6015:
993:
5113:
5135:
33:
6007:
163:
4565:
4533:
3780:
1774:
3509:
5048:
4854:
this holds no longer, and one can find regions of two overlapping locking regions. For the circle map it can be shown that in this region, no more than two stable mode locking regions can overlap, but if there is any limit to the number of overlapping Arnold tongues for general synchronised systems
1370:
99:
triggers in the area a series of substance (mainly proteins) oscillations that interact with each other; simulations show that these interactions cause Arnold tongues to appear, that is, the frequency of some oscillations constrain the others, and this can be used to control tumor growth.
146:
One of the simplest physical models that exhibits mode-locking consists of two rotating disks connected by a weak spring. One disk is allowed to spin freely, and the other is driven by a motor. Mode locking occurs when the freely-spinning disk turns at a frequency that is a
3775:{\displaystyle {\begin{aligned}\theta _{n}'&=\theta _{0}+n\Omega '+{\frac {K}{2\pi }}\sum _{i=0}^{n}\sin(2\pi \theta _{i})\\&=\theta _{0}+n(\Omega +p)+{\frac {K}{2\pi }}\sum _{i=0}^{n}\sin(2\pi \theta _{i})\\&=\theta _{n}+np,\end{aligned}}}
2844:
5452:
Guevara, M.R.; Glass, L. (1982). "Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias".
2463:
371:(the external oscillator) produce periodic electric signals to stimulate heart contractions (the driven oscillator); here, it could be useful to determine the relation between the frequency of the oscillators, possibly to design better
4880:
798:
1769:{\displaystyle {\begin{aligned}0&=y(t_{n-1})-a\cdot (t_{n}-t_{n-1})\\0&=\left-at_{n}+at_{n-1}\\t_{n}&={\frac {1}{a}}\left+t_{n-1}\\t_{n}&=t_{n-1}+{\frac {c}{a}}+{\frac {b}{a}}\sin(2\pi t_{n-1})\end{aligned}}}
3921:
1961:
1375:
4685:
2308:
3499:
4136:
2654:
2226:
552:
4885:
3514:
3185:
3005:
2900:
1295:
3428:
4344:
4015:
4291:
3828:
3125:
5225:
S. Adjan; V. I. Arnol'd; S. P. Demuškin; Ju. S. GureviÄŤ; S. S. Kemhadze; N. I. Klimov; Ju. V. Linnik; A. V. Malyšev; P. S. Novikov; D. A. Suprunenko; V. A. TartakovskiÄ; V. Tašbaev.
3321:
83:
Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and
1138:
2724:
630:
3069:
2345:
4697:
The phase-locked regions, or Arnold tongues, are illustrated in yellow in the figure to the right. Each such V-shaped region touches down to a rational value Ω =
5097:
4502:
4078:
3212:
2947:
2716:
2564:
1811:
946:
852:
4428:
2337:
1851:
1171:
3382:
4522:
4395:
3973:
3448:
3258:
1207:
912:
690:
670:
650:
575:
187:
4196:
2683:
2517:
154:
The simplest mathematical model that exhibits mode-locking is the circle map, which attempts to capture the motion of the spinning disks at discrete time intervals.
1342:
449:
426:
278:
5043:{\displaystyle {\begin{aligned}\theta _{n+1}&=\theta _{n}+p_{n}+{\frac {K}{2\pi }}\sin(2\pi \theta _{n})\\p_{n+1}&=\theta _{n+1}-\theta _{n}\end{aligned}}}
4788:), without disturbing the limiting rotation number. That is, the sequence stays "locked on" to the signal, despite the addition of significant noise to the series
2126:
2054:
1996:
1056:
1023:
4373:
4242:
4041:
3953:
3284:
3035:
2154:
2025:
215:
4468:
4448:
4216:
4159:
3341:
3232:
2920:
2584:
2537:
2491:
2097:
2077:
1362:
1315:
1076:
966:
892:
872:
821:
705:
595:
400:
255:
235:
484:
345:
313:
76:
thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form
5532:
Romeira, B.; Figueiredo, J.M.; Ironside, C.N.; Slight, T. (2009). "Chaotic dynamics in resonant tunneling optoelectronic voltage controlled oscillators".
1859:
359:
another. That is, one oscillator depends on the other but not the other way around, so they do not mutually influence each other as happens in
3833:
1969:
argues that this simple model is applicable to some biological systems, such as regulation of substance concentration in cells or blood, with
6304:
6134:
4631:
2234:
5060:
which is allowed to dynamically vary, rather than being forced fixed, as it is in the circle map. The standard map is studied in
5245:
Jensen, M.H.; Krishna, S. (2012). "Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli".
6014:
5774:
95:, in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a
4085:
4795:. This ability to "lock on" in the presence of noise is central to the utility of the phase-locked loop electronic circuit.
5967:
3453:
2170:
496:
375:. The family of circle maps serves as a useful mathematical model for this biological phenomenon, as well as many others.
5914:
5666:; Guevara, M.R.; Shrier, A.; Perez, R. (1983). "Bifurcation and chaos in a periodically stimulated cardiac oscillator".
3130:
2952:
2855:
6247:
5648:
2592:
428:, representing the angle at which the point is located in the circle. When the modulo is taken with a value other than
1212:
6482:
364:
6289:
5977:
3387:
6164:
4297:
3978:
5703:
McGuinness, M.; Hong, Y.; Galletly, D.; Larsen, P. (2004). "Arnold tongues in human cardiorespiratory systems".
4250:
3787:
3074:
699:
The particular circle map originally studied by Arnold, and which continues to prove useful even nowadays, is:
5982:
5972:
6329:
6237:
6102:
2839:{\displaystyle \theta _{n}=\theta _{0}+n\Omega +{\frac {K}{2\pi }}\sum _{i=0}^{n-1}\sin(2\pi \theta _{i}).}
382:) of the circle to itself. It is mathematically simpler to consider a point in the circle as being a point
4784:
can be perturbed by rather large random disturbances (up to the width of the tongue, for a given value of
2458:{\displaystyle \theta _{i+1}=f(\theta _{i})=\theta _{i}+\Omega +{\frac {K}{2\pi }}\sin(2\pi \theta _{i}).}
1081:
4842:
is plotted as a function of Ω, gives the "Devil's staircase", a shape that is generically similar to the
3289:
2922:. Since the sine oscillates at frequency 1 Hz, the number of oscillations of the sine per cycle of
600:
6242:
5767:
4753:,Ω) in the large V-shaped region in the bottom-center of the figure correspond to a rotation number of
3046:
17:
6309:
6357:
5467:
6061:
1078:. Once it reaches zero, its value is reset to a certain oscillating value, described by a function
6006:
5116:
Circle map showing mode-locked regions or Arnold tongues in black. Ω varies from 0 to 1 along the
6222:
5987:
5894:
4859:
4473:
4049:
3190:
2925:
2694:
2542:
1782:
917:
830:
597:
is a periodic function that yields the influence caused by the external oscillator. Note that if
73:
5962:
4404:
2313:
1816:
6262:
5874:
5462:
1143:
3346:
6581:
6412:
6319:
6117:
5944:
5879:
5854:
5760:
5068:
4871:
4507:
4380:
3958:
3433:
3243:
1179:
897:
675:
655:
635:
560:
172:
128:
4874:
is related to the circle map, having similar recurrence relations, which may be written as
4164:
2662:
2496:
6422:
6174:
6074:
5919:
5712:
5675:
5611:
5541:
5413:
5355:
2566:
only move forward in the circle, never backwards. To see this, note that the derivative of
1320:
431:
408:
372:
260:
6252:
2102:
2030:
1972:
1032:
999:
793:{\displaystyle \theta _{i+1}=\theta _{i}+\Omega +{\frac {K}{2\pi }}\sin(2\pi \theta _{i})}
104:
8:
6382:
6339:
6324:
6169:
6122:
6107:
6092:
5992:
5899:
5884:
5869:
4398:
4352:
4221:
4020:
3932:
3263:
3014:
2310:. Sometimes it will also be convenient to represent the circle map in terms of a mapping
2131:
2004:
981:
969:
693:
192:
166:
84:
5924:
5716:
5679:
5615:
5545:
5417:
5359:
4850:, the circle map is a diffeomorphism, there exist only one stable solution. However, as
6560:
6427:
6257:
6144:
6139:
6031:
5909:
5811:
5627:
5557:
5488:
5379:
5280:
5254:
4453:
4433:
4201:
4144:
3326:
3217:
2905:
2569:
2522:
2476:
2082:
2062:
1347:
1300:
1061:
996:
Depiction of the simple model where the circle map arises 'naturally'. The red line is
977:
951:
877:
857:
806:
580:
451:, the result still represents an angle, but must be normalized so that the whole range
403:
385:
240:
220:
5335:. Conference of Engineering in Medicine and Biology Society. IEEE. pp. 6826–6829.
5053:
with both iterates taken modulo 1. In essence, the standard map introduces a momentum
4725:,Ω) in one of these regions will all result in a motion such that the rotation number
454:
318:
283:
6432:
6397:
6387:
6284:
5904:
5826:
5728:
5687:
5644:
5631:
5580:
5507:
5480:
5371:
5330:
5272:
2468:
We now proceed to listing some interesting properties of these circle endomorphisms.
992:
368:
124:
108:
57:
6447:
5561:
6540:
6452:
6402:
6299:
6227:
6179:
6056:
6041:
6036:
5831:
5720:
5683:
5619:
5549:
5531:
5492:
5472:
5421:
5383:
5363:
5333:
Entrainability of cell cycle oscillator models with exponential growth of cell mass
5311:
5284:
5264:
136:
36:
Rotation number for different values of two parameters of the circle map: Ω on the
5746:
5600:"Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals"
5268:
5112:
6472:
6367:
6294:
6127:
5939:
5929:
5599:
5224:
4843:
4622:
4611:
4568:
973:
148:
77:
69:
65:
6462:
6407:
5425:
6555:
6522:
6517:
6512:
6314:
6204:
6199:
6097:
6046:
5836:
5583:
5083:
4831:
4691:
4595: = 1), and certain values of Ω, the map exhibits a phenomenon called
360:
132:
367:, with a driving force that has a periodic behaviour. As a practical example,
87:. Sometimes the frequency of oscillation depends on, or is constrained (i.e.,
6575:
6550:
6507:
6497:
6492:
6392:
6372:
6232:
6154:
6051:
5553:
5180:
5134:
112:
32:
5702:
5316:
5299:
5091:
351:
Arnold tongues appear most frequently when studying the interaction between
6502:
6467:
6377:
6334:
6189:
6184:
5783:
5732:
5375:
5276:
5228:
Eleven Papers on Number Theory, Algebra and
Functions of a Complex Variable
5065:
4819:
3504:
To see this, note that the recurrence relation in property 2 would become:
1956:{\displaystyle t_{n}=t_{n-1}+\Omega +{\frac {K}{2\pi }}\sin(2\pi t_{n-1}).}
379:
120:
116:
6149:
5484:
5346:
Glass, L. (2001). "Synchronization and rhythmic processes in physiology".
162:
6487:
6477:
6362:
6112:
5934:
5841:
5662:
5087:
4818:. It is sometimes said that the circle map maps the rationals, a set of
352:
53:
3916:{\displaystyle \theta _{n}'-\theta _{0}=\theta _{n}+np-\theta _{0}=M+np}
3187:. Because of this, for many purposes it does not matter if the iterates
6545:
6442:
5889:
5663:
5623:
5476:
5226:
140:
5724:
4777:. One reason the term "locking" is used is that the individual values
1029:
Another way to view the circle map is as follows. Consider a function
874:. This map displays very diverse behavior depending on the parameters
6457:
6417:
6159:
5821:
5806:
5588:
5367:
4680:{\displaystyle \omega =\lim _{n\to \infty }{\frac {\theta _{n}}{n}}.}
1025:
and is reset to the sinusoidal black line every time it reaches zero.
80:
that resemble tongues, in which case they are called Arnold tongues.
5331:
Nakao, M.; Enkhkhudulmur, T.E.; Katayama, N.; Karashima, A. (2014).
5231:. Vol. 46. American Mathematical Society Translations Series 2.
5201:
6194:
4830: ≠0. The largest tongues, ordered by size, occur at the
4564:
5259:
2691:
When expanding the recurrence relation, one obtains a formula for
5864:
5816:
5128:
at the top. The redder the color, the longer the recurrence time.
5061:
4862:
to chaos, that is, period doubling of the form 3, 6, 12, 24,....
4621:
The limiting behavior in the mode-locked regions is given by the
4532:
68:) are a pictorial phenomenon that occur when visualizing how the
5404:
Glass, L.; Perez, R. (1982). "Fine structure of phase locking".
6437:
5752:
96:
4046:
Considering the recurrence relation in property 2, a rational
2303:{\displaystyle g(\theta )=\theta +(K/2\pi )\sin(2\pi \theta )}
5695:
3168:
2883:
1998:
above representing the concentration of a certain substance.
103:
Other examples where Arnold tongues can be found include the
5438:
He studied it using cosine instead of sine; see page 78 of
5206:
Izvestiya
Rossiiskoi Akademii Nauk. Seriya Matematicheskaya
5214:
Section 12 in page 78 has a figure showing Arnold tongues.
4618:, although they may do so chaotically on the small scale.
5138:
Rotation number, with black corresponding to 0, green to
4798:
There is a mode-locked region for every rational number
4536:
Some of the Arnold tongues for the standard circle map,
972:
around this paragraph is obtained, where we can observe
5641:
The
Topology of Chaos: Alice in Stretch and Squeezeland
5297:
5244:
5202:"Small denominators. I. Mapping the circle onto itself"
4838:
and taking a cross-section through this image, so that
4131:{\displaystyle \theta _{n}=\theta _{0}+n{\frac {p}{q}}}
5657:
Provides a brief review of basic facts in section 2.12
5638:
5451:
4524:
is rational, which contradicts the initial hypothesis.
2128:. The rotation number, in turn, would be the quotient
486:
can be represented. With this in mind, the family of
4883:
4634:
4510:
4476:
4456:
4436:
4407:
4383:
4355:
4300:
4253:
4224:
4204:
4167:
4147:
4088:
4052:
4023:
3981:
3961:
3935:
3836:
3830:
due to the original phase-locking, now we would have
3790:
3512:
3456:
3436:
3390:
3349:
3329:
3292:
3266:
3246:
3220:
3193:
3133:
3077:
3049:
3017:
2955:
2928:
2908:
2858:
2727:
2697:
2665:
2595:
2572:
2545:
2525:
2499:
2479:
2348:
2316:
2237:
2173:
2164:
Consider the general family of circle endomorphisms:
2134:
2105:
2085:
2065:
2033:
2007:
1975:
1862:
1819:
1785:
1373:
1350:
1323:
1303:
1215:
1182:
1146:
1084:
1064:
1035:
1002:
954:
920:
900:
880:
860:
833:
809:
708:
678:
658:
638:
603:
583:
563:
499:
457:
434:
411:
388:
321:
286:
263:
243:
223:
195:
175:
5339:
3494:{\displaystyle \Omega '=\Omega +p,p\in \mathbb {N} }
2221:{\displaystyle \theta _{i+1}=g(\theta _{i})+\Omega }
547:{\displaystyle \theta _{i+1}=g(\theta _{i})+\Omega }
5578:
5403:
5079:Arnold tongues have been applied to the study of
5042:
4679:
4516:
4496:
4462:
4442:
4422:
4389:
4367:
4338:
4285:
4236:
4210:
4190:
4153:
4130:
4072:
4035:
4009:
3967:
3947:
3915:
3822:
3774:
3493:
3442:
3422:
3376:
3335:
3315:
3278:
3252:
3226:
3206:
3180:{\displaystyle f(\theta +p)=f(\theta ){\bmod {1}}}
3179:
3119:
3063:
3029:
3000:{\displaystyle M=(\theta _{n}-\theta _{0})\cdot 1}
2999:
2941:
2914:
2895:{\displaystyle \theta _{n}=\theta _{0}{\bmod {1}}}
2894:
2838:
2710:
2677:
2648:
2578:
2558:
2531:
2511:
2485:
2457:
2331:
2302:
2220:
2148:
2120:
2091:
2071:
2048:
2019:
1990:
1955:
1845:
1805:
1768:
1356:
1336:
1309:
1289:
1201:
1165:
1132:
1070:
1050:
1017:
960:
940:
906:
886:
866:
846:
815:
792:
684:
664:
644:
624:
589:
569:
546:
478:
443:
420:
394:
339:
307:
272:
249:
229:
209:
181:
5698:cardiac rhythms in the context of the circle map.
5439:
5199:
4826: = 0, to a set of non-zero measure for
2649:{\displaystyle f'(\theta )=1+K\cos(2\pi \theta )}
2231:where, for the standard circle map, we have that
1140:. We are now interested in the sequence of times
280:at top, and the orbits are shown in the interval
6573:
4642:
1290:{\displaystyle y(t_{n-1})=c+b\sin(2\pi t_{n-1})}
355:, particularly in the case where one oscillator
4690:which is also sometimes referred to as the map
1853:we obtain the circle map discussed previously:
652:the particle simply walks around the circle at
3423:{\displaystyle \theta _{0},\dots ,\theta _{n}}
2902:, so they are periodic fixed points of period
5768:
5597:
5399:
5397:
5395:
5393:
5154:and red to 1. Ω varies from 0 to 1 along the
347:. Black regions correspond to Arnold tongues.
1160:
1147:
577:is the oscillator's "natural" frequency and
402:in the real line that should be interpreted
378:The family of circle maps are functions (or
363:, for example. This is a particular case of
5304:Mathematical Modelling of Natural Phenomena
4339:{\displaystyle (\theta _{q}-\theta _{0})=p}
4010:{\displaystyle \Omega =p/q\in \mathbb {Q} }
3450:, then it is also a periodic orbit for any
5775:
5761:
5390:
5223:Translation to english of Arnold's paper:
987:
5466:
5345:
5315:
5258:
4286:{\displaystyle \theta _{q}=\theta _{0}+p}
4003:
3823:{\displaystyle \theta _{n}-\theta _{0}=M}
3487:
3120:{\displaystyle f(\theta +p)=f(\theta )+p}
3057:
1966:
5240:
5238:
5133:
5111:
4865:
4603:. In a phase-locked region, the values
4563:
4531:
991:
161:
151:multiple of that of the driven rotator.
72:of a dynamical system, or other related
31:
3384:phase-locking. This also means that if
3286:phase-locking in the system. Then, for
14:
6574:
5604:Communications in Mathematical Physics
157:
48:-axis. Some tongue shapes are visible.
5756:
5579:
5505:
5235:
5193:
3955:there will be phase-locking whenever
4583:For small to intermediate values of
1133:{\displaystyle z(t)=c+b\sin(2\pi t)}
692:is irrational the map reduces to an
5915:Measure-preserving dynamical system
5797:
1058:that decreases linearly with slope
672:units at a time; in particular, if
24:
5298:GĂ©rard, C.; Goldbeter, A. (2012).
4652:
4511:
4477:
4411:
4384:
4053:
3982:
3962:
3657:
3554:
3468:
3458:
3437:
3430:is a periodic orbit for parameter
3316:{\displaystyle \Omega '=\Omega +p}
3304:
3294:
3247:
2757:
2403:
2215:
2001:In this model, a phase-locking of
1895:
1786:
1317:will then decrease linearly until
921:
901:
741:
679:
659:
625:{\displaystyle g(\theta )=\theta }
564:
541:
176:
25:
6593:
6483:Oleksandr Mykolayovych Sharkovsky
5740:
5639:Gilmore, R.; Lefranc, M. (2002).
5534:IEEE Photonics Technology Letters
5300:"The cell cycle is a limit cycle"
4401:), it would be necessary to have
3064:{\displaystyle p\in \mathbb {N} }
1176:This model tells us that at time
6013:
6005:
5782:
5694:Performs a detailed analysis of
5162:varies from 0 at the bottom to 2
5124:varies from 0 at the bottom to 4
2493:is monotonically increasing for
5525:
5499:
5455:Journal of Mathematical Biology
5074:
4721: → 0. The values of (
4527:
6248:Rabinovich–Fabrikant equations
5668:Physica D: Nonlinear Phenomena
5445:
5432:
5324:
5291:
5217:
5096:Synchronisation of a resonant
4977:
4958:
4749:. For example, all values of (
4649:
4327:
4301:
4185:
4171:
3733:
3714:
3666:
3654:
3625:
3606:
3371:
3356:
3164:
3158:
3149:
3137:
3108:
3102:
3093:
3081:
2988:
2962:
2830:
2811:
2643:
2631:
2610:
2604:
2449:
2430:
2384:
2371:
2326:
2320:
2297:
2285:
2276:
2259:
2247:
2241:
2209:
2196:
2115:
2109:
2043:
2037:
1985:
1979:
1947:
1922:
1759:
1734:
1638:
1613:
1516:
1491:
1454:
1422:
1410:
1391:
1284:
1259:
1238:
1219:
1127:
1115:
1094:
1088:
1045:
1039:
1012:
1006:
787:
768:
613:
607:
535:
522:
473:
458:
334:
322:
302:
287:
13:
1:
5572:
5269:10.1016/j.febslet.2012.04.044
4858:The circle map also exhibits
4198:is an integer, and the first
3975:is a rational. Moreover, let
2659:which is positive as long as
2159:
854:should be interpreted modulo
5749:with interactive Java applet
5688:10.1016/0167-2789(83)90119-7
5092:McGuinness, M. et al. (2004)
4017:, then the phase-locking is
3238:P5 (translational symmetry).
1173:at which y(t) reaches zero.
978:period-doubling bifurcations
7:
5983:Poincaré recurrence theorem
5426:10.1103/PhysRevLett.48.1772
5174:
5098:tunneling diode oscillators
4497:{\displaystyle \Omega =k/n}
4073:{\displaystyle \Omega =p/q}
3207:{\displaystyle \theta _{i}}
3127:, which in turn means that
2942:{\displaystyle \theta _{i}}
2711:{\displaystyle \theta _{n}}
2559:{\displaystyle \theta _{i}}
1806:{\displaystyle \Omega =c/a}
941:{\displaystyle \Omega =1/3}
847:{\displaystyle \theta _{i}}
10:
6598:
5978:Poincaré–Bendixson theorem
5103:
4587:(that is, in the range of
4423:{\displaystyle n\Omega =k}
2332:{\displaystyle f(\theta )}
2099:periods of the sinusoidal
1846:{\displaystyle K=2\pi b/a}
6531:
6348:
6330:Swinging Atwood's machine
6275:
6213:
6083:
6070:
6022:
6003:
5973:Krylov–Bogolyubov theorem
5953:
5850:
5790:
5643:. John Wiley & Sons.
4610:advance essentially as a
3240:Suppose that for a given
2519:, so for these values of
1166:{\displaystyle \{t_{n}\}}
6238:Lotka–Volterra equations
6062:Synchronization of chaos
5865:axiom A dynamical system
5554:10.1109/LPT.2009.2034129
5186:
4846:. One can show that for
4591: = 0 to about
4571:as a function of Ω with
3377:{\displaystyle n:(M+np)}
3007:, thus characterizing a
1364:is zero, thus yielding:
107:of musical instruments,
6223:Double scroll attractor
5988:Stable manifold theorem
5895:False nearest neighbors
5406:Physical Review Letters
5088:Glass, L. et al. (1983)
4517:{\displaystyle \Omega }
4390:{\displaystyle \Omega }
4218:that satisfies this is
3968:{\displaystyle \Omega }
3443:{\displaystyle \Omega }
3253:{\displaystyle \Omega }
1202:{\displaystyle t_{n-1}}
988:Deriving the circle map
907:{\displaystyle \Omega }
685:{\displaystyle \Omega }
665:{\displaystyle \Omega }
645:{\displaystyle \theta }
570:{\displaystyle \Omega }
182:{\displaystyle \Omega }
6263:Van der Pol oscillator
6243:Mackey–Glass equations
5875:Box-counting dimension
5598:Boyland, P.L. (1986).
5200:Arnol'd, V.I. (1961).
5167:
5129:
5044:
4681:
4580:
4561:
4518:
4498:
4464:
4444:
4424:
4391:
4369:
4340:
4287:
4238:
4212:
4192:
4191:{\displaystyle n(p/q)}
4155:
4132:
4074:
4037:
4011:
3969:
3949:
3917:
3824:
3776:
3707:
3599:
3495:
3444:
3424:
3378:
3337:
3317:
3280:
3254:
3228:
3208:
3181:
3121:
3065:
3031:
3001:
2943:
2916:
2896:
2840:
2804:
2712:
2679:
2678:{\displaystyle K<1}
2650:
2580:
2560:
2533:
2513:
2512:{\displaystyle K<1}
2487:
2459:
2333:
2304:
2222:
2150:
2122:
2093:
2073:
2050:
2021:
1992:
1957:
1847:
1807:
1770:
1358:
1338:
1311:
1291:
1203:
1167:
1134:
1072:
1052:
1026:
1019:
962:
942:
908:
888:
868:
848:
817:
794:
686:
666:
646:
626:
591:
571:
548:
480:
445:
422:
396:
348:
341:
309:
274:
251:
231:
211:
183:
129:electronic oscillators
85:cardiac electric waves
49:
6413:Svetlana Jitomirskaya
6320:Multiscroll attractor
6165:Interval exchange map
6118:Dyadic transformation
6103:Complex quadratic map
5945:Topological conjugacy
5880:Correlation dimension
5855:Anosov diffeomorphism
5317:10.1051/mmnp/20127607
5137:
5115:
5045:
4872:Chirikov standard map
4866:Chirikov standard map
4682:
4567:
4535:
4519:
4499:
4465:
4445:
4425:
4392:
4370:
4341:
4288:
4239:
4213:
4193:
4156:
4141:and equality modulus
4133:
4075:
4038:
4012:
3970:
3950:
3918:
3825:
3777:
3687:
3579:
3496:
3445:
3425:
3379:
3338:
3318:
3281:
3255:
3229:
3209:
3182:
3122:
3066:
3032:
3002:
2944:
2917:
2897:
2841:
2778:
2713:
2680:
2651:
2581:
2561:
2534:
2514:
2488:
2460:
2334:
2305:
2223:
2151:
2123:
2094:
2074:
2051:
2022:
1993:
1958:
1848:
1808:
1771:
1359:
1344:, where the function
1339:
1337:{\displaystyle t_{n}}
1312:
1292:
1204:
1168:
1135:
1073:
1053:
1020:
995:
963:
943:
909:
889:
869:
849:
818:
795:
687:
667:
647:
627:
592:
572:
549:
481:
446:
444:{\displaystyle 2\pi }
423:
421:{\displaystyle 2\pi }
397:
373:artificial pacemakers
342:
310:
275:
273:{\displaystyle 4\pi }
252:
232:
212:
184:
165:
35:
6423:Edward Norton Lorenz
5508:"Map Winding Number"
5440:Arnol'd, V.I. (1961)
4881:
4632:
4508:
4474:
4454:
4434:
4405:
4381:
4353:
4298:
4251:
4222:
4202:
4165:
4161:will hold only when
4145:
4086:
4050:
4021:
3979:
3959:
3933:
3834:
3788:
3510:
3454:
3434:
3388:
3347:
3327:
3290:
3264:
3244:
3218:
3191:
3131:
3075:
3047:
3015:
2953:
2926:
2906:
2856:
2725:
2695:
2663:
2593:
2570:
2543:
2523:
2497:
2477:
2346:
2314:
2235:
2171:
2132:
2121:{\displaystyle z(t)}
2103:
2083:
2063:
2049:{\displaystyle y(t)}
2031:
2005:
1991:{\displaystyle y(t)}
1973:
1860:
1817:
1783:
1371:
1348:
1321:
1301:
1213:
1180:
1144:
1082:
1062:
1051:{\displaystyle y(t)}
1033:
1018:{\displaystyle y(t)}
1000:
980:as well as possible
952:
918:
898:
878:
858:
831:
807:
706:
676:
656:
636:
601:
581:
561:
497:
455:
432:
409:
386:
319:
284:
261:
241:
221:
193:
173:
6383:Mitchell Feigenbaum
6325:Population dynamics
6310:Hénon–Heiles system
6170:Irrational rotation
6123:Dynamical billiards
6108:Coupled map lattice
5968:Liouville's theorem
5900:Hausdorff dimension
5885:Conservative system
5870:Bifurcation diagram
5717:2004Chaos..14....1M
5680:1983PhyD....7...89G
5616:1986CMaPh.106..353B
5546:2009IPTL...21.1819R
5418:1982PhRvL..48.1772G
5360:2001Natur.410..277G
4399:irrational rotation
4397:(which leads to an
4368:{\displaystyle q:p}
4237:{\displaystyle n=q}
4036:{\displaystyle q:p}
3948:{\displaystyle K=0}
3849:
3529:
3343:, there would be a
3279:{\displaystyle n:M}
3030:{\displaystyle n:M}
2149:{\displaystyle N/M}
2020:{\displaystyle N:M}
1297:. From this point,
970:bifurcation diagram
694:irrational rotation
210:{\displaystyle 1/3}
167:Bifurcation diagram
158:Standard circle map
115:of orbiting moons,
27:Phenomenon in maths
6561:Santa Fe Institute
6428:Aleksandr Lyapunov
6258:Three-body problem
6145:Gingerbreadman map
6032:Bifurcation theory
5910:Lyapunov stability
5624:10.1007/BF01207252
5581:Weisstein, Eric W.
5477:10.1007/BF02154750
5168:
5130:
5040:
5038:
4860:subharmonic routes
4677:
4656:
4581:
4562:
4514:
4494:
4460:
4440:
4420:
4387:
4365:
4336:
4283:
4234:
4208:
4188:
4151:
4128:
4070:
4033:
4007:
3965:
3945:
3913:
3837:
3820:
3772:
3770:
3517:
3491:
3440:
3420:
3374:
3333:
3313:
3276:
3250:
3224:
3214:are taken modulus
3204:
3177:
3117:
3071:, it is true that
3061:
3027:
2997:
2939:
2912:
2892:
2836:
2708:
2675:
2646:
2576:
2556:
2529:
2509:
2483:
2455:
2329:
2300:
2218:
2146:
2118:
2089:
2069:
2046:
2017:
1988:
1953:
1843:
1803:
1766:
1764:
1354:
1334:
1307:
1287:
1199:
1163:
1130:
1068:
1048:
1027:
1015:
958:
938:
904:
884:
864:
844:
813:
790:
682:
662:
642:
622:
587:
567:
544:
476:
441:
418:
392:
365:driven oscillators
349:
337:
305:
270:
247:
227:
207:
179:
125:phase-locked loops
74:invariant property
56:, particularly in
50:
6569:
6568:
6433:Benoît Mandelbrot
6398:Martin Gutzwiller
6388:Peter Grassberger
6271:
6270:
6253:Rössler attractor
6001:
6000:
5905:Invariant measure
5827:Lyapunov exponent
5725:10.1063/1.1620990
5540:(24): 1819–1821.
5506:Weisstein, Eric.
5354:(6825): 277–284.
5253:(11): 1664–1668.
5172:
5171:
4950:
4672:
4641:
4612:rational multiple
4575:held constant at
4463:{\displaystyle k}
4443:{\displaystyle n}
4211:{\displaystyle n}
4154:{\displaystyle 1}
4126:
3685:
3577:
3336:{\displaystyle p}
3227:{\displaystyle 1}
2915:{\displaystyle n}
2776:
2579:{\displaystyle f}
2532:{\displaystyle K}
2486:{\displaystyle f}
2422:
2092:{\displaystyle M}
2072:{\displaystyle N}
1914:
1726:
1713:
1591:
1357:{\displaystyle y}
1310:{\displaystyle y}
1209:it is valid that
1071:{\displaystyle a}
961:{\displaystyle K}
887:{\displaystyle K}
867:{\displaystyle 1}
825:coupling strength
816:{\displaystyle K}
760:
590:{\displaystyle g}
395:{\displaystyle x}
250:{\displaystyle 0}
230:{\displaystyle K}
137:heart arrhythmias
109:orbital resonance
58:dynamical systems
16:(Redirected from
6589:
6541:Butterfly effect
6453:Itamar Procaccia
6403:Brosl Hasslacher
6300:Elastic pendulum
6228:Duffing equation
6175:Kaplan–Yorke map
6093:Arnold's cat map
6081:
6080:
6057:Stability theory
6042:Dynamical system
6037:Control of chaos
6017:
6009:
5993:Takens's theorem
5925:Poincaré section
5795:
5794:
5777:
5770:
5763:
5754:
5753:
5736:
5691:
5654:
5635:
5594:
5593:
5566:
5565:
5529:
5523:
5522:
5520:
5518:
5503:
5497:
5496:
5470:
5449:
5443:
5436:
5430:
5429:
5401:
5388:
5387:
5368:10.1038/35065745
5343:
5337:
5336:
5328:
5322:
5321:
5319:
5295:
5289:
5288:
5262:
5242:
5233:
5232:
5221:
5215:
5213:
5197:
5165:
5153:
5151:
5150:
5147:
5144:
5127:
5108:
5107:
5064:by means of the
5049:
5047:
5046:
5041:
5039:
5035:
5034:
5022:
5021:
4999:
4998:
4976:
4975:
4951:
4949:
4938:
4933:
4932:
4920:
4919:
4903:
4902:
4817:
4815:
4814:
4809:
4806:
4776:
4774:
4773:
4768:
4765:
4748:
4746:
4745:
4740:
4737:
4717:in the limit of
4716:
4714:
4713:
4708:
4705:
4686:
4684:
4683:
4678:
4673:
4668:
4667:
4658:
4655:
4560:
4558:
4557:
4556:
4551:
4548:
4523:
4521:
4520:
4515:
4503:
4501:
4500:
4495:
4490:
4469:
4467:
4466:
4461:
4449:
4447:
4446:
4441:
4429:
4427:
4426:
4421:
4396:
4394:
4393:
4388:
4374:
4372:
4371:
4366:
4345:
4343:
4342:
4337:
4326:
4325:
4313:
4312:
4292:
4290:
4289:
4284:
4276:
4275:
4263:
4262:
4244:. Consequently:
4243:
4241:
4240:
4235:
4217:
4215:
4214:
4209:
4197:
4195:
4194:
4189:
4181:
4160:
4158:
4157:
4152:
4137:
4135:
4134:
4129:
4127:
4119:
4111:
4110:
4098:
4097:
4079:
4077:
4076:
4071:
4066:
4042:
4040:
4039:
4034:
4016:
4014:
4013:
4008:
4006:
3995:
3974:
3972:
3971:
3966:
3954:
3952:
3951:
3946:
3922:
3920:
3919:
3914:
3897:
3896:
3875:
3874:
3862:
3861:
3845:
3829:
3827:
3826:
3821:
3813:
3812:
3800:
3799:
3781:
3779:
3778:
3773:
3771:
3755:
3754:
3739:
3732:
3731:
3706:
3701:
3686:
3684:
3673:
3647:
3646:
3631:
3624:
3623:
3598:
3593:
3578:
3576:
3565:
3560:
3546:
3545:
3525:
3500:
3498:
3497:
3492:
3490:
3464:
3449:
3447:
3446:
3441:
3429:
3427:
3426:
3421:
3419:
3418:
3400:
3399:
3383:
3381:
3380:
3375:
3342:
3340:
3339:
3334:
3322:
3320:
3319:
3314:
3300:
3285:
3283:
3282:
3277:
3259:
3257:
3256:
3251:
3233:
3231:
3230:
3225:
3213:
3211:
3210:
3205:
3203:
3202:
3186:
3184:
3183:
3178:
3176:
3175:
3126:
3124:
3123:
3118:
3070:
3068:
3067:
3062:
3060:
3036:
3034:
3033:
3028:
3006:
3004:
3003:
2998:
2987:
2986:
2974:
2973:
2948:
2946:
2945:
2940:
2938:
2937:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2891:
2890:
2881:
2880:
2868:
2867:
2845:
2843:
2842:
2837:
2829:
2828:
2803:
2792:
2777:
2775:
2764:
2750:
2749:
2737:
2736:
2717:
2715:
2714:
2709:
2707:
2706:
2684:
2682:
2681:
2676:
2655:
2653:
2652:
2647:
2603:
2585:
2583:
2582:
2577:
2565:
2563:
2562:
2557:
2555:
2554:
2538:
2536:
2535:
2530:
2518:
2516:
2515:
2510:
2492:
2490:
2489:
2484:
2464:
2462:
2461:
2456:
2448:
2447:
2423:
2421:
2410:
2399:
2398:
2383:
2382:
2364:
2363:
2338:
2336:
2335:
2330:
2309:
2307:
2306:
2301:
2269:
2227:
2225:
2224:
2219:
2208:
2207:
2189:
2188:
2155:
2153:
2152:
2147:
2142:
2127:
2125:
2124:
2119:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2055:
2053:
2052:
2047:
2027:would mean that
2026:
2024:
2023:
2018:
1997:
1995:
1994:
1989:
1967:Glass, L. (2001)
1962:
1960:
1959:
1954:
1946:
1945:
1915:
1913:
1902:
1891:
1890:
1872:
1871:
1852:
1850:
1849:
1844:
1839:
1812:
1810:
1809:
1804:
1799:
1779:and by choosing
1775:
1773:
1772:
1767:
1765:
1758:
1757:
1727:
1719:
1714:
1706:
1701:
1700:
1678:
1677:
1664:
1663:
1645:
1641:
1637:
1636:
1592:
1584:
1575:
1574:
1561:
1560:
1539:
1538:
1523:
1519:
1515:
1514:
1453:
1452:
1434:
1433:
1409:
1408:
1363:
1361:
1360:
1355:
1343:
1341:
1340:
1335:
1333:
1332:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1283:
1282:
1237:
1236:
1208:
1206:
1205:
1200:
1198:
1197:
1172:
1170:
1169:
1164:
1159:
1158:
1139:
1137:
1136:
1131:
1077:
1075:
1074:
1069:
1057:
1055:
1054:
1049:
1024:
1022:
1021:
1016:
982:chaotic behavior
967:
965:
964:
959:
947:
945:
944:
939:
934:
913:
911:
910:
905:
893:
891:
890:
885:
873:
871:
870:
865:
853:
851:
850:
845:
843:
842:
822:
820:
819:
814:
799:
797:
796:
791:
786:
785:
761:
759:
748:
737:
736:
724:
723:
691:
689:
688:
683:
671:
669:
668:
663:
651:
649:
648:
643:
631:
629:
628:
623:
596:
594:
593:
588:
576:
574:
573:
568:
553:
551:
550:
545:
534:
533:
515:
514:
485:
483:
482:
479:{\displaystyle }
477:
450:
448:
447:
442:
427:
425:
424:
419:
401:
399:
398:
393:
346:
344:
343:
340:{\displaystyle }
338:
314:
312:
311:
308:{\displaystyle }
306:
279:
277:
276:
271:
256:
254:
253:
248:
236:
234:
233:
228:
216:
214:
213:
208:
203:
188:
186:
185:
180:
131:, as well as in
78:geometric shapes
21:
6597:
6596:
6592:
6591:
6590:
6588:
6587:
6586:
6572:
6571:
6570:
6565:
6533:
6527:
6473:Caroline Series
6368:Mary Cartwright
6350:
6344:
6295:Double pendulum
6277:
6267:
6216:
6209:
6135:Exponential map
6086:
6072:
6066:
6024:
6018:
6011:
5997:
5963:Ergodic theorem
5956:
5949:
5940:Stable manifold
5930:Recurrence plot
5846:
5800:
5786:
5781:
5743:
5674:(1–3): 89–101.
5651:
5575:
5570:
5569:
5530:
5526:
5516:
5514:
5504:
5500:
5468:10.1.1.476.8649
5450:
5446:
5437:
5433:
5402:
5391:
5344:
5340:
5329:
5325:
5296:
5292:
5243:
5236:
5222:
5218:
5198:
5194:
5189:
5177:
5163:
5148:
5145:
5142:
5141:
5139:
5125:
5106:
5084:Cardiac rhythms
5077:
5058:
5037:
5036:
5030:
5026:
5011:
5007:
5000:
4988:
4984:
4981:
4980:
4971:
4967:
4942:
4937:
4928:
4924:
4915:
4911:
4904:
4892:
4888:
4884:
4882:
4879:
4878:
4868:
4844:Cantor function
4832:Farey fractions
4810:
4807:
4802:
4801:
4799:
4793:
4782:
4769:
4766:
4761:
4760:
4758:
4741:
4738:
4733:
4732:
4730:
4709:
4706:
4701:
4700:
4698:
4663:
4659:
4657:
4645:
4633:
4630:
4629:
4623:rotation number
4608:
4569:Rotation number
4554:
4552:
4549:
4544:
4543:
4541:
4530:
4525:
4509:
4506:
4505:
4486:
4475:
4472:
4471:
4455:
4452:
4451:
4435:
4432:
4431:
4406:
4403:
4402:
4382:
4379:
4378:
4377:For irrational
4375:phase-locking.
4354:
4351:
4350:
4321:
4317:
4308:
4304:
4299:
4296:
4295:
4271:
4267:
4258:
4254:
4252:
4249:
4248:
4223:
4220:
4219:
4203:
4200:
4199:
4177:
4166:
4163:
4162:
4146:
4143:
4142:
4118:
4106:
4102:
4093:
4089:
4087:
4084:
4083:
4062:
4051:
4048:
4047:
4022:
4019:
4018:
4002:
3991:
3980:
3977:
3976:
3960:
3957:
3956:
3934:
3931:
3930:
3924:
3892:
3888:
3870:
3866:
3857:
3853:
3841:
3835:
3832:
3831:
3808:
3804:
3795:
3791:
3789:
3786:
3785:
3769:
3768:
3750:
3746:
3737:
3736:
3727:
3723:
3702:
3691:
3677:
3672:
3642:
3638:
3629:
3628:
3619:
3615:
3594:
3583:
3569:
3564:
3553:
3541:
3537:
3530:
3521:
3513:
3511:
3508:
3507:
3486:
3457:
3455:
3452:
3451:
3435:
3432:
3431:
3414:
3410:
3395:
3391:
3389:
3386:
3385:
3348:
3345:
3344:
3328:
3325:
3324:
3293:
3291:
3288:
3287:
3265:
3262:
3261:
3245:
3242:
3241:
3219:
3216:
3215:
3198:
3194:
3192:
3189:
3188:
3171:
3167:
3132:
3129:
3128:
3076:
3073:
3072:
3056:
3048:
3045:
3044:
3016:
3013:
3012:
2982:
2978:
2969:
2965:
2954:
2951:
2950:
2933:
2929:
2927:
2924:
2923:
2907:
2904:
2903:
2886:
2882:
2876:
2872:
2863:
2859:
2857:
2854:
2853:
2824:
2820:
2793:
2782:
2768:
2763:
2745:
2741:
2732:
2728:
2726:
2723:
2722:
2702:
2698:
2696:
2693:
2692:
2664:
2661:
2660:
2596:
2594:
2591:
2590:
2571:
2568:
2567:
2550:
2546:
2544:
2541:
2540:
2524:
2521:
2520:
2498:
2495:
2494:
2478:
2475:
2474:
2443:
2439:
2414:
2409:
2394:
2390:
2378:
2374:
2353:
2349:
2347:
2344:
2343:
2315:
2312:
2311:
2265:
2236:
2233:
2232:
2203:
2199:
2178:
2174:
2172:
2169:
2168:
2162:
2138:
2133:
2130:
2129:
2104:
2101:
2100:
2084:
2081:
2080:
2064:
2061:
2060:
2032:
2029:
2028:
2006:
2003:
2002:
1974:
1971:
1970:
1935:
1931:
1906:
1901:
1880:
1876:
1867:
1863:
1861:
1858:
1857:
1835:
1818:
1815:
1814:
1795:
1784:
1781:
1780:
1763:
1762:
1747:
1743:
1718:
1705:
1690:
1686:
1679:
1673:
1669:
1666:
1665:
1653:
1649:
1626:
1622:
1597:
1593:
1583:
1576:
1570:
1566:
1563:
1562:
1550:
1546:
1534:
1530:
1504:
1500:
1475:
1471:
1464:
1458:
1457:
1442:
1438:
1429:
1425:
1398:
1394:
1381:
1374:
1372:
1369:
1368:
1349:
1346:
1345:
1328:
1324:
1322:
1319:
1318:
1302:
1299:
1298:
1272:
1268:
1226:
1222:
1214:
1211:
1210:
1187:
1183:
1181:
1178:
1177:
1154:
1150:
1145:
1142:
1141:
1083:
1080:
1079:
1063:
1060:
1059:
1034:
1031:
1030:
1001:
998:
997:
990:
974:periodic orbits
953:
950:
949:
930:
919:
916:
915:
899:
896:
895:
879:
876:
875:
859:
856:
855:
838:
834:
832:
829:
828:
808:
805:
804:
781:
777:
752:
747:
732:
728:
713:
709:
707:
704:
703:
677:
674:
673:
657:
654:
653:
637:
634:
633:
602:
599:
598:
582:
579:
578:
562:
559:
558:
529:
525:
504:
500:
498:
495:
494:
456:
453:
452:
433:
430:
429:
410:
407:
406:
387:
384:
383:
361:Kuramoto models
320:
317:
316:
285:
282:
281:
262:
259:
258:
242:
239:
238:
222:
219:
218:
199:
194:
191:
190:
174:
171:
170:
160:
133:cardiac rhythms
70:rotation number
66:Vladimir Arnold
28:
23:
22:
15:
12:
11:
5:
6595:
6585:
6584:
6567:
6566:
6564:
6563:
6558:
6556:Predictability
6553:
6548:
6543:
6537:
6535:
6529:
6528:
6526:
6525:
6523:Lai-Sang Young
6520:
6518:James A. Yorke
6515:
6513:Amie Wilkinson
6510:
6505:
6500:
6495:
6490:
6485:
6480:
6475:
6470:
6465:
6460:
6455:
6450:
6448:Henri Poincaré
6445:
6440:
6435:
6430:
6425:
6420:
6415:
6410:
6405:
6400:
6395:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6354:
6352:
6346:
6345:
6343:
6342:
6337:
6332:
6327:
6322:
6317:
6315:Kicked rotator
6312:
6307:
6302:
6297:
6292:
6287:
6285:Chua's circuit
6281:
6279:
6273:
6272:
6269:
6268:
6266:
6265:
6260:
6255:
6250:
6245:
6240:
6235:
6230:
6225:
6219:
6217:
6214:
6211:
6210:
6208:
6207:
6205:Zaslavskii map
6202:
6200:Tinkerbell map
6197:
6192:
6187:
6182:
6177:
6172:
6167:
6162:
6157:
6152:
6147:
6142:
6137:
6132:
6131:
6130:
6120:
6115:
6110:
6105:
6100:
6095:
6089:
6087:
6084:
6078:
6068:
6067:
6065:
6064:
6059:
6054:
6049:
6047:Ergodic theory
6044:
6039:
6034:
6028:
6026:
6020:
6019:
6004:
6002:
5999:
5998:
5996:
5995:
5990:
5985:
5980:
5975:
5970:
5965:
5959:
5957:
5954:
5951:
5950:
5948:
5947:
5942:
5937:
5932:
5927:
5922:
5917:
5912:
5907:
5902:
5897:
5892:
5887:
5882:
5877:
5872:
5867:
5862:
5857:
5851:
5848:
5847:
5845:
5844:
5839:
5837:Periodic point
5834:
5829:
5824:
5819:
5814:
5809:
5803:
5801:
5798:
5792:
5788:
5787:
5780:
5779:
5772:
5765:
5757:
5751:
5750:
5742:
5741:External links
5739:
5738:
5737:
5700:
5660:
5650:0-471-40816--6
5649:
5636:
5610:(3): 353–381.
5595:
5574:
5571:
5568:
5567:
5524:
5498:
5444:
5431:
5389:
5338:
5323:
5310:(6): 126–166.
5290:
5234:
5216:
5191:
5190:
5188:
5185:
5184:
5183:
5176:
5173:
5170:
5169:
5131:
5105:
5102:
5101:
5100:
5094:
5076:
5073:
5056:
5051:
5050:
5033:
5029:
5025:
5020:
5017:
5014:
5010:
5006:
5003:
5001:
4997:
4994:
4991:
4987:
4983:
4982:
4979:
4974:
4970:
4966:
4963:
4960:
4957:
4954:
4948:
4945:
4941:
4936:
4931:
4927:
4923:
4918:
4914:
4910:
4907:
4905:
4901:
4898:
4895:
4891:
4887:
4886:
4867:
4864:
4855:is not known.
4791:
4780:
4692:winding number
4688:
4687:
4676:
4671:
4666:
4662:
4654:
4651:
4648:
4644:
4640:
4637:
4606:
4579: = 1
4529:
4526:
4513:
4493:
4489:
4485:
4482:
4479:
4459:
4439:
4419:
4416:
4413:
4410:
4386:
4364:
4361:
4358:
4347:
4346:
4335:
4332:
4329:
4324:
4320:
4316:
4311:
4307:
4303:
4293:
4282:
4279:
4274:
4270:
4266:
4261:
4257:
4233:
4230:
4227:
4207:
4187:
4184:
4180:
4176:
4173:
4170:
4150:
4139:
4138:
4125:
4122:
4117:
4114:
4109:
4105:
4101:
4096:
4092:
4069:
4065:
4061:
4058:
4055:
4045:
4032:
4029:
4026:
4005:
4001:
3998:
3994:
3990:
3987:
3984:
3964:
3944:
3941:
3938:
3912:
3909:
3906:
3903:
3900:
3895:
3891:
3887:
3884:
3881:
3878:
3873:
3869:
3865:
3860:
3856:
3852:
3848:
3844:
3840:
3819:
3816:
3811:
3807:
3803:
3798:
3794:
3783:
3782:
3767:
3764:
3761:
3758:
3753:
3749:
3745:
3742:
3740:
3738:
3735:
3730:
3726:
3722:
3719:
3716:
3713:
3710:
3705:
3700:
3697:
3694:
3690:
3683:
3680:
3676:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3645:
3641:
3637:
3634:
3632:
3630:
3627:
3622:
3618:
3614:
3611:
3608:
3605:
3602:
3597:
3592:
3589:
3586:
3582:
3575:
3572:
3568:
3563:
3559:
3556:
3552:
3549:
3544:
3540:
3536:
3533:
3531:
3528:
3524:
3520:
3516:
3515:
3503:
3489:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3463:
3460:
3439:
3417:
3413:
3409:
3406:
3403:
3398:
3394:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3332:
3312:
3309:
3306:
3303:
3299:
3296:
3275:
3272:
3269:
3249:
3223:
3201:
3197:
3174:
3170:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3059:
3055:
3052:
3026:
3023:
3020:
2996:
2993:
2990:
2985:
2981:
2977:
2972:
2968:
2964:
2961:
2958:
2936:
2932:
2911:
2889:
2885:
2879:
2875:
2871:
2866:
2862:
2847:
2846:
2835:
2832:
2827:
2823:
2819:
2816:
2813:
2810:
2807:
2802:
2799:
2796:
2791:
2788:
2785:
2781:
2774:
2771:
2767:
2762:
2759:
2756:
2753:
2748:
2744:
2740:
2735:
2731:
2705:
2701:
2674:
2671:
2668:
2657:
2656:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2602:
2599:
2575:
2553:
2549:
2528:
2508:
2505:
2502:
2482:
2466:
2465:
2454:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2426:
2420:
2417:
2413:
2408:
2405:
2402:
2397:
2393:
2389:
2386:
2381:
2377:
2373:
2370:
2367:
2362:
2359:
2356:
2352:
2328:
2325:
2322:
2319:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2268:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2229:
2228:
2217:
2214:
2211:
2206:
2202:
2198:
2195:
2192:
2187:
2184:
2181:
2177:
2161:
2158:
2145:
2141:
2137:
2117:
2114:
2111:
2108:
2088:
2068:
2045:
2042:
2039:
2036:
2016:
2013:
2010:
1987:
1984:
1981:
1978:
1964:
1963:
1952:
1949:
1944:
1941:
1938:
1934:
1930:
1927:
1924:
1921:
1918:
1912:
1909:
1905:
1900:
1897:
1894:
1889:
1886:
1883:
1879:
1875:
1870:
1866:
1842:
1838:
1834:
1831:
1828:
1825:
1822:
1802:
1798:
1794:
1791:
1788:
1777:
1776:
1761:
1756:
1753:
1750:
1746:
1742:
1739:
1736:
1733:
1730:
1725:
1722:
1717:
1712:
1709:
1704:
1699:
1696:
1693:
1689:
1685:
1682:
1680:
1676:
1672:
1668:
1667:
1662:
1659:
1656:
1652:
1648:
1644:
1640:
1635:
1632:
1629:
1625:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1596:
1590:
1587:
1582:
1579:
1577:
1573:
1569:
1565:
1564:
1559:
1556:
1553:
1549:
1545:
1542:
1537:
1533:
1529:
1526:
1522:
1518:
1513:
1510:
1507:
1503:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1474:
1470:
1467:
1465:
1463:
1460:
1459:
1456:
1451:
1448:
1445:
1441:
1437:
1432:
1428:
1424:
1421:
1418:
1415:
1412:
1407:
1404:
1401:
1397:
1393:
1390:
1387:
1384:
1382:
1380:
1377:
1376:
1353:
1331:
1327:
1306:
1286:
1281:
1278:
1275:
1271:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1235:
1232:
1229:
1225:
1221:
1218:
1196:
1193:
1190:
1186:
1162:
1157:
1153:
1149:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1067:
1047:
1044:
1041:
1038:
1014:
1011:
1008:
1005:
989:
986:
957:
937:
933:
929:
926:
923:
903:
883:
863:
841:
837:
812:
801:
800:
789:
784:
780:
776:
773:
770:
767:
764:
758:
755:
751:
746:
743:
740:
735:
731:
727:
722:
719:
716:
712:
681:
661:
641:
621:
618:
615:
612:
609:
606:
586:
566:
555:
554:
543:
540:
537:
532:
528:
524:
521:
518:
513:
510:
507:
503:
475:
472:
469:
466:
463:
460:
440:
437:
417:
414:
391:
336:
333:
330:
327:
324:
304:
301:
298:
295:
292:
289:
269:
266:
246:
226:
206:
202:
198:
189:held fixed at
178:
159:
156:
62:Arnold tongues
26:
9:
6:
4:
3:
2:
6594:
6583:
6580:
6579:
6577:
6562:
6559:
6557:
6554:
6552:
6551:Edge of chaos
6549:
6547:
6544:
6542:
6539:
6538:
6536:
6530:
6524:
6521:
6519:
6516:
6514:
6511:
6509:
6508:Marcelo Viana
6506:
6504:
6501:
6499:
6498:Audrey Terras
6496:
6494:
6493:Floris Takens
6491:
6489:
6486:
6484:
6481:
6479:
6476:
6474:
6471:
6469:
6466:
6464:
6461:
6459:
6456:
6454:
6451:
6449:
6446:
6444:
6441:
6439:
6436:
6434:
6431:
6429:
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6406:
6404:
6401:
6399:
6396:
6394:
6393:Celso Grebogi
6391:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6373:Chen Guanrong
6371:
6369:
6366:
6364:
6361:
6359:
6358:Michael Berry
6356:
6355:
6353:
6347:
6341:
6338:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6301:
6298:
6296:
6293:
6291:
6288:
6286:
6283:
6282:
6280:
6274:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6233:Lorenz system
6231:
6229:
6226:
6224:
6221:
6220:
6218:
6212:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6180:Langton's ant
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6155:Horseshoe map
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6129:
6126:
6125:
6124:
6121:
6119:
6116:
6114:
6111:
6109:
6106:
6104:
6101:
6099:
6096:
6094:
6091:
6090:
6088:
6082:
6079:
6076:
6069:
6063:
6060:
6058:
6055:
6053:
6052:Quantum chaos
6050:
6048:
6045:
6043:
6040:
6038:
6035:
6033:
6030:
6029:
6027:
6021:
6016:
6012:
6008:
5994:
5991:
5989:
5986:
5984:
5981:
5979:
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5960:
5958:
5952:
5946:
5943:
5941:
5938:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5860:Arnold tongue
5858:
5856:
5853:
5852:
5849:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5804:
5802:
5796:
5793:
5789:
5785:
5778:
5773:
5771:
5766:
5764:
5759:
5758:
5755:
5748:
5745:
5744:
5734:
5730:
5726:
5722:
5718:
5714:
5710:
5706:
5701:
5699:
5697:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5658:
5652:
5646:
5642:
5637:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5605:
5601:
5596:
5591:
5590:
5585:
5582:
5577:
5576:
5563:
5559:
5555:
5551:
5547:
5543:
5539:
5535:
5528:
5513:
5509:
5502:
5494:
5490:
5486:
5482:
5478:
5474:
5469:
5464:
5460:
5456:
5448:
5441:
5435:
5427:
5423:
5419:
5415:
5411:
5407:
5400:
5398:
5396:
5394:
5385:
5381:
5377:
5373:
5369:
5365:
5361:
5357:
5353:
5349:
5342:
5334:
5327:
5318:
5313:
5309:
5305:
5301:
5294:
5286:
5282:
5278:
5274:
5270:
5266:
5261:
5256:
5252:
5248:
5241:
5239:
5230:
5229:
5220:
5211:
5207:
5203:
5196:
5192:
5182:
5181:Sturmian word
5179:
5178:
5161:
5157:
5136:
5132:
5123:
5119:
5114:
5110:
5109:
5099:
5095:
5093:
5089:
5085:
5082:
5081:
5080:
5072:
5070:
5067:
5063:
5059:
5031:
5027:
5023:
5018:
5015:
5012:
5008:
5004:
5002:
4995:
4992:
4989:
4985:
4972:
4968:
4964:
4961:
4955:
4952:
4946:
4943:
4939:
4934:
4929:
4925:
4921:
4916:
4912:
4908:
4906:
4899:
4896:
4893:
4889:
4877:
4876:
4875:
4873:
4863:
4861:
4856:
4853:
4849:
4845:
4841:
4837:
4833:
4829:
4825:
4821:
4813:
4805:
4796:
4794:
4787:
4783:
4772:
4764:
4757: =
4756:
4752:
4744:
4736:
4729: =
4728:
4724:
4720:
4712:
4704:
4695:
4693:
4674:
4669:
4664:
4660:
4646:
4638:
4635:
4628:
4627:
4626:
4624:
4619:
4617:
4613:
4609:
4602:
4601:phase locking
4598:
4594:
4590:
4586:
4578:
4574:
4570:
4566:
4547:
4539:
4534:
4491:
4487:
4483:
4480:
4457:
4437:
4430:for integers
4417:
4414:
4408:
4400:
4376:
4362:
4359:
4356:
4333:
4330:
4322:
4318:
4314:
4309:
4305:
4294:
4280:
4277:
4272:
4268:
4264:
4259:
4255:
4247:
4246:
4245:
4231:
4228:
4225:
4205:
4182:
4178:
4174:
4168:
4148:
4123:
4120:
4115:
4112:
4107:
4103:
4099:
4094:
4090:
4082:
4081:
4067:
4063:
4059:
4056:
4044:
4030:
4027:
4024:
3999:
3996:
3992:
3988:
3985:
3942:
3939:
3936:
3928:
3910:
3907:
3904:
3901:
3898:
3893:
3889:
3885:
3882:
3879:
3876:
3871:
3867:
3863:
3858:
3854:
3850:
3846:
3842:
3838:
3817:
3814:
3809:
3805:
3801:
3796:
3792:
3765:
3762:
3759:
3756:
3751:
3747:
3743:
3741:
3728:
3724:
3720:
3717:
3711:
3708:
3703:
3698:
3695:
3692:
3688:
3681:
3678:
3674:
3669:
3663:
3660:
3651:
3648:
3643:
3639:
3635:
3633:
3620:
3616:
3612:
3609:
3603:
3600:
3595:
3590:
3587:
3584:
3580:
3573:
3570:
3566:
3561:
3557:
3550:
3547:
3542:
3538:
3534:
3532:
3526:
3522:
3518:
3506:
3505:
3502:
3483:
3480:
3477:
3474:
3471:
3465:
3461:
3415:
3411:
3407:
3404:
3401:
3396:
3392:
3368:
3365:
3362:
3359:
3353:
3350:
3330:
3323:with integer
3310:
3307:
3301:
3297:
3273:
3270:
3267:
3239:
3235:
3221:
3199:
3195:
3172:
3161:
3155:
3152:
3146:
3143:
3140:
3134:
3114:
3111:
3105:
3099:
3096:
3090:
3087:
3084:
3078:
3053:
3050:
3042:
3038:
3024:
3021:
3018:
3010:
3009:phase-locking
2994:
2991:
2983:
2979:
2975:
2970:
2966:
2959:
2956:
2934:
2930:
2909:
2887:
2877:
2873:
2869:
2864:
2860:
2852:Suppose that
2851:
2833:
2825:
2821:
2817:
2814:
2808:
2805:
2800:
2797:
2794:
2789:
2786:
2783:
2779:
2772:
2769:
2765:
2760:
2754:
2751:
2746:
2742:
2738:
2733:
2729:
2721:
2720:
2719:
2703:
2699:
2690:
2686:
2672:
2669:
2666:
2640:
2637:
2634:
2628:
2625:
2622:
2619:
2616:
2613:
2607:
2600:
2597:
2589:
2588:
2587:
2573:
2551:
2547:
2539:the iterates
2526:
2506:
2503:
2500:
2480:
2473:
2469:
2452:
2444:
2440:
2436:
2433:
2427:
2424:
2418:
2415:
2411:
2406:
2400:
2395:
2391:
2387:
2379:
2375:
2368:
2365:
2360:
2357:
2354:
2350:
2342:
2341:
2340:
2323:
2317:
2294:
2291:
2288:
2282:
2279:
2273:
2270:
2266:
2262:
2256:
2253:
2250:
2244:
2238:
2212:
2204:
2200:
2193:
2190:
2185:
2182:
2179:
2175:
2167:
2166:
2165:
2157:
2143:
2139:
2135:
2112:
2106:
2086:
2066:
2059:
2040:
2034:
2014:
2011:
2008:
1999:
1982:
1976:
1968:
1950:
1942:
1939:
1936:
1932:
1928:
1925:
1919:
1916:
1910:
1907:
1903:
1898:
1892:
1887:
1884:
1881:
1877:
1873:
1868:
1864:
1856:
1855:
1854:
1840:
1836:
1832:
1829:
1826:
1823:
1820:
1800:
1796:
1792:
1789:
1754:
1751:
1748:
1744:
1740:
1737:
1731:
1728:
1723:
1720:
1715:
1710:
1707:
1702:
1697:
1694:
1691:
1687:
1683:
1681:
1674:
1670:
1660:
1657:
1654:
1650:
1646:
1642:
1633:
1630:
1627:
1623:
1619:
1616:
1610:
1607:
1604:
1601:
1598:
1594:
1588:
1585:
1580:
1578:
1571:
1567:
1557:
1554:
1551:
1547:
1543:
1540:
1535:
1531:
1527:
1524:
1520:
1511:
1508:
1505:
1501:
1497:
1494:
1488:
1485:
1482:
1479:
1476:
1472:
1468:
1466:
1461:
1449:
1446:
1443:
1439:
1435:
1430:
1426:
1419:
1416:
1413:
1405:
1402:
1399:
1395:
1388:
1385:
1383:
1378:
1367:
1366:
1365:
1351:
1329:
1325:
1304:
1279:
1276:
1273:
1269:
1265:
1262:
1256:
1253:
1250:
1247:
1244:
1241:
1233:
1230:
1227:
1223:
1216:
1194:
1191:
1188:
1184:
1174:
1155:
1151:
1124:
1121:
1118:
1112:
1109:
1106:
1103:
1100:
1097:
1091:
1085:
1065:
1042:
1036:
1009:
1003:
994:
985:
983:
979:
975:
971:
955:
935:
931:
927:
924:
881:
861:
839:
835:
826:
810:
782:
778:
774:
771:
765:
762:
756:
753:
749:
744:
738:
733:
729:
725:
720:
717:
714:
710:
702:
701:
700:
697:
695:
639:
619:
616:
610:
604:
584:
538:
530:
526:
519:
516:
511:
508:
505:
501:
493:
492:
491:
490:is given by:
489:
470:
467:
464:
461:
438:
435:
415:
412:
405:
389:
381:
380:endomorphisms
376:
374:
370:
366:
362:
358:
354:
331:
328:
325:
299:
296:
293:
290:
267:
264:
257:at bottom to
244:
224:
204:
200:
196:
168:
164:
155:
152:
150:
144:
142:
138:
134:
130:
126:
122:
118:
114:
113:tidal locking
110:
106:
105:inharmonicity
101:
98:
94:
90:
86:
81:
79:
75:
71:
67:
64:(named after
63:
59:
55:
47:
43:
39:
34:
30:
19:
6582:Chaotic maps
6503:Mary Tsingou
6468:David Ruelle
6463:Otto Rössler
6408:Michel HĂ©non
6378:Leon O. Chua
6335:Tilt-A-Whirl
6305:FPUT problem
6190:Standard map
6185:Logistic map
6010:
5859:
5784:Chaos theory
5708:
5704:
5693:
5671:
5667:
5656:
5640:
5607:
5603:
5587:
5584:"Circle Map"
5537:
5533:
5527:
5515:. Retrieved
5511:
5501:
5458:
5454:
5447:
5434:
5412:(26): 1772.
5409:
5405:
5351:
5347:
5341:
5332:
5326:
5307:
5303:
5293:
5250:
5247:FEBS Letters
5246:
5227:
5219:
5209:
5205:
5195:
5159:
5155:
5121:
5117:
5078:
5075:Applications
5066:kicked rotor
5054:
5052:
4869:
4857:
4851:
4847:
4839:
4835:
4827:
4823:
4820:measure zero
4811:
4803:
4797:
4789:
4785:
4778:
4770:
4762:
4754:
4750:
4742:
4734:
4726:
4722:
4718:
4710:
4702:
4696:
4689:
4620:
4615:
4604:
4600:
4597:mode locking
4596:
4592:
4588:
4584:
4582:
4576:
4572:
4545:
4537:
4528:Mode locking
4348:
4140:
3926:
3925:
3237:
3236:
3040:
3039:
3008:
2849:
2848:
2688:
2687:
2658:
2471:
2470:
2467:
2230:
2163:
2079:times every
2057:
2000:
1965:
1778:
1175:
1028:
914:; if we fix
824:
802:
698:
556:
487:
377:
356:
350:
153:
145:
121:fiber optics
117:mode-locking
102:
92:
89:phase-locked
88:
82:
61:
51:
45:
41:
37:
29:
6488:Nina Snaith
6478:Yakov Sinai
6363:Rufus Bowen
6113:Duffing map
6098:Baker's map
6023:Theoretical
5935:SRB measure
5842:Phase space
5812:Bifurcation
5461:(1): 1–23.
5212:(1): 21–86.
5166:at the top.
5158:-axis, and
5120:-axis, and
5069:Hamiltonian
4470:, but then
3260:there is a
488:circle maps
369:heart cells
353:oscillators
315:instead of
93:mode-locked
54:mathematics
6546:Complexity
6443:Edward Ott
6290:Convection
6215:Continuous
5890:Ergodicity
5747:Circle map
5711:(1): 1–6.
5573:References
4349:meaning a
2160:Properties
823:is called
237:goes from
141:cell cycle
127:and other
40:-axis and
18:Phase lock
6458:Mary Rees
6418:Bryna Kra
6351:theorists
6160:Ikeda map
6150:HĂ©non map
6140:Gauss map
5822:Limit set
5807:Attractor
5664:Glass, L.
5632:121088353
5589:MathWorld
5512:MathWorld
5463:CiteSeerX
5260:1112.6093
5028:θ
5024:−
5009:θ
4969:θ
4965:π
4956:
4947:π
4913:θ
4890:θ
4834:. Fixing
4661:θ
4653:∞
4650:→
4636:ω
4512:Ω
4478:Ω
4412:Ω
4385:Ω
4319:θ
4315:−
4306:θ
4269:θ
4256:θ
4104:θ
4091:θ
4080:implies:
4054:Ω
4000:∈
3983:Ω
3963:Ω
3890:θ
3886:−
3868:θ
3855:θ
3851:−
3839:θ
3806:θ
3802:−
3793:θ
3784:so since
3748:θ
3725:θ
3721:π
3712:
3689:∑
3682:π
3658:Ω
3640:θ
3617:θ
3613:π
3604:
3581:∑
3574:π
3555:Ω
3539:θ
3519:θ
3484:∈
3469:Ω
3459:Ω
3438:Ω
3412:θ
3405:…
3393:θ
3305:Ω
3295:Ω
3248:Ω
3196:θ
3162:θ
3141:θ
3106:θ
3085:θ
3054:∈
2992:⋅
2980:θ
2976:−
2967:θ
2931:θ
2874:θ
2861:θ
2822:θ
2818:π
2809:
2798:−
2780:∑
2773:π
2758:Ω
2743:θ
2730:θ
2700:θ
2641:θ
2638:π
2629:
2608:θ
2548:θ
2441:θ
2437:π
2428:
2419:π
2404:Ω
2392:θ
2376:θ
2351:θ
2324:θ
2295:θ
2292:π
2283:
2274:π
2254:θ
2245:θ
2216:Ω
2201:θ
2176:θ
2056:is reset
1940:−
1929:π
1920:
1911:π
1896:Ω
1885:−
1830:π
1787:Ω
1752:−
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