine
8134:
8114:
8058:
7937:
7895:
7762:
7645:
7583:
7528:
7486:
7441:
7406:
7293:
7156:
7107:
7081:
7048:
6912:
6842:
6803:
6729:
6709:
6636:
6585:
6565:
6536:
6510:
6505:Feigenbaum constants
6447:
6408:
6357:
6321:
6263:
6236:
6209:
6163:
6084:
6052:
6008:
5972:
5914:
5887:
5860:
5814:
5774:
5738:
5644:
5618:
5592:
5572:
5564:Other scaling limits
5531:
5483:
5446:
5426:
5393:
5323:
5287:
5250:
5230:
5185:
5107:
5087:
5017:
4997:
4977:
4829:
4790:
4754:
4605:
4569:
4525:
4505:
4444:
4409:
4383:
4343:
4308:
4282:
4234:
4208:
4181:
4148:
4106:
4080:
4053:
4020:
3978:
3952:
3925:
3892:
3848:
3822:
3802:
3773:
3747:
3743:increases to beyond
3727:
3686:
3660:
3619:
3587:
3567:
3541:
3509:
3489:
3429:
3285:
3141:
2976:
2719:
2603:
2505:
2368:
2286:
2258:The special case of
2095:
1568:
1449:
1426:
1392:
1366:
1328:
1321:in early computers.
1273:). However, we can
902:
860:
791:cascade of harmonics
709:
685:islands of stability
515:
183:
10105:Mitchell Feigenbaum
10047:Population dynamics
10032:Hénon–Heiles system
9892:Irrational rotation
9845:Dynamical billiards
9830:Coupled map lattice
9690:Liouville's theorem
9622:Hausdorff dimension
9607:Conservative system
9592:Bifurcation diagram
9300:1981JSP....26..173G
9268:1983PhyD....9..189G
9219:1986JOSAB...3..741O
9174:2023Atmos..14.1279S
9135:1984QuEle..14.1235O
9082:1978JSP....19...25F
8992:1964Tell...16....1L
8827:1981CMaPh..81...39J
8551:1976Natur.261..459M
8505:1982PhRvA..26.2117J
8378:1997ZNatA..52..513T
8333:"Logistic Equation"
8281:1976Natur.261..459M
8233:1964Tell...16....1L
8192:Schröder's equation
7999:
7960:
7402:. At approximately
7192:
7096:{\displaystyle x=p}
6615:bifurcation diagram
6069:
5712:Feigenbaum function
5548:
5500:
5175:Feigenbaum function
4953:
4928:
4903:
4878:
4853:
4814:
4729:
4704:
4679:
4654:
4629:
4484:Feigenbaum function
4095:{\displaystyle a=3}
3604:
3526:
3300:{\displaystyle r=4}
2927:(also known as the
1648:bifurcation diagram
1526:probability measure
1486:Hausdorff dimension
1381:{\displaystyle y=x}
1243:difference equation
1041:Bifurcation diagram
1019:bifurcation diagram
640:Feigenbaum constant
480:rate of convergence
144:recurrence relation
10283:Santa Fe Institute
10150:Aleksandr Lyapunov
9980:Three-body problem
9867:Gingerbreadman map
9754:Bifurcation theory
9632:Lyapunov stability
9308:10.1007/BF01106792
9207:J. Opt. Soc. Am. B
9100:10.1007/BF01020332
8986:(February): 1–11.
8901:10.1007/BF01443992
8835:10.1007/BF01941800
8675:. Amer. Math Soc.
8626:10.1257/jep.3.1.77
8599:Baumol, William J.
8568:10338.dmlcz/104555
8330:Weisstein, Eric W.
8298:10338.dmlcz/104555
8140:
8120:
8100:
8027:
7985:
7946:
7923:
7878:
7774:
7748:
7693:
7670:
7631:
7569:
7514:
7469:
7427:
7392:
7279:
7178:
7142:
7093:
7067:
7034:
7029:
6898:
6828:
6786:
6715:
6695:
6632:Define the family
6603:
6571:
6548:
6522:
6485:
6433:
6387:
6343:
6307:
6249:
6222:
6195:
6149:
6070:
6055:
6038:
5994:
5958:
5900:
5873:
5846:
5808:
5796:
5760:
5700:
5630:
5604:
5578:
5561:
5549:
5534:
5501:
5486:
5461:
5432:
5420:
5408:
5379:
5309:
5277:
5265:
5236:
5200:
5163:
5093:
5073:
5003:
4983:
4960:
4932:
4907:
4882:
4857:
4832:
4815:
4793:
4776:
4748:
4736:
4708:
4683:
4658:
4633:
4608:
4591:
4559:
4547:
4511:
4456:
4421:{\displaystyle +1}
4418:
4395:
4355:
4320:{\displaystyle +1}
4317:
4294:
4250:
4230:. The fixed point
4220:
4194:
4167:
4122:
4092:
4066:
4039:
3994:
3964:
3938:
3911:
3860:
3834:
3808:
3785:
3759:
3733:
3710:
3672:
3656:For example, when
3643:
3605:
3590:
3573:
3553:
3527:
3512:
3495:
3475:
3297:
3199:
3118:
3113:
3087:
3050:
2805:
2760:
2745:
2680:
2579:
2430:
2387:
2344:
2234:
1639:
1633:Logistic map with
1608:
1455:
1435:
1412:
1378:
1352:
1302:Lyapunov exponents
1298:
1230:
1179:
1100:
1088:
1057:
978:
884:
737:
602:
491:Bifurcation memory
450:
397:
243:
97:interlanguage link
10291:
10290:
10155:Benoît Mandelbrot
10120:Martin Gutzwiller
10110:Peter Grassberger
9993:
9992:
9975:Rössler attractor
9723:
9722:
9627:Invariant measure
9549:Lyapunov exponent
9413:by Elmer G. Wiens
9380:978-0-201-55441-0
9357:978-0-7382-0453-6
9334:978-0-19-850840-3
9024:Physical Review A
8773:978-0-367-09206-1
8743:978-0-14-009250-9
8715:978-3-7643-3026-2
8682:978-0-8218-8328-0
8605:(February 1989).
8492:Physical Review A
8275:(5560): 459–467.
8160:Logistic function
8143:{\displaystyle G}
8123:{\displaystyle x}
8016:
7915:
7876:
7833:
7765:
7737:
7727:
7684:
7661:
7623:
7390:
7022:
6991:
6968:
6937:
6718:{\displaystyle r}
6574:{\displaystyle G}
6317:, with the limit
5968:, with the limit
5731:
5710:is also the same
5581:{\displaystyle r}
5524:
5223:
5096:{\displaystyle g}
4514:{\displaystyle r}
4498:
3811:{\displaystyle r}
3736:{\displaystyle r}
3576:{\displaystyle n}
3537:and the graph of
3498:{\displaystyle n}
3086:
3049:
2839:for any value of
2759:
2744:
2678:
2577:
2421:
2386:
2229:
2132:
2045:Upper bound when
1674:matplotlib.pyplot
1635:Lyapunov exponent
1603:
1600:
1545:beta distribution
1488:of about 0.538 (
1458:{\displaystyle r}
1186:Logistic function
1006:For any value of
896:Misiurewicz point
729:
595:
544:
267:
266:
168:logistic equation
129:
128:
45:
10311:
10263:Butterfly effect
10175:Itamar Procaccia
10125:Brosl Hasslacher
10022:Elastic pendulum
9950:Duffing equation
9897:Kaplan–Yorke map
9815:Arnold's cat map
9803:
9802:
9779:Stability theory
9764:Dynamical system
9759:Control of chaos
9739:
9731:
9715:Takens's theorem
9647:Poincaré section
9517:
9516:
9499:
9492:
9485:
9476:
9475:
9384:
9372:
9361:
9349:
9338:
9319:
9279:
9262:(1–2): 189–208.
9239:
9238:
9202:
9196:
9195:
9185:
9153:
9147:
9146:
9129:(2): 1235–1237.
9118:
9112:
9111:
9093:
9065:
9059:
9054:
9048:
9047:
9015:
9006:
9005:
9003:
8971:
8965:
8964:
8946:
8928:
8919:
8913:
8912:
8881:
8868:
8867:
8865:
8853:
8847:
8846:
8806:
8800:
8799:
8793:
8785:
8757:
8748:
8747:
8729:
8720:
8719:
8707:
8696:Collet, Pierre;
8693:
8687:
8686:
8674:
8661:
8655:
8654:
8652:
8651:
8637:
8631:
8630:
8628:
8595:
8589:
8588:
8570:
8559:10.1038/261459a0
8545:(5560): 459–67.
8534:
8525:
8524:
8499:(4): 2117–2122.
8482:
8476:
8475:
8443:
8437:
8436:
8406:
8400:
8399:
8389:
8372:(6–7): 513–516.
8357:
8344:
8343:
8342:
8325:
8319:
8318:
8300:
8289:10.1038/261459a0
8262:
8253:
8252:
8212:
8149:
8147:
8146:
8141:
8129:
8127:
8126:
8121:
8109:
8107:
8106:
8101:
8036:
8034:
8033:
8028:
8017:
8015:
8014:
7998:
7993:
7984:
7976:
7975:
7959:
7954:
7932:
7930:
7929:
7924:
7916:
7911:
7887:
7885:
7884:
7879:
7877:
7872:
7858:
7857:
7845:
7834:
7832:
7831:
7830:
7812:
7811:
7801:
7800:
7799:
7787:
7786:
7776:
7773:
7757:
7755:
7754:
7749:
7738:
7733:
7728:
7720:
7706:
7705:
7692:
7680:
7679:
7669:
7657:
7656:
7640:
7638:
7637:
7632:
7624:
7616:
7615:
7606:
7601:
7600:
7578:
7576:
7575:
7570:
7565:
7564:
7540:
7539:
7523:
7521:
7520:
7515:
7513:
7512:
7478:
7476:
7475:
7470:
7459:
7458:
7436:
7434:
7433:
7428:
7401:
7399:
7398:
7393:
7391:
7371:
7354:
7353:
7320:
7319:
7288:
7286:
7285:
7280:
7275:
7274:
7256:
7255:
7191:
7186:
7174:
7173:
7151:
7149:
7148:
7143:
7132:
7102:
7100:
7099:
7094:
7076:
7074:
7073:
7068:
7066:
7065:
7043:
7041:
7040:
7035:
7033:
7032:
7023:
7003:
6992:
6984:
6969:
6949:
6938:
6930:
6907:
6905:
6904:
6899:
6882:
6881:
6854:
6853:
6837:
6835:
6834:
6829:
6821:
6820:
6795:
6793:
6792:
6787:
6773:
6772:
6760:
6759:
6747:
6746:
6724:
6722:
6721:
6716:
6704:
6702:
6701:
6696:
6694:
6693:
6648:
6647:
6612:
6610:
6609:
6604:
6580:
6578:
6577:
6572:
6557:
6555:
6554:
6549:
6531:
6529:
6528:
6523:
6494:
6492:
6491:
6486:
6481:
6470:
6469:
6460:
6442:
6440:
6439:
6434:
6432:
6431:
6396:
6394:
6393:
6388:
6352:
6350:
6349:
6344:
6333:
6332:
6316:
6314:
6313:
6308:
6294:
6293:
6275:
6274:
6258:
6256:
6255:
6250:
6248:
6247:
6231:
6229:
6228:
6223:
6221:
6220:
6204:
6202:
6201:
6196:
6188:
6187:
6175:
6174:
6158:
6156:
6155:
6150:
6136:
6079:
6077:
6076:
6071:
6068:
6063:
6047:
6045:
6044:
6039:
6003:
6001:
6000:
5995:
5984:
5983:
5967:
5965:
5964:
5959:
5945:
5944:
5926:
5925:
5909:
5907:
5906:
5901:
5899:
5898:
5882:
5880:
5879:
5874:
5872:
5871:
5855:
5853:
5852:
5847:
5839:
5838:
5826:
5825:
5805:
5803:
5802:
5797:
5786:
5785:
5769:
5767:
5766:
5761:
5750:
5749:
5732:
5709:
5707:
5706:
5701:
5690:
5639:
5637:
5636:
5631:
5613:
5611:
5610:
5605:
5587:
5585:
5584:
5579:
5558:
5556:
5555:
5550:
5547:
5542:
5525:
5510:
5508:
5507:
5502:
5499:
5494:
5470:
5468:
5467:
5462:
5441:
5439:
5438:
5433:
5417:
5415:
5414:
5409:
5388:
5386:
5385:
5380:
5369:
5318:
5316:
5315:
5310:
5299:
5298:
5274:
5272:
5271:
5266:
5245:
5243:
5242:
5237:
5224:
5209:
5207:
5206:
5201:
5172:
5170:
5169:
5164:
5153:
5102:
5100:
5099:
5094:
5082:
5080:
5079:
5074:
5063:
5012:
5010:
5009:
5004:
4992:
4990:
4989:
4984:
4969:
4967:
4966:
4961:
4952:
4947:
4946:
4945:
4927:
4922:
4921:
4920:
4902:
4897:
4896:
4895:
4877:
4872:
4871:
4870:
4852:
4847:
4846:
4845:
4824:
4822:
4821:
4816:
4813:
4808:
4807:
4806:
4785:
4783:
4782:
4777:
4766:
4765:
4745:
4743:
4742:
4737:
4728:
4723:
4722:
4721:
4703:
4698:
4697:
4696:
4678:
4673:
4672:
4671:
4653:
4648:
4647:
4646:
4628:
4623:
4622:
4621:
4600:
4598:
4597:
4592:
4581:
4580:
4556:
4554:
4553:
4548:
4537:
4536:
4520:
4518:
4517:
4512:
4499:
4465:
4463:
4462:
4457:
4437:
4427:
4425:
4424:
4419:
4404:
4402:
4401:
4396:
4376:
4364:
4362:
4361:
4356:
4336:
4326:
4324:
4323:
4318:
4303:
4301:
4300:
4295:
4275:
4259:
4257:
4256:
4251:
4249:
4248:
4229:
4227:
4226:
4221:
4203:
4201:
4200:
4195:
4193:
4192:
4176:
4174:
4173:
4168:
4166:
4165:
4141:
4131:
4129:
4128:
4123:
4121:
4120:
4101:
4099:
4098:
4093:
4075:
4073:
4072:
4067:
4065:
4064:
4048:
4046:
4045:
4040:
4038:
4037:
4013:
4003:
4001:
4000:
3995:
3993:
3992:
3973:
3971:
3970:
3965:
3947:
3945:
3944:
3939:
3937:
3936:
3920:
3918:
3917:
3912:
3910:
3909:
3885:
3869:
3867:
3866:
3861:
3843:
3841:
3840:
3835:
3817:
3815:
3814:
3809:
3794:
3792:
3791:
3786:
3768:
3766:
3765:
3760:
3742:
3740:
3739:
3734:
3719:
3717:
3716:
3711:
3681:
3679:
3678:
3673:
3652:
3650:
3649:
3644:
3614:
3612:
3611:
3606:
3603:
3598:
3582:
3580:
3579:
3574:
3562:
3560:
3559:
3554:
3536:
3534:
3533:
3528:
3525:
3520:
3504:
3502:
3501:
3496:
3484:
3482:
3481:
3476:
3441:
3440:
3411:
3409:
3408:
3405:
3402:
3395:
3394:
3392:
3391:
3386:
3383:
3374:
3370:
3359:
3353:
3342:
3335:
3325:
3318:
3311:
3306:
3304:
3303:
3298:
3280:
3278:
3277:
3274:
3271:
3264:
3262:
3261:
3258:
3255:
3248:
3246:
3245:
3242:
3239:
3232:
3230:
3229:
3226:
3223:
3215:
3208:
3206:
3205:
3200:
3195:
3191:
3190:
3189:
3166:
3165:
3153:
3152:
3127:
3125:
3124:
3119:
3117:
3116:
3101:
3100:
3088:
3079:
3068:
3067:
3051:
3042:
3036:
3035:
3018:
3017:
2994:
2993:
2965:
2961:
2934:
2922:
2911:
2906:
2896:
2885:
2883:
2882:
2879:
2876:
2869:
2862:
2858:
2847:
2838:
2827:
2814:
2812:
2811:
2806:
2804:
2803:
2802:
2801:
2791:
2787:
2786:
2785:
2761:
2752:
2746:
2737:
2731:
2730:
2711:
2701:
2689:
2687:
2686:
2681:
2679:
2671:
2670:
2665:
2661:
2660:
2659:
2635:
2630:
2629:
2595:
2588:
2586:
2585:
2580:
2578:
2573:
2566:
2565:
2564:
2563:
2543:
2542:
2541:
2540:
2522:
2517:
2516:
2493:
2483:
2472:
2468:
2461:
2457:
2453:
2446:
2439:
2437:
2436:
2431:
2426:
2422:
2420:
2419:
2410:
2401:
2400:
2388:
2379:
2360:
2353:
2351:
2350:
2345:
2340:
2336:
2329:
2328:
2311:
2310:
2298:
2297:
2278:
2271:
2264:
2254:
2243:
2241:
2240:
2235:
2230:
2228:
2224:
2223:
2211:
2210:
2197:
2196:
2187:
2182:
2181:
2144:
2143:
2133:
2130:
2087:
2076:
2067:
2063:
2052:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1617:
1615:
1614:
1609:
1604:
1602:
1601:
1581:
1572:
1560:
1553:
1547:with parameters
1542:
1538:
1534:
1518:dynamical system
1506:
1479:
1477:
1464:
1462:
1461:
1456:
1444:
1442:
1441:
1436:
1421:
1419:
1418:
1413:
1411:
1410:
1409:
1408:
1387:
1385:
1384:
1379:
1361:
1359:
1358:
1353:
1310:unpredictability
1284:
1266:
1248:
1236:
1227:
1221:
1214:
1208:
1202:
1196:
1190:
1177:
1153:
1152:
1146:
1142:
1138:
1131:
1127:
1126:
1120:
1113:
1105:
1069:
1054:
1050:
1032:
1028:
1024:
1013:
1009:
997:
987:
985:
984:
979:
974:
930:
929:
914:
913:
893:
891:
890:
885:
852:
848:
844:
840:
833:
829:
821:
817:
806:
799:
788:
781:
777:
773:
769:
761:
753:
746:
744:
743:
738:
730:
725:
701:
697:
693:
692:
682:
678:
667:
661:
647:
636:
625:
618:
611:
609:
608:
603:
601:
597:
596:
564:
545:
543:
532:
527:
526:
510:
506:
505:
499:
488:
477:
476:
474:
473:
468:
465:
454:
446:
433:
432:
430:
429:
424:
421:
410:
403:
394:
383:
377:
366:
351:
344:
336:
325:
314:
307:
300:
275:
261:
252:
250:
249:
244:
236:
235:
217:
216:
201:
200:
177:
170:written down by
108:
102:
76:
75:|topic=
73:, and specifying
58:Google Translate
43:
27:
26:
19:
10319:
10318:
10314:
10313:
10312:
10310:
10309:
10308:
10294:
10293:
10292:
10287:
10255:
10249:
10195:Caroline Series
10090:Mary Cartwright
10072:
10066:
10017:Double pendulum
9999:
9989:
9938:
9931:
9857:Exponential map
9808:
9794:
9788:
9746:
9740:
9733:
9719:
9685:Ergodic theorem
9678:
9671:
9662:Stable manifold
9652:Recurrence plot
9568:
9522:
9508:
9503:
9434:Stephen Wolfram
9391:
9381:
9358:
9335:
9284:Grassberger, P.
9252:Grassberger, P.
9247:
9242:
9203:
9199:
9154:
9150:
9119:
9115:
9091:10.1.1.418.9339
9066:
9062:
9055:
9051:
9016:
9009:
8972:
8968:
8937:(11): 949–953.
8926:
8920:
8916:
8885:Schröder, Ernst
8882:
8871:
8854:
8850:
8807:
8803:
8787:
8786:
8774:
8758:
8751:
8744:
8730:
8723:
8716:
8694:
8690:
8683:
8662:
8658:
8649:
8647:
8639:
8638:
8634:
8596:
8592:
8535:
8528:
8483:
8479:
8444:
8440:
8407:
8403:
8358:
8347:
8326:
8322:
8263:
8256:
8213:
8209:
8205:
8156:
8135:
8132:
8131:
8115:
8112:
8111:
8059:
8056:
8055:
8052:
8043:
8010:
8006:
7994:
7989:
7983:
7971:
7967:
7955:
7950:
7938:
7935:
7934:
7910:
7896:
7893:
7892:
7871:
7853:
7849:
7838:
7820:
7816:
7807:
7803:
7802:
7795:
7791:
7782:
7778:
7777:
7775:
7769:
7763:
7760:
7759:
7732:
7719:
7698:
7694:
7688:
7675:
7671:
7665:
7652:
7648:
7646:
7643:
7642:
7611:
7607:
7605:
7590:
7586:
7584:
7581:
7580:
7554:
7550:
7535:
7531:
7529:
7526:
7525:
7508:
7504:
7487:
7484:
7483:
7454:
7450:
7442:
7439:
7438:
7407:
7404:
7403:
7370:
7349:
7345:
7315:
7311:
7294:
7291:
7290:
7270:
7266:
7251:
7247:
7187:
7182:
7166:
7162:
7157:
7154:
7153:
7128:
7108:
7105:
7104:
7082:
7079:
7078:
7061:
7057:
7049:
7046:
7045:
7028:
7027:
7002:
6983:
6974:
6973:
6948:
6929:
6916:
6915:
6913:
6910:
6909:
6908:, which yields
6877:
6873:
6849:
6845:
6843:
6840:
6839:
6816:
6812:
6804:
6801:
6800:
6768:
6764:
6755:
6751:
6742:
6738:
6730:
6727:
6726:
6710:
6707:
6706:
6689:
6685:
6643:
6639:
6637:
6634:
6633:
6623:
6586:
6583:
6582:
6566:
6563:
6562:
6537:
6534:
6533:
6511:
6508:
6507:
6501:
6477:
6465:
6461:
6456:
6448:
6445:
6444:
6427:
6423:
6409:
6406:
6405:
6358:
6355:
6354:
6328:
6324:
6322:
6319:
6318:
6289:
6285:
6270:
6266:
6264:
6261:
6260:
6243:
6239:
6237:
6234:
6233:
6216:
6212:
6210:
6207:
6206:
6183:
6179:
6170:
6166:
6164:
6161:
6160:
6132:
6085:
6082:
6081:
6064:
6059:
6053:
6050:
6049:
6009:
6006:
6005:
5979:
5975:
5973:
5970:
5969:
5940:
5936:
5921:
5917:
5915:
5912:
5911:
5894:
5890:
5888:
5885:
5884:
5867:
5863:
5861:
5858:
5857:
5834:
5830:
5821:
5817:
5815:
5812:
5811:
5781:
5777:
5775:
5772:
5771:
5745:
5741:
5739:
5736:
5735:
5721:
5686:
5645:
5642:
5641:
5640:. The limit of
5619:
5616:
5615:
5593:
5590:
5589:
5573:
5570:
5569:
5566:
5543:
5538:
5532:
5529:
5528:
5514:
5495:
5490:
5484:
5481:
5480:
5477:
5447:
5444:
5443:
5427:
5424:
5423:
5394:
5391:
5390:
5365:
5324:
5321:
5320:
5294:
5290:
5288:
5285:
5284:
5275:, it converges.
5251:
5248:
5247:
5231:
5228:
5227:
5213:
5186:
5183:
5182:
5149:
5108:
5105:
5104:
5103:that satisfies
5088:
5085:
5084:
5059:
5018:
5015:
5014:
4998:
4995:
4994:
4978:
4975:
4974:
4948:
4941:
4937:
4936:
4923:
4916:
4912:
4911:
4898:
4891:
4887:
4886:
4873:
4866:
4862:
4861:
4848:
4841:
4837:
4836:
4830:
4827:
4826:
4809:
4802:
4798:
4797:
4791:
4788:
4787:
4786:, the curve of
4761:
4757:
4755:
4752:
4751:
4724:
4717:
4713:
4712:
4699:
4692:
4688:
4687:
4674:
4667:
4663:
4662:
4649:
4642:
4638:
4637:
4624:
4617:
4613:
4612:
4606:
4603:
4602:
4576:
4572:
4570:
4567:
4566:
4532:
4528:
4526:
4523:
4522:
4506:
4503:
4502:
4488:
4486:
4480:
4475:
4474:
4473:
4472:
4471:
4445:
4442:
4441:
4438:
4430:
4429:
4410:
4407:
4406:
4384:
4381:
4380:
4377:
4368:
4367:
4366:
4344:
4341:
4340:
4337:
4329:
4328:
4309:
4306:
4305:
4283:
4280:
4279:
4276:
4265:
4264:
4263:
4262:
4261:
4241:
4237:
4235:
4232:
4231:
4209:
4206:
4205:
4188:
4184:
4182:
4179:
4178:
4155:
4151:
4149:
4146:
4145:
4142:
4134:
4133:
4113:
4109:
4107:
4104:
4103:
4081:
4078:
4077:
4060:
4056:
4054:
4051:
4050:
4027:
4023:
4021:
4018:
4017:
4014:
4006:
4005:
3985:
3981:
3979:
3976:
3975:
3953:
3950:
3949:
3932:
3928:
3926:
3923:
3922:
3899:
3895:
3893:
3890:
3889:
3886:
3849:
3846:
3845:
3823:
3820:
3819:
3803:
3800:
3799:
3774:
3771:
3770:
3748:
3745:
3744:
3728:
3725:
3724:
3687:
3684:
3683:
3661:
3658:
3657:
3620:
3617:
3616:
3599:
3594:
3588:
3585:
3584:
3568:
3565:
3564:
3542:
3539:
3538:
3521:
3516:
3510:
3507:
3506:
3490:
3487:
3486:
3436:
3432:
3430:
3427:
3426:
3423:
3418:
3406:
3403:
3400:
3399:
3397:
3387:
3384:
3381:
3380:
3378:
3376:
3372:
3365:
3355:
3348:
3337:
3330:
3320:
3316:
3309:
3286:
3283:
3282:
3275:
3272:
3269:
3268:
3266:
3259:
3256:
3253:
3252:
3250:
3243:
3240:
3237:
3236:
3234:
3227:
3224:
3221:
3220:
3218:
3213:
3185:
3181:
3174:
3170:
3161:
3157:
3148:
3144:
3142:
3139:
3138:
3112:
3111:
3096:
3092:
3077:
3075:
3063:
3059:
3053:
3052:
3040:
3031:
3027:
3019:
3013:
3009:
2999:
2998:
2983:
2979:
2977:
2974:
2973:
2963:
2958:
2951:
2945:
2936:
2932:
2917:
2909:
2901:
2898:
2891:
2880:
2877:
2874:
2873:
2871:
2868:
2864:
2860:
2856:
2849:
2846:
2840:
2836:
2829:
2825:
2819:
2797:
2793:
2792:
2781:
2777:
2767:
2763:
2762:
2750:
2735:
2726:
2722:
2720:
2717:
2716:
2706:
2699:
2666:
2655:
2651:
2641:
2637:
2636:
2634:
2625:
2621:
2604:
2601:
2600:
2593:
2559:
2555:
2551:
2547:
2536:
2532:
2531:
2527:
2523:
2521:
2512:
2508:
2506:
2503:
2502:
2496:complex numbers
2488:
2482:
2478:
2470:
2467:
2463:
2459:
2455:
2452:
2448:
2444:
2415:
2411:
2409:
2405:
2393:
2389:
2377:
2369:
2366:
2365:
2358:
2324:
2320:
2319:
2315:
2306:
2302:
2293:
2289:
2287:
2284:
2283:
2273:
2266:
2259:
2256:
2249:
2219:
2215:
2203:
2199:
2198:
2192:
2188:
2186:
2177:
2173:
2139:
2135:
2129:
2096:
2093:
2092:
2084:
2078:
2075:
2069:
2065:
2057:
2054:
2046:
2042:
2037:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1791:set_size_inches
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1644:
1580:
1576:
1571:
1569:
1566:
1565:
1555:
1548:
1540:
1536:
1529:
1501:
1475:
1473:
1450:
1447:
1446:
1427:
1424:
1423:
1404:
1400:
1399:
1395:
1393:
1390:
1389:
1367:
1364:
1363:
1329:
1326:
1325:
1283:
1279:
1261:
1246:
1234:
1223:
1216:
1210:
1204:
1198:
1192:
1188:
1172:
1161:
1150:
1148:
1144:
1140:
1136:
1134:
1129:
1124:
1122:
1118:
1111:
1103:
1064:
1052:
1048:
1030:
1026:
1022:
1011:
1007:
992:
970:
925:
921:
909:
905:
903:
900:
899:
861:
858:
857:
850:
846:
842:
835:
831:
827:
819:
808:
801:
794:
783:
779:
775:
771:
767:
766:with parameter
759:
751:
724:
710:
707:
706:
699:
695:
690:
688:
680:
676:
675:Most values of
663:
656:
642:
634:
621:
616:
563:
550:
546:
536:
531:
522:
518:
516:
513:
512:
508:
503:
501:
497:
483:
469:
466:
460:
459:
457:
456:
452:
444:
425:
422:
416:
415:
413:
412:
408:
401:
392:
381:
368:
362:
358:
350:
346:
342:
331:
320:
309:
306:
302:
298:
274:
270:
231:
227:
212:
208:
190:
186:
184:
181:
180:
142:(equivalently,
125:
124:
123:
106:
100:
74:
46:
28:
24:
17:
12:
11:
5:
10317:
10307:
10306:
10289:
10288:
10286:
10285:
10280:
10278:Predictability
10275:
10270:
10265:
10259:
10257:
10251:
10250:
10248:
10247:
10245:Lai-Sang Young
10242:
10240:James A. Yorke
10237:
10235:Amie Wilkinson
10232:
10227:
10222:
10217:
10212:
10207:
10202:
10197:
10192:
10187:
10182:
10177:
10172:
10170:Henri Poincaré
10167:
10162:
10157:
10152:
10147:
10142:
10137:
10132:
10127:
10122:
10117:
10112:
10107:
10102:
10097:
10092:
10087:
10082:
10076:
10074:
10068:
10067:
10065:
10064:
10059:
10054:
10049:
10044:
10039:
10037:Kicked rotator
10034:
10029:
10024:
10019:
10014:
10009:
10007:Chua's circuit
10003:
10001:
9995:
9994:
9991:
9990:
9988:
9987:
9982:
9977:
9972:
9967:
9962:
9957:
9952:
9947:
9941:
9939:
9936:
9933:
9932:
9930:
9929:
9927:Zaslavskii map
9924:
9922:Tinkerbell map
9919:
9914:
9909:
9904:
9899:
9894:
9889:
9884:
9879:
9874:
9869:
9864:
9859:
9854:
9853:
9852:
9842:
9837:
9832:
9827:
9822:
9817:
9811:
9809:
9806:
9800:
9790:
9789:
9787:
9786:
9781:
9776:
9771:
9769:Ergodic theory
9766:
9761:
9756:
9750:
9748:
9742:
9741:
9726:
9724:
9721:
9720:
9718:
9717:
9712:
9707:
9702:
9697:
9692:
9687:
9681:
9679:
9676:
9673:
9672:
9670:
9669:
9664:
9659:
9654:
9649:
9644:
9639:
9634:
9629:
9624:
9619:
9614:
9609:
9604:
9599:
9594:
9589:
9584:
9579:
9573:
9570:
9569:
9567:
9566:
9561:
9559:Periodic point
9556:
9551:
9546:
9541:
9536:
9531:
9525:
9523:
9520:
9514:
9510:
9509:
9502:
9501:
9494:
9487:
9479:
9473:
9472:
9467:
9457:
9447:
9442:
9437:
9420:
9414:
9408:
9398:
9390:
9389:External links
9387:
9386:
9385:
9379:
9362:
9356:
9339:
9333:
9320:
9294:(1): 173–179.
9280:
9246:
9243:
9241:
9240:
9213:(5): 741–746.
9197:
9148:
9113:
9060:
9049:
9030:(1): 514–516.
9007:
8966:
8914:
8895:(2): 296–322.
8869:
8848:
8801:
8772:
8749:
8742:
8721:
8714:
8708:. Birkhauser.
8688:
8681:
8665:Teschl, Gerald
8656:
8632:
8603:Benhabib, Jess
8590:
8526:
8477:
8458:(2): 115–118.
8438:
8419:(4): 295–297.
8401:
8366:Z. Naturforsch
8345:
8320:
8254:
8206:
8204:
8201:
8200:
8199:
8197:Stiff equation
8194:
8189:
8183:
8177:
8172:
8167:
8155:
8152:
8139:
8119:
8099:
8096:
8093:
8090:
8087:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8051:
8048:
8042:
8039:
8026:
8023:
8020:
8013:
8009:
8005:
8002:
7997:
7992:
7988:
7982:
7979:
7974:
7970:
7966:
7963:
7958:
7953:
7949:
7945:
7942:
7922:
7919:
7914:
7909:
7906:
7903:
7900:
7875:
7870:
7867:
7864:
7861:
7856:
7852:
7848:
7844:
7841:
7837:
7829:
7826:
7823:
7819:
7815:
7810:
7806:
7798:
7794:
7790:
7785:
7781:
7772:
7768:
7747:
7744:
7741:
7736:
7731:
7726:
7723:
7718:
7715:
7712:
7709:
7704:
7701:
7697:
7691:
7687:
7683:
7678:
7674:
7668:
7664:
7660:
7655:
7651:
7630:
7627:
7622:
7619:
7614:
7610:
7604:
7599:
7596:
7593:
7589:
7568:
7563:
7560:
7557:
7553:
7549:
7546:
7543:
7538:
7534:
7511:
7507:
7503:
7500:
7497:
7494:
7491:
7468:
7465:
7462:
7457:
7453:
7449:
7446:
7426:
7423:
7420:
7417:
7414:
7411:
7389:
7386:
7383:
7380:
7377:
7374:
7369:
7366:
7363:
7360:
7357:
7352:
7348:
7344:
7341:
7338:
7335:
7332:
7329:
7326:
7323:
7318:
7314:
7310:
7307:
7304:
7301:
7298:
7278:
7273:
7269:
7265:
7262:
7259:
7254:
7250:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7190:
7185:
7181:
7177:
7172:
7169:
7165:
7161:
7141:
7138:
7135:
7131:
7127:
7124:
7121:
7118:
7115:
7112:
7092:
7089:
7086:
7064:
7060:
7056:
7053:
7044:At some point
7031:
7026:
7021:
7018:
7015:
7012:
7009:
7006:
7001:
6998:
6995:
6990:
6987:
6982:
6979:
6976:
6975:
6972:
6967:
6964:
6961:
6958:
6955:
6952:
6947:
6944:
6941:
6936:
6933:
6928:
6925:
6922:
6921:
6919:
6897:
6894:
6891:
6888:
6885:
6880:
6876:
6872:
6869:
6866:
6863:
6860:
6857:
6852:
6848:
6827:
6824:
6819:
6815:
6811:
6808:
6785:
6782:
6779:
6776:
6771:
6767:
6763:
6758:
6754:
6750:
6745:
6741:
6737:
6734:
6714:
6692:
6688:
6684:
6681:
6678:
6675:
6672:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6646:
6642:
6622:
6619:
6602:
6599:
6596:
6593:
6590:
6570:
6547:
6544:
6541:
6521:
6518:
6515:
6500:
6497:
6484:
6480:
6476:
6473:
6468:
6464:
6459:
6455:
6452:
6430:
6426:
6422:
6419:
6416:
6413:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6342:
6339:
6336:
6331:
6327:
6306:
6303:
6300:
6297:
6292:
6288:
6284:
6281:
6278:
6273:
6269:
6246:
6242:
6219:
6215:
6194:
6191:
6186:
6182:
6178:
6173:
6169:
6148:
6145:
6142:
6139:
6135:
6131:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6067:
6062:
6058:
6037:
6034:
6031:
6028:
6025:
6022:
6019:
6016:
6013:
5993:
5990:
5987:
5982:
5978:
5957:
5954:
5951:
5948:
5943:
5939:
5935:
5932:
5929:
5924:
5920:
5897:
5893:
5870:
5866:
5845:
5842:
5837:
5833:
5829:
5824:
5820:
5795:
5792:
5789:
5784:
5780:
5759:
5756:
5753:
5748:
5744:
5699:
5696:
5693:
5689:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5629:
5626:
5623:
5603:
5600:
5597:
5577:
5565:
5562:
5546:
5541:
5537:
5498:
5493:
5489:
5476:
5475:Chaotic regime
5473:
5460:
5457:
5454:
5451:
5431:
5407:
5404:
5401:
5398:
5378:
5375:
5372:
5368:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5308:
5305:
5302:
5297:
5293:
5264:
5261:
5258:
5255:
5235:
5199:
5196:
5193:
5190:
5162:
5159:
5156:
5152:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5092:
5072:
5069:
5066:
5062:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5002:
4982:
4959:
4956:
4951:
4944:
4940:
4935:
4931:
4926:
4919:
4915:
4910:
4906:
4901:
4894:
4890:
4885:
4881:
4876:
4869:
4865:
4860:
4856:
4851:
4844:
4840:
4835:
4812:
4805:
4801:
4796:
4775:
4772:
4769:
4764:
4760:
4735:
4732:
4727:
4720:
4716:
4711:
4707:
4702:
4695:
4691:
4686:
4682:
4677:
4670:
4666:
4661:
4657:
4652:
4645:
4641:
4636:
4632:
4627:
4620:
4616:
4611:
4590:
4587:
4584:
4579:
4575:
4546:
4543:
4540:
4535:
4531:
4510:
4482:Main article:
4479:
4476:
4455:
4452:
4449:
4439:
4432:
4431:
4417:
4414:
4394:
4391:
4388:
4378:
4371:
4370:
4369:
4354:
4351:
4348:
4338:
4331:
4330:
4316:
4313:
4293:
4290:
4287:
4277:
4270:
4269:
4268:
4267:
4266:
4247:
4244:
4240:
4219:
4216:
4213:
4191:
4187:
4164:
4161:
4158:
4154:
4143:
4136:
4135:
4119:
4116:
4112:
4091:
4088:
4085:
4063:
4059:
4036:
4033:
4030:
4026:
4015:
4008:
4007:
3991:
3988:
3984:
3963:
3960:
3957:
3935:
3931:
3908:
3905:
3902:
3898:
3887:
3880:
3879:
3878:
3877:
3876:
3859:
3856:
3853:
3833:
3830:
3827:
3807:
3784:
3781:
3778:
3758:
3755:
3752:
3732:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3671:
3668:
3665:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3615:is bounded in
3602:
3597:
3593:
3572:
3563:intersects at
3552:
3549:
3546:
3524:
3519:
3515:
3494:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3439:
3435:
3422:
3419:
3417:
3414:
3296:
3293:
3290:
3210:
3209:
3198:
3194:
3188:
3184:
3180:
3177:
3173:
3169:
3164:
3160:
3156:
3151:
3147:
3129:
3128:
3115:
3110:
3107:
3104:
3099:
3095:
3091:
3085:
3082:
3076:
3074:
3071:
3066:
3062:
3058:
3055:
3054:
3048:
3045:
3039:
3034:
3030:
3026:
3023:
3020:
3016:
3012:
3008:
3005:
3004:
3002:
2997:
2992:
2989:
2986:
2982:
2956:
2949:
2940:
2897:
2888:
2866:
2854:
2844:
2834:
2823:
2816:
2815:
2800:
2796:
2790:
2784:
2780:
2776:
2773:
2770:
2766:
2758:
2755:
2749:
2743:
2740:
2734:
2729:
2725:
2691:
2690:
2677:
2674:
2669:
2664:
2658:
2654:
2650:
2647:
2644:
2640:
2633:
2628:
2624:
2620:
2617:
2614:
2611:
2608:
2590:
2589:
2576:
2572:
2569:
2562:
2558:
2554:
2550:
2546:
2539:
2535:
2530:
2526:
2520:
2515:
2511:
2480:
2465:
2450:
2441:
2440:
2429:
2425:
2418:
2414:
2408:
2404:
2399:
2396:
2392:
2385:
2382:
2376:
2373:
2355:
2354:
2343:
2339:
2335:
2332:
2327:
2323:
2318:
2314:
2309:
2305:
2301:
2296:
2292:
2255:
2248:Solution when
2246:
2245:
2244:
2233:
2227:
2222:
2218:
2214:
2209:
2206:
2202:
2195:
2191:
2185:
2180:
2176:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2142:
2138:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2082:
2073:
2053:
2043:
2041:
2038:
1959:"b."
1657:
1643:
1640:
1619:
1618:
1607:
1599:
1596:
1593:
1590:
1587:
1584:
1579:
1575:
1492:1981), and an
1454:
1434:
1431:
1407:
1403:
1398:
1377:
1374:
1371:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1294:Poincaré plots
1281:
1169:cobweb diagram
1160:
1157:
1156:
1155:
1132:
1115:
1004:
1003:
989:
977:
973:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
928:
924:
920:
917:
912:
908:
883:
880:
877:
874:
871:
868:
865:
854:
770:is a range of
748:
736:
733:
728:
723:
720:
717:
714:
703:
673:
653:
631:
613:
600:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
562:
559:
556:
553:
549:
542:
539:
535:
530:
525:
521:
494:
435:
405:
367:
359:
357:
354:
348:
304:
295:
294:
288:
272:
265:
264:
255:
253:
242:
239:
234:
230:
226:
223:
220:
215:
211:
207:
204:
199:
196:
193:
189:
127:
126:
122:
121:
114:
103:
81:
78:
66:adding a topic
61:
50:
47:
33:
32:
31:
29:
22:
15:
9:
6:
4:
3:
2:
10316:
10305:
10302:
10301:
10299:
10284:
10281:
10279:
10276:
10274:
10273:Edge of chaos
10271:
10269:
10266:
10264:
10261:
10260:
10258:
10252:
10246:
10243:
10241:
10238:
10236:
10233:
10231:
10230:Marcelo Viana
10228:
10226:
10223:
10221:
10220:Audrey Terras
10218:
10216:
10215:Floris Takens
10213:
10211:
10208:
10206:
10203:
10201:
10198:
10196:
10193:
10191:
10188:
10186:
10183:
10181:
10178:
10176:
10173:
10171:
10168:
10166:
10163:
10161:
10158:
10156:
10153:
10151:
10148:
10146:
10143:
10141:
10138:
10136:
10133:
10131:
10128:
10126:
10123:
10121:
10118:
10116:
10115:Celso Grebogi
10113:
10111:
10108:
10106:
10103:
10101:
10098:
10096:
10095:Chen Guanrong
10093:
10091:
10088:
10086:
10083:
10081:
10080:Michael Berry
10078:
10077:
10075:
10069:
10063:
10060:
10058:
10055:
10053:
10050:
10048:
10045:
10043:
10040:
10038:
10035:
10033:
10030:
10028:
10025:
10023:
10020:
10018:
10015:
10013:
10010:
10008:
10005:
10004:
10002:
9996:
9986:
9983:
9981:
9978:
9976:
9973:
9971:
9968:
9966:
9963:
9961:
9958:
9956:
9955:Lorenz system
9953:
9951:
9948:
9946:
9943:
9942:
9940:
9934:
9928:
9925:
9923:
9920:
9918:
9915:
9913:
9910:
9908:
9905:
9903:
9902:Langton's ant
9900:
9898:
9895:
9893:
9890:
9888:
9885:
9883:
9880:
9878:
9877:Horseshoe map
9875:
9873:
9870:
9868:
9865:
9863:
9860:
9858:
9855:
9851:
9848:
9847:
9846:
9843:
9841:
9838:
9836:
9833:
9831:
9828:
9826:
9823:
9821:
9818:
9816:
9813:
9812:
9810:
9804:
9801:
9798:
9791:
9785:
9782:
9780:
9777:
9775:
9774:Quantum chaos
9772:
9770:
9767:
9765:
9762:
9760:
9757:
9755:
9752:
9751:
9749:
9743:
9738:
9734:
9730:
9716:
9713:
9711:
9708:
9706:
9703:
9701:
9698:
9696:
9693:
9691:
9688:
9686:
9683:
9682:
9680:
9674:
9668:
9665:
9663:
9660:
9658:
9655:
9653:
9650:
9648:
9645:
9643:
9640:
9638:
9635:
9633:
9630:
9628:
9625:
9623:
9620:
9618:
9615:
9613:
9610:
9608:
9605:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9582:Arnold tongue
9580:
9578:
9575:
9574:
9571:
9565:
9562:
9560:
9557:
9555:
9552:
9550:
9547:
9545:
9542:
9540:
9537:
9535:
9532:
9530:
9527:
9526:
9524:
9518:
9515:
9511:
9507:
9500:
9495:
9493:
9488:
9486:
9481:
9480:
9477:
9471:
9468:
9465:
9461:
9458:
9455:
9451:
9448:
9446:
9443:
9441:
9438:
9435:
9431:
9430:
9425:
9421:
9418:
9415:
9412:
9409:
9406:
9402:
9399:
9396:
9393:
9392:
9382:
9376:
9371:
9370:
9363:
9359:
9353:
9348:
9347:
9340:
9336:
9330:
9326:
9321:
9317:
9313:
9309:
9305:
9301:
9297:
9293:
9289:
9285:
9281:
9277:
9273:
9269:
9265:
9261:
9257:
9253:
9249:
9248:
9236:
9232:
9228:
9224:
9220:
9216:
9212:
9208:
9201:
9193:
9189:
9184:
9179:
9175:
9171:
9167:
9163:
9159:
9152:
9144:
9140:
9136:
9132:
9128:
9124:
9117:
9109:
9105:
9101:
9097:
9092:
9087:
9083:
9079:
9075:
9071:
9064:
9058:
9053:
9045:
9041:
9037:
9033:
9029:
9025:
9021:
9014:
9012:
9002:
8997:
8993:
8989:
8985:
8981:
8977:
8970:
8962:
8958:
8954:
8950:
8945:
8940:
8936:
8932:
8925:
8918:
8910:
8906:
8902:
8898:
8894:
8890:
8886:
8880:
8878:
8876:
8874:
8864:
8859:
8852:
8844:
8840:
8836:
8832:
8828:
8824:
8820:
8816:
8812:
8805:
8797:
8791:
8783:
8779:
8775:
8769:
8765:
8764:
8756:
8754:
8745:
8739:
8735:
8728:
8726:
8717:
8711:
8706:
8705:
8699:
8692:
8684:
8678:
8673:
8672:
8666:
8660:
8646:
8642:
8636:
8627:
8622:
8619:(1): 77–105.
8618:
8614:
8613:
8608:
8604:
8600:
8594:
8586:
8582:
8578:
8574:
8569:
8564:
8560:
8556:
8552:
8548:
8544:
8540:
8533:
8531:
8522:
8518:
8514:
8510:
8506:
8502:
8498:
8494:
8493:
8488:
8481:
8473:
8469:
8465:
8461:
8457:
8453:
8449:
8442:
8434:
8430:
8426:
8422:
8418:
8414:
8413:
8405:
8397:
8393:
8388:
8383:
8379:
8375:
8371:
8367:
8363:
8356:
8354:
8352:
8350:
8340:
8339:
8334:
8331:
8324:
8316:
8312:
8308:
8304:
8299:
8294:
8290:
8286:
8282:
8278:
8274:
8270:
8269:
8261:
8259:
8250:
8246:
8242:
8238:
8234:
8230:
8226:
8222:
8218:
8211:
8207:
8198:
8195:
8193:
8190:
8187:
8184:
8181:
8178:
8176:
8173:
8171:
8168:
8165:
8161:
8158:
8157:
8151:
8137:
8117:
8091:
8085:
8082:
8079:
8076:
8070:
8067:
8061:
8047:
8038:
8024:
8021:
8018:
8007:
8003:
8000:
7995:
7986:
7980:
7977:
7968:
7964:
7961:
7956:
7947:
7943:
7940:
7920:
7917:
7912:
7907:
7904:
7901:
7898:
7889:
7873:
7868:
7865:
7862:
7850:
7842:
7839:
7835:
7827:
7824:
7821:
7817:
7813:
7804:
7796:
7792:
7788:
7779:
7770:
7742:
7739:
7734:
7724:
7721:
7716:
7710:
7702:
7699:
7695:
7689:
7681:
7676:
7672:
7666:
7658:
7649:
7628:
7625:
7620:
7617:
7612:
7608:
7602:
7597:
7594:
7591:
7587:
7561:
7558:
7555:
7551:
7544:
7541:
7536:
7532:
7509:
7505:
7501:
7495:
7489:
7480:
7466:
7463:
7455:
7451:
7444:
7424:
7421:
7415:
7409:
7384:
7381:
7378:
7372:
7367:
7364:
7361:
7358:
7355:
7350:
7346:
7342:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7316:
7312:
7308:
7302:
7296:
7271:
7267:
7260:
7257:
7252:
7248:
7244:
7241:
7232:
7226:
7223:
7220:
7214:
7211:
7205:
7196:
7193:
7188:
7183:
7179:
7175:
7170:
7167:
7163:
7139:
7136:
7133:
7129:
7125:
7122:
7116:
7110:
7090:
7087:
7084:
7062:
7058:
7054:
7051:
7016:
7013:
7010:
7004:
6999:
6996:
6988:
6985:
6980:
6977:
6962:
6959:
6956:
6950:
6945:
6942:
6934:
6931:
6926:
6923:
6917:
6895:
6892:
6886:
6878:
6874:
6870:
6867:
6864:
6858:
6850:
6846:
6825:
6822:
6817:
6813:
6809:
6806:
6797:
6783:
6780:
6777:
6774:
6769:
6765:
6761:
6756:
6752:
6748:
6743:
6739:
6735:
6732:
6712:
6690:
6686:
6682:
6679:
6673:
6670:
6667:
6661:
6658:
6652:
6644:
6640:
6630:
6626:
6618:
6616:
6594:
6591:
6568:
6559:
6545:
6542:
6539:
6519:
6516:
6513:
6506:
6496:
6482:
6478:
6474:
6471:
6466:
6462:
6457:
6453:
6428:
6424:
6420:
6417:
6414:
6411:
6402:
6398:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6340:
6338:3.96155658717
6337:
6334:
6325:
6304:
6301:
6298:
6295:
6290:
6286:
6282:
6279:
6276:
6271:
6267:
6244:
6240:
6217:
6213:
6192:
6189:
6184:
6180:
6176:
6171:
6167:
6137:
6133:
6129:
6126:
6120:
6114:
6108:
6105:
6102:
6093:
6087:
6060:
6056:
6035:
6032:
6029:
6026:
6023:
6020:
6017:
6014:
6011:
5991:
5988:
5985:
5976:
5955:
5952:
5949:
5946:
5941:
5937:
5933:
5930:
5927:
5922:
5918:
5895:
5891:
5868:
5864:
5843:
5840:
5835:
5831:
5827:
5822:
5818:
5793:
5790:
5787:
5782:
5778:
5757:
5754:
5751:
5746:
5742:
5719:
5717:
5713:
5691:
5687:
5683:
5680:
5674:
5668:
5665:
5662:
5653:
5647:
5627:
5624:
5621:
5601:
5598:
5595:
5575:
5539:
5535:
5512:
5491:
5487:
5472:
5458:
5455:
5452:
5449:
5429:
5422:The constant
5405:
5402:
5399:
5396:
5370:
5366:
5362:
5359:
5353:
5347:
5344:
5341:
5332:
5326:
5306:
5303:
5300:
5295:
5291:
5281:
5262:
5259:
5256:
5253:
5233:
5211:
5197:
5194:
5191:
5188:
5180:
5176:
5154:
5150:
5146:
5143:
5137:
5131:
5128:
5125:
5122:
5116:
5110:
5090:
5064:
5060:
5056:
5053:
5047:
5041:
5038:
5035:
5026:
5020:
5000:
4980:
4971:
4957:
4954:
4949:
4942:
4938:
4933:
4929:
4924:
4917:
4913:
4908:
4904:
4899:
4892:
4888:
4883:
4879:
4874:
4867:
4863:
4858:
4854:
4849:
4842:
4838:
4833:
4803:
4799:
4794:
4773:
4770:
4767:
4762:
4758:
4733:
4730:
4725:
4718:
4714:
4709:
4705:
4700:
4693:
4689:
4684:
4680:
4675:
4668:
4664:
4659:
4655:
4650:
4643:
4639:
4634:
4630:
4625:
4618:
4614:
4609:
4588:
4585:
4582:
4577:
4573:
4563:
4544:
4541:
4538:
4533:
4529:
4508:
4485:
4478:Scaling limit
4469:
4453:
4450:
4447:
4436:
4415:
4412:
4392:
4389:
4386:
4375:
4352:
4349:
4346:
4335:
4314:
4311:
4291:
4288:
4285:
4274:
4245:
4242:
4238:
4217:
4214:
4211:
4189:
4185:
4162:
4159:
4156:
4152:
4140:
4117:
4114:
4110:
4089:
4086:
4083:
4061:
4057:
4034:
4031:
4028:
4024:
4012:
3989:
3986:
3982:
3961:
3958:
3955:
3933:
3929:
3906:
3903:
3900:
3896:
3884:
3875:
3873:
3857:
3854:
3851:
3831:
3828:
3825:
3805:
3796:
3782:
3779:
3776:
3756:
3753:
3750:
3730:
3721:
3704:
3701:
3698:
3695:
3692:
3669:
3666:
3663:
3654:
3637:
3634:
3631:
3628:
3625:
3600:
3595:
3591:
3570:
3550:
3544:
3522:
3517:
3513:
3492:
3469:
3466:
3463:
3457:
3454:
3451:
3445:
3437:
3433:
3413:
3390:
3368:
3363:
3358:
3351:
3346:
3340:
3333:
3327:
3323:
3313:
3294:
3291:
3288:
3196:
3192:
3186:
3182:
3178:
3175:
3171:
3167:
3162:
3158:
3154:
3149:
3145:
3137:
3136:
3135:
3134:
3133:homeomorphism
3108:
3105:
3102:
3097:
3093:
3089:
3083:
3080:
3072:
3069:
3064:
3060:
3056:
3046:
3043:
3037:
3032:
3028:
3024:
3021:
3014:
3010:
3006:
3000:
2995:
2990:
2987:
2984:
2980:
2972:
2971:
2970:
2969:
2959:
2952:
2943:
2939:
2930:
2929:bit-shift map
2926:
2920:
2915:
2904:
2894:
2887:
2859:goes to 0 as
2853:
2843:
2833:
2822:
2798:
2794:
2788:
2782:
2778:
2774:
2771:
2768:
2764:
2756:
2753:
2747:
2741:
2738:
2732:
2727:
2723:
2715:
2714:
2713:
2709:
2703:
2696:
2675:
2672:
2667:
2662:
2656:
2652:
2648:
2645:
2642:
2638:
2631:
2626:
2622:
2618:
2615:
2612:
2609:
2606:
2599:
2598:
2597:
2574:
2570:
2567:
2560:
2556:
2552:
2548:
2544:
2537:
2533:
2528:
2524:
2518:
2513:
2509:
2501:
2500:
2499:
2497:
2491:
2485:
2476:
2443:For rational
2427:
2423:
2416:
2412:
2406:
2402:
2397:
2394:
2390:
2383:
2380:
2374:
2371:
2364:
2363:
2362:
2341:
2337:
2333:
2330:
2325:
2321:
2316:
2312:
2307:
2303:
2299:
2294:
2290:
2282:
2281:
2280:
2276:
2269:
2262:
2252:
2231:
2225:
2220:
2216:
2212:
2207:
2204:
2200:
2193:
2189:
2183:
2178:
2174:
2169:
2163:
2160:
2157:
2151:
2148:
2145:
2140:
2136:
2122:
2119:
2116:
2113:
2110:
2104:
2101:
2091:
2090:
2089:
2085:
2072:
2061:
2050:
2007:"x"
1995:"r"
1655:
1653:
1649:
1636:
1631:
1627:
1625:
1605:
1594:
1591:
1588:
1582:
1577:
1573:
1564:
1563:
1562:
1558:
1551:
1546:
1532:
1527:
1523:
1519:
1515:
1514:
1508:
1504:
1499:
1495:
1491:
1487:
1483:
1471:
1466:
1452:
1429:
1405:
1401:
1396:
1375:
1372:
1369:
1346:
1343:
1340:
1334:
1331:
1322:
1320:
1316:
1315:exponentially
1311:
1307:
1303:
1295:
1290:
1286:
1276:
1272:
1271:
1264:
1259:
1255:
1254:Poincaré plot
1250:
1244:
1241:
1226:
1219:
1213:
1207:
1201:
1195:
1187:
1183:
1175:
1170:
1165:
1116:
1109:
1108:
1107:
1097:
1092:
1084:
1080:
1078:
1074:
1067:
1062:
1046:
1042:
1038:
1034:
1020:
1015:
1001:
995:
990:
975:
971:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
926:
922:
918:
915:
910:
906:
897:
881:
878:
875:
872:
869:
866:
863:
855:
838:
825:
815:
811:
805:
798:
792:
787:
765:
757:
749:
734:
731:
726:
721:
718:
715:
712:
704:
686:
674:
671:
666:
659:
654:
651:
645:
641:
632:
629:
624:
614:
598:
589:
586:
583:
574:
571:
568:
560:
557:
554:
551:
547:
540:
537:
533:
528:
523:
519:
511:and given by
495:
492:
486:
481:
472:
463:
447:
440:
436:
428:
419:
406:
399:
398:
389:
385:
378:
376:
371:
365:
353:
340:
334:
329:
323:
318:
317:bit-shift map
312:
292:
289:
286:
282:
279:
278:
277:
263:
256:
254:
240:
232:
228:
224:
221:
213:
209:
205:
202:
197:
194:
191:
187:
179:
178:
175:
173:
169:
165:
161:
160:Edward Lorenz
157:
153:
149:
148:degree 2
145:
141:
138:
134:
119:
115:
112:
104:
98:
94:
90:
86:
82:
79:
72:
71:main category
68:
67:
62:
59:
55:
51:
49:
48:
41:
39:
34:You can help
30:
21:
20:
10304:Chaotic maps
10225:Mary Tsingou
10190:David Ruelle
10185:Otto Rössler
10130:Michel Hénon
10100:Leon O. Chua
10057:Tilt-A-Whirl
10027:FPUT problem
9912:Standard map
9907:Logistic map
9906:
9732:
9506:Chaos theory
9427:
9368:
9345:
9324:
9291:
9287:
9259:
9255:
9210:
9206:
9200:
9165:
9161:
9151:
9126:
9122:
9116:
9076:(1): 25–52.
9073:
9069:
9063:
9052:
9027:
9023:
8983:
8979:
8969:
8944:nlin/0407042
8934:
8930:
8917:
8892:
8888:
8851:
8821:(1): 39–88.
8818:
8814:
8804:
8762:
8733:
8703:
8691:
8670:
8659:
8648:. Retrieved
8644:
8635:
8616:
8610:
8593:
8542:
8538:
8496:
8490:
8480:
8455:
8451:
8441:
8416:
8410:
8404:
8369:
8365:
8336:
8323:
8272:
8266:
8224:
8220:
8210:
8053:
8044:
7890:
7481:
6798:
6631:
6627:
6624:
6581:at interval
6560:
6502:
6403:
6399:
5809:
5716:universality
5715:
5567:
5478:
5421:
5179:universality
5178:
5173:. This is a
4972:
4749:
3797:
3722:
3655:
3424:
3416:Universality
3388:
3366:
3349:
3338:
3331:
3328:
3321:
3314:
3211:
3130:
2967:
2966:follows the
2954:
2947:
2941:
2937:
2928:
2918:
2913:
2902:
2899:
2892:
2851:
2841:
2831:
2820:
2817:
2707:
2704:
2692:
2591:
2489:
2486:
2442:
2361:is given by
2356:
2274:
2267:
2260:
2257:
2250:
2080:
2070:
2059:
2055:
2048:
1704:# start, end
1645:
1620:
1556:
1549:
1530:
1511:
1509:
1502:
1467:
1323:
1299:
1268:
1262:
1251:
1231:
1224:
1217:
1211:
1205:
1199:
1193:
1173:
1101:
1065:
1061:self-similar
1058:
1016:
1005:
993:
836:
823:
813:
809:
803:
796:
790:
785:
763:
684:
657:
643:
484:
470:
461:
443:
426:
417:
379:
372:
369:
363:
339:Ricker model
332:
326:case of the
321:
310:
296:
290:
285:proportional
281:reproduction
280:
268:
257:
133:logistic map
132:
130:
93:edit summary
84:
64:
35:
10210:Nina Snaith
10200:Yakov Sinai
10085:Rufus Bowen
9835:Duffing map
9820:Baker's map
9745:Theoretical
9657:SRB measure
9564:Phase space
9534:Bifurcation
9168:(8): 1279.
8227:(1): 1–11.
8050:Occurrences
6546:2.502907...
6520:4.669201...
6404:Generally,
5989:3.854077963
5588:approaches
4557:from below.
4521:approaches
3818:approaches
3334:= 1, 2, 3,…
1505:≈ 3.5699456
1498:Grassberger
1490:Grassberger
1484:, 1983), a
1482:Grassberger
1362:intersects
1258:state space
40:in Japanese
10268:Complexity
10165:Edward Ott
10012:Convection
9937:Continuous
9612:Ergodicity
9245:References
9162:Atmosphere
8863:1710.05053
8782:1112373147
8650:2023-05-08
6205:such that
5856:such that
4454:3.56994567
3858:3.56994567
2837:) ∈ (−1,1)
1965:markersize
1513:likelihood
1500:1983) for
1306:complexity
1135:between 1/
1000:Cantor set
662:(sequence
291:starvation
164:Robert May
137:polynomial
10180:Mary Rees
10140:Bryna Kra
10073:theorists
9882:Ikeda map
9872:Hénon map
9862:Gauss map
9544:Limit set
9529:Attractor
9316:119833080
9256:Physica D
9235:124347430
9192:2073-4433
9108:124498882
9086:CiteSeerX
9044:0556-2791
8961:122705492
8909:116998358
8843:119956479
8790:cite book
8521:119466337
8472:0025-570X
8433:123124113
8396:101491730
8338:MathWorld
8249:0040-2826
8086:
8080:−
8065:→
8022:−
8019:≈
8012:∞
7991:∞
7978:−
7973:∞
7952:∞
7944:≈
7941:α
7902:≈
7899:δ
7863:≈
7855:∞
7836:≈
7814:−
7809:∞
7789:−
7784:∞
7740:−
7700:−
7682:≈
7654:∞
7626:−
7603:≈
7542:≈
7502:≈
7464:≈
7365:−
7331:−
7215:−
7194:∘
7176:∘
7168:−
7000:−
6662:−
6598:∞
6540:α
6514:δ
6463:α
6454:δ
6425:α
6418:≈
6415:δ
6385:…
6376:α
6370:…
6361:δ
6341:…
6330:∞
6305:…
6299:3.9615554
6193:…
6138:α
6127:−
6106:α
6103:−
6100:↦
6066:∞
6036:…
6027:α
6021:…
6012:δ
5992:…
5981:∞
5956:…
5844:…
5794:⋯
5783:∗
5758:…
5747:∗
5692:α
5681:−
5666:α
5663:−
5660:↦
5628:α
5622:δ
5602:3.8494344
5599:≈
5545:∞
5497:∞
5459:…
5450:α
5430:α
5406:…
5397:α
5371:α
5360:−
5345:α
5342:−
5339:↦
5307:⋯
5296:∗
5263:…
5254:α
5234:α
5198:⋯
5195:4.6692016
5189:δ
5155:α
5144:−
5129:α
5126:−
5065:α
5054:−
5039:α
5036:−
5033:↦
5001:α
4981:α
4958:…
4943:∗
4918:∗
4893:∗
4868:∗
4843:∗
4811:∞
4804:∗
4774:⋯
4763:∗
4734:…
4719:∗
4694:∗
4669:∗
4644:∗
4619:∗
4589:⋯
4578:∗
4545:⋯
4534:∗
4451:≈
3855:≈
3693:−
3626:−
3548:↦
3467:−
3179:π
3168:
3090:≤
3070:−
3025:≤
2916:integers
2828:. Since
2772:−
2748:−
2673:−
2646:−
2632:±
2616:−
2607:α
2553:−
2549:α
2545:−
2529:α
2525:−
2403:
2395:−
2384:π
2372:θ
2334:π
2331:θ
2313:
2205:−
2184:≤
2152:∈
2123:…
2105:∈
2099:∀
1947:numtoplot
1728:numtoplot
1637:function.
1624:decisions
1592:−
1578:π
1522:attractor
1433:∞
1430:−
1397:δ
1344:−
1240:quadratic
1176:> 3.57
1068:≈ 3.82843
1045:attractor
965:−
941:−
932:−
916:−
735:3.8284...
660:≈ 3.56995
646:≈ 4.66920
572:−
561:±
524:±
225:−
156:nonlinear
111:talk page
63:Consider
10298:Category
10256:articles
9998:Physical
9917:Tent map
9807:Discrete
9747:branches
9677:Theorems
9513:Concepts
9407:notebook
8700:(1980).
8667:(2012).
8154:See also
8110:, where
7843:′
6280:3.960102
3345:tent map
2900:For the
1866:accuracy
1860:interval
1854:interval
1779:subplots
1707:accuracy
1683:interval
1139:and 1-1/
1128:and for
1077:fractals
882:0.728...
870:3.678...
807:for all
328:tent map
319:and the
87:provide
10254:Related
10062:Weather
10000:systems
9793:Chaotic
9539:Fractal
9405:Jupyter
9296:Bibcode
9264:Bibcode
9215:Bibcode
9170:Bibcode
9131:Bibcode
9078:Bibcode
8988:Bibcode
8823:Bibcode
8585:2243371
8547:Bibcode
8501:Bibcode
8374:Bibcode
8315:2243371
8277:Bibcode
8229:Bibcode
8025:2.24...
7921:5.12...
5950:3.85361
5755:3.84943
3410:
3398:
3393:
3379:
3360:in the
3357:A001037
3279:
3267:
3263:
3251:
3247:
3235:
3231:
3219:
2884:
2872:
2826:∈ [0,1)
2695:modulus
1520:has an
1149:√
1123:√
1033:value.
991:Beyond
818:. The
689:√
668:in the
665:A098587
626:in the
623:A086181
502:√
475:
458:
431:
414:
152:chaotic
140:mapping
109:to the
91:in the
10160:Hee Oh
9795:maps (
9642:Mixing
9426:," in
9377:
9354:
9331:
9314:
9233:
9190:
9106:
9088:
9042:
8980:Tellus
8959:
8907:
8841:
8780:
8770:
8740:
8712:
8679:
8583:
8577:934280
8575:
8539:Nature
8519:
8470:
8431:
8394:
8313:
8307:934280
8305:
8268:Nature
8247:
8221:Tellus
7933:, and
7758:, and
7289:where
6048:. And
5931:3.8284
5791:3.5699
5456:2.5029
5403:2.5029
5304:3.5699
5260:2.5029
4771:3.5699
4586:3.5699
4542:3.5699
2921:> 0
2850:(1 − 2
2830:(1 − 2
2592:where
2001:ylabel
1989:xlabel
1848:arange
1821:random
1713:0.0001
1671:import
1659:import
1654:code:
1652:Python
335:> 4
269:where
10071:Chaos
9850:outer
9554:Orbit
9403:as a
9312:S2CID
9231:S2CID
9104:S2CID
8957:S2CID
8939:arXiv
8927:(PDF)
8905:S2CID
8858:arXiv
8839:S2CID
8581:S2CID
8517:S2CID
8429:S2CID
8392:S2CID
8311:S2CID
8203:Notes
7579:, or
6382:38.82
6367:981.6
6033:9.277
6018:55.26
5568:When
5389:with
4440:When
4379:When
4339:When
4278:When
4204:when
4076:when
3948:when
3401:2 − 1
3382:2 − 1
3347:with
3317:[0,1)
2953:(1 −
2693:with
2079:(1 −
2013:title
1881:range
1749:zeros
1662:numpy
1559:= 0.5
1552:= 0.5
1541:(0,1)
1537:(0,1)
1478:0.005
1474:0.500
1275:embed
1247:(0,1)
1220:= 3.5
1117:With
1073:chaos
812:<
762:. A
633:With
615:With
496:With
451:With
407:With
400:With
330:. If
146:) of
135:is a
54:DeepL
9797:list
9521:Core
9375:ISBN
9352:ISBN
9329:ISBN
9188:ISSN
9040:ISSN
8796:link
8778:OCLC
8768:ISBN
8738:ISBN
8710:ISBN
8677:ISBN
8573:PMID
8468:ISSN
8303:PMID
8245:ISSN
8083:tanh
4393:3.45
4177:and
4049:and
3921:and
3832:3.45
3377:2 ⋅
3362:OEIS
3103:<
3038:<
2962:and
2912:for
2818:for
2487:For
2279:is,
2058:0 ≤
2047:0 ≤
2031:show
1977:biax
1971:0.02
1953:lims
1938:plot
1932:biax
1926:lims
1911:lims
1899:lims
1887:reps
1827:rand
1809:lims
1767:biax
1755:reps
1737:lims
1716:reps
1646:The
1554:and
1308:and
1260:for
1215:for
1209:and
1110:For
1075:and
1017:The
670:OEIS
628:OEIS
131:The
85:must
83:You
9432:by
9304:doi
9272:doi
9223:doi
9178:doi
9139:doi
9096:doi
9032:doi
8996:doi
8949:doi
8897:doi
8831:doi
8621:doi
8563:hdl
8555:doi
8543:261
8509:doi
8460:doi
8421:doi
8382:doi
8370:52a
8293:hdl
8285:doi
8273:261
8237:doi
7767:lim
7686:lim
7663:lim
6451:lim
6397:.
4353:3.4
4292:3.0
4218:3.3
3962:2.7
3798:As
3783:3.4
3757:3.0
3723:As
3670:3.0
3369:= 4
3352:= 2
3341:= 4
3324:= 4
3159:sin
2946:= 4
2944:+ 1
2914:all
2905:= 4
2895:= 4
2712:is
2710:= 2
2492:= 4
2391:sin
2304:sin
2277:= 4
2270:= 2
2263:= 4
2253:= 4
2131:and
2062:≤ 1
2051:≤ 1
2025:plt
1983:set
1872:for
1833:for
1785:fig
1773:plt
1761:fig
1734:200
1722:600
1692:2.8
1680:plt
1533:= 4
1472:of
1265:= 4
996:= 4
856:At
839:= 1
705:At
655:At
487:= 3
464:− 1
420:− 1
324:= 2
313:= 4
56:or
42:.
10300::
9310:.
9302:.
9292:26
9290:.
9270:.
9258:.
9229:.
9221:.
9209:.
9186:.
9176:.
9166:14
9164:.
9160:.
9137:.
9127:14
9125:.
9102:.
9094:.
9084:.
9074:19
9072:.
9038:.
9028:31
9026:.
9022:.
9010:^
8994:.
8984:16
8982:.
8978:.
8955:.
8947:.
8935:10
8933:.
8929:.
8903:.
8891:.
8872:^
8837:.
8829:.
8819:81
8817:.
8813:.
8792:}}
8788:{{
8776:.
8752:^
8724:^
8643:.
8615:.
8609:.
8601:;
8579:.
8571:.
8561:.
8553:.
8541:.
8529:^
8515:.
8507:.
8497:26
8495:.
8489:.
8466:.
8456:69
8454:.
8450:.
8427:.
8417:83
8415:.
8390:.
8380:.
8368:.
8364:.
8348:^
8335:.
8309:.
8301:.
8291:.
8283:.
8271:.
8257:^
8243:.
8235:.
8225:16
8223:.
8219:.
7913:17
7888:.
7874:17
7735:17
7479:.
6796:.
6558:.
6532:,
6495:.
4950:16
4726:16
3874:.
3407:13
3375::
3265:→
3249:→
3233:→
2886:.
2702:.
2462:,
2034:()
1896:):
1878:in
1869:):
1842:np
1839:in
1830:()
1815:np
1797:16
1782:()
1743:np
1677:as
1668:np
1665:as
1249:.
1203:,
1197:,
1167:A
1106::
1079:.
630:).
493:).
9799:)
9498:e
9491:t
9484:v
9466:.
9456:.
9422:"
9383:.
9360:.
9337:.
9318:.
9306::
9298::
9278:.
9274::
9266::
9260:9
9237:.
9225::
9217::
9211:3
9194:.
9180::
9172::
9145:.
9141::
9133::
9110:.
9098::
9080::
9046:.
9034::
9004:.
8998::
8990::
8963:.
8951::
8941::
8911:.
8899::
8893:3
8866:.
8860::
8845:.
8833::
8825::
8798:)
8784:.
8746:.
8718:.
8685:.
8653:.
8629:.
8623::
8617:3
8587:.
8565::
8557::
8549::
8523:.
8511::
8503::
8474:.
8462::
8435:.
8423::
8398:.
8384::
8376::
8341:.
8317:.
8295::
8287::
8279::
8251:.
8239::
8231::
8166:.
8138:G
8118:x
8098:)
8095:)
8092:x
8089:(
8077:1
8074:(
8071:x
8068:G
8062:x
8008:r
8004:4
8001:+
7996:2
7987:r
7981:3
7969:r
7965:4
7962:+
7957:2
7948:r
7918:=
7908:+
7905:1
7869:+
7866:1
7860:)
7851:r
7847:(
7840:S
7828:1
7825:+
7822:n
7818:r
7805:r
7797:n
7793:r
7780:r
7771:n
7746:)
7743:3
7730:(
7725:2
7722:1
7717:=
7714:)
7711:0
7708:(
7703:n
7696:S
7690:n
7677:n
7673:r
7667:n
7659:=
7650:r
7629:2
7621:6
7618:+
7613:n
7609:r
7598:1
7595:+
7592:n
7588:r
7567:)
7562:1
7559:+
7556:n
7552:r
7548:(
7545:S
7537:n
7533:r
7510:1
7506:r
7499:)
7496:r
7493:(
7490:S
7467:0
7461:)
7456:1
7452:r
7448:(
7445:S
7425:0
7422:=
7419:)
7416:r
7413:(
7410:S
7388:)
7385:4
7382:+
7379:r
7376:(
7373:r
7368:3
7362:r
7359:4
7356:+
7351:2
7347:r
7343:=
7340:c
7337:,
7334:2
7328:r
7325:4
7322:+
7317:2
7313:r
7309:=
7306:)
7303:r
7300:(
7297:S
7277:)
7272:3
7268:x
7264:(
7261:O
7258:+
7253:2
7249:x
7245:+
7242:x
7239:)
7236:)
7233:r
7230:(
7227:S
7224:+
7221:1
7218:(
7212:=
7209:)
7206:x
7203:(
7200:)
7197:T
7189:2
7184:r
7180:f
7171:1
7164:T
7160:(
7140:p
7137:+
7134:c
7130:/
7126:x
7123:=
7120:)
7117:x
7114:(
7111:T
7091:p
7088:=
7085:x
7063:1
7059:r
7055:=
7052:r
7025:)
7020:)
7017:4
7014:+
7011:r
7008:(
7005:r
6997:r
6994:(
6989:2
6986:1
6981:=
6978:q
6971:)
6966:)
6963:4
6960:+
6957:r
6954:(
6951:r
6946:+
6943:r
6940:(
6935:2
6932:1
6927:=
6924:p
6918:{
6896:p
6893:=
6890:)
6887:q
6884:(
6879:r
6875:f
6871:,
6868:q
6865:=
6862:)
6859:p
6856:(
6851:r
6847:f
6826:0
6823:=
6818:0
6814:r
6810:=
6807:r
6784:.
6781:.
6778:.
6775:,
6770:2
6766:r
6762:,
6757:1
6753:r
6749:,
6744:0
6740:r
6736:=
6733:r
6713:r
6691:2
6687:x
6683:+
6680:x
6677:)
6674:r
6671:+
6668:1
6665:(
6659:=
6656:)
6653:x
6650:(
6645:r
6641:f
6601:)
6595:,
6592:0
6589:[
6569:G
6543:=
6517:=
6483:3
6479:/
6475:2
6472:=
6467:2
6458:/
6429:2
6421:2
6412:3
6379:=
6373:,
6364:=
6335:=
6326:r
6302:,
6296:=
6291:2
6287:r
6283:,
6277:=
6272:1
6268:r
6245:n
6241:4
6218:n
6214:r
6190:,
6185:2
6181:r
6177:,
6172:1
6168:r
6147:)
6144:)
6141:)
6134:/
6130:x
6124:(
6121:f
6118:(
6115:f
6112:(
6109:f
6097:)
6094:x
6091:(
6088:f
6061:r
6057:f
6030:=
6024:,
6015:=
5986:=
5977:r
5953:,
5947:=
5942:2
5938:r
5934:,
5928:=
5923:1
5919:r
5896:n
5892:3
5869:n
5865:r
5841:,
5836:2
5832:r
5828:,
5823:1
5819:r
5788:=
5779:r
5752:=
5743:r
5718:.
5698:)
5695:)
5688:/
5684:x
5678:(
5675:f
5672:(
5669:f
5657:)
5654:x
5651:(
5648:f
5625:,
5596:r
5576:r
5540:r
5536:f
5492:r
5488:f
5453:=
5400:=
5377:)
5374:)
5367:/
5363:x
5357:(
5354:f
5351:(
5348:f
5336:)
5333:x
5330:(
5327:f
5301:=
5292:r
5257:=
5210:.
5192:=
5161:)
5158:)
5151:/
5147:x
5141:(
5138:g
5135:(
5132:g
5123:=
5120:)
5117:x
5114:(
5111:g
5091:g
5071:)
5068:)
5061:/
5057:x
5051:(
5048:f
5045:(
5042:f
5030:)
5027:x
5024:(
5021:f
5013::
4955:,
4939:r
4934:f
4930:,
4925:8
4914:r
4909:f
4905:,
4900:4
4889:r
4884:f
4880:,
4875:2
4864:r
4859:f
4855:,
4850:1
4839:r
4834:f
4800:r
4795:f
4768:=
4759:r
4731:,
4715:r
4710:f
4706:,
4701:8
4690:r
4685:f
4681:,
4676:4
4665:r
4660:f
4656:,
4651:2
4640:r
4635:f
4631:,
4626:1
4615:r
4610:f
4583:=
4574:r
4539:=
4530:r
4509:r
4470:.
4448:r
4416:1
4413:+
4390:=
4387:r
4350:=
4347:r
4315:1
4312:+
4289:=
4286:r
4246:2
4243:f
4239:x
4215:=
4212:a
4190:n
4186:x
4163:2
4160:+
4157:n
4153:x
4118:2
4115:f
4111:x
4090:3
4087:=
4084:a
4062:n
4058:x
4035:2
4032:+
4029:n
4025:x
4004:.
3990:2
3987:f
3983:x
3959:=
3956:a
3934:n
3930:x
3907:2
3904:+
3901:n
3897:x
3852:r
3829:=
3826:r
3806:r
3780:=
3777:r
3754:=
3751:r
3731:r
3708:)
3705:1
3702:+
3699:,
3696:1
3690:(
3667:=
3664:r
3641:)
3638:1
3635:+
3632:,
3629:1
3623:(
3601:n
3596:r
3592:f
3571:n
3551:x
3545:x
3523:n
3518:r
3514:f
3493:n
3473:)
3470:x
3464:1
3461:(
3458:x
3455:r
3452:=
3449:)
3446:x
3443:(
3438:r
3434:f
3404:/
3389:k
3385:/
3373:k
3367:r
3350:μ
3343:(
3339:r
3332:k
3322:r
3310:k
3295:4
3292:=
3289:r
3276:7
3273:/
3270:1
3260:7
3257:/
3254:4
3244:7
3241:/
3238:2
3228:7
3225:/
3222:1
3214:y
3197:.
3193:)
3187:n
3183:y
3176:2
3172:(
3163:2
3155:=
3150:n
3146:x
3109:,
3106:1
3098:n
3094:y
3084:2
3081:1
3073:1
3065:n
3061:y
3057:2
3047:2
3044:1
3033:n
3029:y
3022:0
3015:n
3011:y
3007:2
3001:{
2996:=
2991:1
2988:+
2985:n
2981:y
2964:y
2960:)
2957:n
2955:x
2950:n
2948:x
2942:n
2938:x
2933:x
2919:k
2910:k
2903:r
2893:r
2881:2
2878:/
2875:1
2867:n
2865:x
2861:n
2857:)
2855:0
2852:x
2845:0
2842:x
2835:0
2832:x
2824:0
2821:x
2799:n
2795:2
2789:)
2783:0
2779:x
2775:2
2769:1
2765:(
2757:2
2754:1
2742:2
2739:1
2733:=
2728:n
2724:x
2708:r
2700:α
2676:1
2668:2
2663:)
2657:0
2653:x
2649:2
2643:1
2639:(
2627:0
2623:x
2619:2
2613:1
2610:=
2594:α
2575:4
2571:2
2568:+
2561:n
2557:2
2538:n
2534:2
2519:=
2514:n
2510:x
2490:r
2481:n
2479:x
2471:2
2466:n
2464:x
2460:θ
2456:θ
2451:n
2449:x
2445:θ
2428:.
2424:)
2417:0
2413:x
2407:(
2398:1
2381:1
2375:=
2359:θ
2342:,
2338:)
2326:n
2322:2
2317:(
2308:2
2300:=
2295:n
2291:x
2275:r
2268:r
2261:r
2251:r
2232:.
2226:n
2221:0
2217:x
2213:+
2208:n
2201:r
2194:0
2190:x
2179:n
2175:x
2170:,
2167:]
2164:1
2161:,
2158:0
2155:[
2149:r
2146:,
2141:0
2137:x
2126:}
2120:,
2117:1
2114:,
2111:0
2108:{
2102:n
2086:)
2083:n
2081:x
2074:0
2071:x
2066:r
2060:r
2049:r
2028:.
2022:)
2016:=
2010:,
2004:=
1998:,
1992:=
1986:(
1980:.
1974:)
1968:=
1962:,
1956:,
1950:,
1944:*
1941:(
1935:.
1929:)
1923:-
1920:1
1917:(
1914:*
1908:*
1905:r
1902:=
1893:1
1890:-
1884:(
1875:i
1863:,
1857:,
1851:(
1845:.
1836:r
1824:.
1818:.
1812:=
1806:)
1803:9
1800:,
1794:(
1788:.
1776:.
1770:=
1764:,
1758:)
1752:(
1746:.
1740:=
1731:=
1719:=
1710:=
1701:)
1698:4
1695:,
1689:(
1686:=
1606:.
1598:)
1595:x
1589:1
1586:(
1583:x
1574:1
1557:b
1550:a
1531:r
1503:r
1480:(
1476:±
1453:r
1406:n
1402:2
1376:x
1373:=
1370:y
1350:)
1347:x
1341:1
1338:(
1335:x
1332:r
1282:t
1280:x
1270:1
1263:r
1235:r
1225:f
1218:r
1212:f
1206:f
1200:f
1194:f
1189:f
1174:r
1151:6
1145:r
1141:r
1137:r
1133:0
1130:x
1125:6
1119:r
1112:r
1104:r
1098:.
1066:r
1055:.
1053:r
1049:r
1031:r
1027:x
1023:r
1012:r
1008:r
994:r
988:.
976:r
972:/
968:1
962:1
959:=
956:x
953:,
950:0
947:=
944:8
938:r
935:4
927:2
923:r
919:2
911:3
907:r
879:=
876:x
873:,
867:=
864:r
853:.
851:c
847:c
843:c
837:c
832:c
828:r
820:r
816:*
814:k
810:k
804:c
802:2
797:c
795:2
786:c
784:2
780:r
776:k
772:r
768:c
760:r
752:r
732:=
727:8
722:+
719:1
716:=
713:r
700:r
696:r
691:8
681:r
677:r
658:r
652:.
644:δ
635:r
617:r
612:.
599:)
593:)
590:1
587:+
584:r
581:(
578:)
575:3
569:r
566:(
558:1
555:+
552:r
548:(
541:r
538:2
534:1
529:=
520:x
509:r
504:6
498:r
485:r
471:r
467:/
462:r
453:r
445:r
427:r
423:/
418:r
409:r
402:r
393:r
382:r
364:r
349:n
347:x
343:r
333:r
322:μ
311:r
305:n
303:x
299:r
273:n
271:x
262:)
260:1
258:(
241:,
238:)
233:n
229:x
222:1
219:(
214:n
210:x
206:r
203:=
198:1
195:+
192:n
188:x
120:.
113:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
