2447:. The StuartâLandau equation in fact describes an entire class of limit-cycle oscillators in the weakly-nonlinear limit. The form of the classical StuartâLandau equation is much simpler, and perhaps not surprisingly, can be quantized by a Lindblad equation which is also simpler than the Lindblad equation for the van der Pol oscillator. The quantum StuartâLandau model has played an important role in the study of quantum synchronisation (where it has often been called a van der Pol oscillator although it cannot be uniquely associated with the van der Pol oscillator). The relationship between the classical StuartâLandau model (
2464:
4318:
257:
4310:
2683:
760:
236:
265:
203:
239:
243:
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244:
1578:
1417:
241:
1946:
1185:
1736:
206:
2645:
210:
209:
205:
204:
2439:. Note the above Hamiltonian approach with an auxiliary second-order equation produces unbounded phase-space trajectories and hence cannot be used to quantize the van der Pol oscillator. In the limit of weak nonlinearity (i.e.
211:
1057:
1935:
2279:
1422:
169:
558:
1750:
moves from less than zero to more than zero, the spiral sink at origin becomes a spiral source, and a limit cycle appears "out of the blue" with radius two. This is because the transition is not generic: when
208:
1271:
2124:
2042:
1077:
2355:
848:
604:
1804:
748:
476:
1957:
formalism for the Van der Pol oscillator by augmenting it to a four-dimensional autonomous dynamical system using an auxiliary second-order nonlinear differential equation as follows:
1239:
1627:
1851:
922:
240:
2397:
681:
643:
952:
3554:
3389:
4607:
4437:
1856:
2512:
207:
60:
4894:
4317:
4077:
1070:, the behavior of the oscillator has a slow buildup, fast release cycle (a cycle of building up the tension and releasing the tension, thus a
4046:
3530:
3365:
3299:
3229:
Panayotounakos, D.E.; Panayotounakou, N.D.; Vakakis, A.F. (2003-09-01). "On the lack of analytic solutions of the Van der Pol oscillator".
2147:
4270:
484:
2129:
Note that the dynamics of the original Van der Pol oscillator is not affected due to the one-way coupling between the time-evolutions of
4217:
4550:
3987:(1945). "On Non-Linear Differential Equations of the Second Order: I. the Equation yš â k (1-y 2 )yË + y = b λk cos(λl + α), k Large".
3770:
Walter, Stefan; Nunnenkamp, Andreas; Bruder, Christoph (2014-03-06). "Quantum
Synchronization of a Driven Self-Sustained Oscillator".
2867:
B. van der Pol: "A theory of the amplitude of free and forced triode vibrations", Radio Review (later
Wireless World) 1 701â710 (1920)
55:
4592:
4280:
4785:
3938:
3414:
3216:
790:
1060:
4467:
2801:, British mathematician, one of the first to study the theory of deterministic chaos, particularly as applied to this oscillator.
2411:, respectively. This may, in principle, lead to quantization of the Van der Pol oscillator. Such a Hamiltonian also connects the
927:
The Van der Pol oscillator does not have an exact, analytic solution. However, such a solution does exist for the limit cycle if
1063:
for a derivation to order 2. See chapter 10 of for a derivation up to order 3, and for a numerical derivation up to order 164.
4899:
330:
3839:
Lee, Tony E.; Sadeghpour, H. R. (2013-12-04). "Quantum
Synchronization of Quantum van der Pol Oscillators with Trapped Ions".
2048:
1963:
1573:{\displaystyle T=(3-2\ln 2)\mu +3\alpha \mu ^{-1/3}-{\frac {22}{9\mu }}\ln \mu +{\frac {0.0087\cdots }{\mu }}+O(\mu ^{-4/3}).}
3916:
4285:
4275:
3955:
4031:
4540:
4632:
4405:
3701:
Chia, A.; Kwek, L. C.; Noh, C. (2020-10-16). "Relaxation oscillations and frequency entrainment in quantum mechanics".
4026:
2779:
478:, where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:
333:
pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the
September 1927 issue of
4545:
2444:
2287:
1412:{\displaystyle T=2\int dt=2\int \mu {\frac {dy}{x}}=2\mu \int _{2}^{1}{\frac {dy}{dx}}{\frac {dx}{x}}=(3-2\ln 2)\mu }
4612:
4904:
4070:
564:
1268:
The leading term in the period of the cycle is due to the slow ascending and descending, which can be computed as:
370:
4889:
4660:
1768:
687:
411:
4364:
4309:
3357:
Advanced mathematical methods for scientists and engineers I : asymptotic methods and perturbation theory
2463:
1193:
2436:
4525:
4290:
4197:
3097:; EguĂluz, VĂctor M.; HernĂĄndez-GarcĂa, Emilio; Piro, Oreste (1999). "Dynamics of Elastic Excitable Media".
1809:
873:
405:
256:
3159:
4177:
4021:
3051:
Nagumo, J.; Arimoto, S.; Yoshizawa, S. (1962). "An Active Pulse
Transmission Line Simulating Nerve Axon".
1180:{\displaystyle {\dot {x}}=\mu \left(x-{\frac {1}{3}}x^{3}-y\right)\!,\quad {\dot {y}}={\frac {1}{\mu }}x.}
855:
763:
294:
180:
4884:
4715:
4622:
4420:
4247:
4182:
4157:
4063:
3439:
2659:
2495:
The forced, or driven, Van der Pol oscillator takes the 'original' function and adds a driving function
2400:
326:
306:
298:
281:
39:
4477:
2360:
4725:
4555:
4377:
4222:
3858:
3789:
3720:
3656:
3591:
3522:
Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering
3317:"Power Series Expansions for the Frequency and Period of the Limit Cycle of the Van Der Pol Equation"
3238:
3116:
2999:
2921:
192:
766:
in the Van der Pol oscillator without external forcing. The nonlinear damping parameter is equal to
4685:
4642:
4627:
4472:
4425:
4410:
4395:
4295:
4227:
4202:
4187:
4172:
3634:
3569:
2470:
in the Van der Pol oscillator with sinusoidal forcing. The nonlinear damping parameter is equal to
1618:
851:
366:
249:
4043:
3635:"Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory"
651:
613:
4863:
4730:
4560:
4447:
4442:
4334:
4212:
4114:
3890:
3848:
3821:
3779:
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3710:
3680:
3646:
3615:
3581:
3548:
3383:
3262:
3140:
3106:
3076:
2945:
2686:
2428:
1954:
344:
314:
43:
1731:{\displaystyle 2+{\frac {\alpha }{3}}\mu ^{-4/3}-{\frac {16}{27}}\mu ^{-2}\ln \mu +O(\mu ^{-2})}
4750:
4735:
4700:
4690:
4587:
4207:
4129:
4000:
3934:
3912:
3882:
3874:
3813:
3805:
3744:
3736:
3672:
3607:
3536:
3526:
3501:
3420:
3410:
3371:
3361:
3336:
3295:
3254:
3212:
3173:
3132:
3094:
3068:
3033:
3015:
2937:
2894:
2850:
2719:
2432:
176:
2963:
2834:
4843:
4755:
4705:
4602:
4530:
4482:
4359:
4344:
4339:
4134:
3992:
3984:
3866:
3797:
3728:
3664:
3599:
3491:
3328:
3287:
3246:
3165:
3124:
3060:
3023:
3007:
2929:
2886:
2842:
2671:
1762:
939:
408:
can be used to prove that the system has a limit cycle. Applying the Liénard transformation
386:
374:
355:
335:
31:
2789:
article, Gleick received a modern electronic Van der Pol oscillator from a reader in 1988.
4775:
4670:
4597:
4430:
4242:
4232:
4050:
3980:
2798:
2712:
2412:
4765:
4710:
4038:
1758:, both the differential equation becomes linear, and the origin becomes a circular node.
3862:
3793:
3724:
3660:
3595:
3242:
3120:
3003:
2925:
4858:
4825:
4820:
4815:
4617:
4507:
4502:
4400:
4349:
4265:
4139:
3028:
2987:
2785:
2756:
216:
3496:
3479:
3462:
3011:
1806:
we may attempt to convert this to a Hopf bifurcation by using the change of variables
4878:
4853:
4810:
4800:
4795:
4695:
4675:
4535:
4457:
4354:
4162:
3756:
3281:
3266:
2949:
2663:
2640:{\displaystyle {d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x-A\sin(\omega t)=0,}
1600:
3894:
3684:
3619:
3080:
870:, all initial conditions converge to a globally unique limit cycle. Near the origin
17:
4805:
4770:
4680:
4637:
4492:
4487:
4086:
3870:
3825:
3801:
3144:
2968:
2770:
2752:
2693:, resulting in a forced Van der Pol oscillator. The circuit contains: a triode, a
2467:
2416:
348:
196:
4452:
3568:
Shah, Tirth; Chattopadhyay, Rohitashwa; Vaidya, Kedar; Chakraborty, Sagar (2015).
2682:
4790:
4780:
4665:
4415:
4237:
4144:
3732:
2774:
2726:
394:
322:
318:
310:
273:
269:
47:
3996:
3668:
3603:
3064:
2879:
The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science
2846:
1945:
759:
4848:
4745:
4192:
3540:
3291:
3128:
2890:
2455:) has also been demonstrated numerically in the corresponding quantum models.
382:
4004:
3878:
3809:
3740:
3505:
3375:
3340:
3258:
3177:
3136:
3072:
3019:
2941:
2898:
2854:
1052:{\displaystyle T={\frac {2\pi }{1-\mu ^{2}/16+17\mu ^{4}/3072+O(\mu ^{6})}}.}
4760:
4720:
4462:
4124:
4109:
3424:
3316:
3208:
2701:
2655:
390:
340:
228:
3886:
3817:
3748:
3676:
3611:
3520:
3250:
3059:(10). Institute of Electrical and Electronics Engineers (IEEE): 2061â2070.
3037:
2988:"Impulses and Physiological States in Theoretical Models of Nerve Membrane"
2912:
2431:
version of the classical van der Pol oscillator, has been proposed using a
365:. For instance, in biology, Fitzhugh and Nagumo extended the equation in a
3355:
264:
4497:
3404:
3111:
2821:
2708:
2694:
4167:
4119:
378:
362:
359:
302:
51:
3169:
3164:. Proceedings of Meetings on Acoustics. Vol. 19. p. 060165.
1930:{\displaystyle {\ddot {u}}+u+u^{2}{\dot {u}}-\varepsilon {\dot {u}}=0}
924:
the system is unstable, and far from the origin, the system is damped.
354:
The Van der Pol equation has a long history of being used in both the
305:. Van der Pol found stable oscillations, which he subsequently called
4740:
2933:
2690:
2274:{\displaystyle H(x,y,p_{x},p_{y})=p_{x}p_{y}+xy-\mu (1-x^{2})yp_{y},}
164:{\displaystyle {d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0,}
3332:
3161:
Modeling vocal fold asymmetries with coupled van der Pol oscillators
2415:
of the limit cycle system having time dependent parameters with the
553:{\displaystyle {\dot {x}}=\mu \left(x-{\tfrac {1}{3}}x^{3}-y\right)}
260:
The Van der Pol
Oscillator simulated with the Brain Dynamics Toolbox
3907:
K. Tomita (1986): "Periodically forced nonlinear oscillators". In:
3715:
3651:
3633:
Chattopadhyay, Rohitashwa; Shah, Tirth; Chakraborty, Sagar (2018).
3586:
1949:
Randomly chosen initial conditions are attracted to a stable orbit.
3853:
3784:
2734:
2681:
2462:
1944:
758:
263:
255:
234:
201:
3286:. Universitext. Berlin, Heidelberg: Springer Berlin Heidelberg.
2817:
Heitmann, S., Breakspear, M (2017â2022) Brain
Dynamics Toolbox.
1603:. (Section 9.7 ) ( contains a derivation, but has a misprint of
293:
The Van der Pol oscillator was originally proposed by the Dutch
4059:
4055:
3570:"Conservative perturbation theory for nonconservative systems"
1187:
In this form, the oscillator completes one cycle as follows:
2877:
van der Pol, Balth. (1926). "On "relaxation-oscillations"".
1252:
Rapidly moving to the left branch of the cubic curve, from
2920:(3019). Springer Science and Business Media LLC: 363â364.
787:, i.e. there is no damping function, the equation becomes
2818:
2119:{\displaystyle {\ddot {y}}+\mu (1-x^{2}){\dot {y}}+y=0.}
2037:{\displaystyle {\ddot {x}}-\mu (1-x^{2}){\dot {x}}+x=0,}
2451:
0) and more general limit-cycle oscillators (arbitrary
610:
Another commonly used form based on the transformation
276:. The limit cycle begins as a circle and, with varying
3467:. NIST Research Library. National Bureau of Standards.
3283:
1210:
884:
518:
2515:
2363:
2290:
2150:
2051:
1966:
1859:
1812:
1771:
1630:
1425:
1274:
1196:
1190:
Slowly ascending the right branch of the cubic curve
1080:
955:
876:
793:
690:
654:
616:
567:
487:
414:
347:
was heard, which was later found to be the result of
63:
4834:
4651:
4578:
4516:
4386:
4373:
4325:
4256:
4100:
4093:
3911:, Ed. Arun V. Holden. Manchester University Press,
27:
Oscillating dynamical system with nonlinear damping
2639:
2391:
2349:
2273:
2118:
2036:
1929:
1845:
1798:
1730:
1572:
1411:
1233:
1179:
1051:
916:
842:
742:
675:
637:
598:
552:
470:
163:
4044:Van Der Pol Oscillator Interactive Demonstrations
3409:. Boca Raton, Fla.: CRC Press. pp. 161â163.
2427:The quantum van der Pol oscillator, which is the
1141:
223:varying from 0.1 to 3.0. The green lines are the
3480:"Periodic solutions of the Van der Pol equation"
2755:. The Van der Pol oscillator is forced by an AC
2141:for this system of equations can be shown to be
3440:"Asymptotic solution of van der Pol's equation"
2350:{\displaystyle p_{x}={\dot {y}}+\mu (1-x^{2})y}
3099:International Journal of Bifurcation and Chaos
280:, becomes increasingly sharp. An example of a
4071:
3464:Asymptotic solution of Van Der Pol's equation
3315:Andersen, C. M.; Geer, James F. (June 1982).
2777:Van der Pol oscillator in his book from 1987
2506:to give a differential equation of the form:
2443:0) the van der Pol oscillator reduces to the
843:{\displaystyle {\frac {d^{2}x}{dt^{2}}}+x=0.}
599:{\displaystyle {\dot {y}}={\frac {1}{\mu }}x}
8:
1419:Higher orders of the period of the cycle is
3933:. New York: Penguin Books. pp. 41â43.
3158:Lucero, Jorge C.; Schoentgen, Jean (2013).
1799:{\displaystyle \propto \varepsilon ^{1/2},}
743:{\displaystyle {\dot {y}}=\mu (1-x^{2})y-x}
471:{\displaystyle y=x-x^{3}/3-{\dot {x}}/\mu }
4383:
4097:
4078:
4064:
4056:
3989:Journal of the London Mathematical Society
3553:: CS1 maint: location missing publisher (
3519:Strogatz, Steven (2019). "Example 8.4.1".
3388:: CS1 maint: location missing publisher (
2839:Journal of the London Mathematical Society
321:. When these circuits are driven near the
3852:
3783:
3714:
3650:
3585:
3495:
3406:Nonlinear ordinary differential equations
3205:Nonlinear ordinary differential equations
3110:
3027:
2578:
2569:
2541:
2523:
2516:
2514:
2419:of the corresponding Hamiltonian system.
2378:
2377:
2368:
2362:
2335:
2305:
2304:
2295:
2289:
2262:
2246:
2212:
2202:
2186:
2173:
2149:
2093:
2092:
2083:
2053:
2052:
2050:
2008:
2007:
1998:
1968:
1967:
1965:
1910:
1909:
1892:
1891:
1885:
1861:
1860:
1858:
1827:
1823:
1811:
1783:
1779:
1770:
1716:
1685:
1671:
1658:
1651:
1637:
1629:
1554:
1547:
1519:
1492:
1479:
1472:
1424:
1364:
1344:
1338:
1333:
1305:
1273:
1216:
1209:
1195:
1161:
1147:
1146:
1124:
1110:
1082:
1081:
1079:
1034:
1013:
1007:
989:
983:
962:
954:
883:
875:
819:
801:
794:
792:
722:
692:
691:
689:
656:
655:
653:
624:
623:
615:
583:
569:
568:
566:
533:
517:
489:
488:
486:
460:
449:
448:
437:
431:
413:
381:. The equation has also been utilised in
248:The same oscillator phase plot, but with
126:
117:
89:
71:
64:
62:
3696:
3694:
1263:Repeat the two steps on the left branch.
1234:{\displaystyle y=x-{\tfrac {x^{3}}{3}},}
1074:). This is most easily seen in the form
3484:Indagationes Mathematicae (Proceedings)
2810:
4039:Van der Pol oscillator on Scholarpedia
3546:
3381:
1953:One can also write a time-independent
1941:Hamiltonian for Van der Pol oscillator
1846:{\displaystyle u=\varepsilon ^{1/2}x,}
917:{\displaystyle x={\tfrac {dx}{dt}}=0,}
54:. It evolves in time according to the
7:
4218:Measure-preserving dynamical system
3321:SIAM Journal on Applied Mathematics
2885:(11). Informa UK Limited: 978â992.
1937:This indeed is a Hopf bifurcation.
1765:, the limit cycle should have size
755:Results for the unforced oscillator
3360:. Steven A. Orszag. New York, NY.
2477:, while the forcing has amplitude
2435:to study its quantum dynamics and
942:is a constant piece-wise function.
56:second-order differential equation
25:
4786:Oleksandr Mykolayovych Sharkovsky
199:and the strength of the damping.
4316:
4308:
3956:"There's No Quiet Without Noise"
3192:Understanding Nonlinear Dynamics
2392:{\displaystyle p_{y}={\dot {x}}}
4895:Ordinary differential equations
1624:The amplitude of the cycle is
1145:
309:and are now known as a type of
4551:RabinovichâFabrikant equations
3954:Colman, David (11 July 2011).
3871:10.1103/physrevlett.111.234101
3802:10.1103/physrevlett.112.094102
2822:doi.org/10.5281/zenodo.5625923
2625:
2616:
2575:
2556:
2341:
2322:
2252:
2233:
2192:
2154:
2089:
2070:
2004:
1985:
1725:
1709:
1564:
1540:
1453:
1432:
1403:
1382:
1040:
1027:
728:
709:
123:
104:
1:
3991:. s1-20 (3). Wiley: 180â189.
3497:10.1016/s1385-7258(69)50068-x
3012:10.1016/s0006-3495(61)86902-6
2841:. s1-35 (3). Wiley: 367â376.
2459:Forced Van der Pol oscillator
385:to model the two plates in a
3525:(2nd ed.). Boca Raton.
3280:Verhulst, Ferdinand (1996).
3194:, Springer, 240â244, (1995).
676:{\displaystyle {\dot {x}}=y}
638:{\displaystyle y={\dot {x}}}
393:to model the right and left
4286:Poincaré recurrence theorem
4027:Encyclopedia of Mathematics
3931:Chaos: Making a New Science
3733:10.1103/physreve.102.042213
2998:(6). Elsevier BV: 445â466.
2780:Chaos: Making a New Science
2744:, while there is a voltage
38:(named for Dutch physicist
4921:
4281:PoincarĂ©âBendixson theorem
3669:10.1103/PhysRevE.97.062209
3604:10.1103/physreve.92.062927
3190:Kaplan, D. and Glass, L.,
3065:10.1109/jrproc.1962.288235
2986:FitzHugh, Richard (1961).
2833:Cartwright, M. L. (1960).
852:simple harmonic oscillator
4633:Swinging Atwood's machine
4306:
4276:KrylovâBogolyubov theorem
4153:
3438:A, Dorodnicyn A. (1947).
3292:10.1007/978-3-642-61453-8
3129:10.1142/s0218127499001620
2891:10.1080/14786442608564127
2733:, and towards the triode
2137:variables. A Hamiltonian
1061:PoincarĂ©âLindstedt method
195:parameter indicating the
4541:LotkaâVolterra equations
4365:Synchronization of chaos
4168:axiom A dynamical system
3997:10.1112/jlms/s1-20.3.180
3478:Zonneveld, J.A. (1966).
3354:Bender, Carl M. (1999).
3095:Cartwright, Julyan H. E.
2964:"Van der Pol oscillator"
2847:10.1112/jlms/s1-35.3.367
1587:is the smallest root of
301:while he was working at
4526:Double scroll attractor
4291:Stable manifold theorem
4198:False nearest neighbors
3841:Physical Review Letters
3772:Physical Review Letters
2835:"Balthazar Van Der Pol"
2437:quantum synchronization
1617:.) This was derived by
307:relaxation-oscillations
215:van der Pol oscillator
50:system with non-linear
4900:Electronic oscillators
4566:Van der Pol oscillator
4546:MackeyâGlass equations
4178:Box-counting dimension
4022:"Van der Pol equation"
3929:Gleick, James (1987).
3251:10.1002/zamm.200310040
3053:Proceedings of the IRE
2766:
2641:
2492:
2484:and angular frequency
2445:StuartâLandau equation
2393:
2351:
2275:
2120:
2038:
1950:
1931:
1847:
1800:
1732:
1574:
1413:
1235:
1181:
1072:relaxation oscillation
1053:
918:
856:conservation of energy
854:, and there is always
850:This is a form of the
844:
774:
764:Relaxation oscillation
744:
677:
639:
600:
554:
472:
339:that at certain drive
285:
261:
253:
232:
165:
36:van der Pol oscillator
4716:Svetlana Jitomirskaya
4623:Multiscroll attractor
4468:Interval exchange map
4421:Dyadic transformation
4406:Complex quadratic map
4248:Topological conjugacy
4183:Correlation dimension
4158:Anosov diffeomorphism
3403:Grimshaw, R. (1993).
3237:(9). Wiley: 611â615.
2685:
2642:
2466:
2394:
2352:
2276:
2121:
2039:
1948:
1932:
1848:
1801:
1733:
1575:
1414:
1236:
1182:
1054:
949:has serial expansion
919:
845:
762:
745:
678:
640:
601:
555:
473:
299:Balthasar van der Pol
282:relaxation oscillator
267:
259:
247:
214:
166:
40:Balthasar van der Pol
4726:Edward Norton Lorenz
2737:("plate") a current
2513:
2361:
2288:
2148:
2049:
1964:
1857:
1810:
1769:
1628:
1423:
1272:
1194:
1078:
953:
945:The period at small
874:
791:
688:
652:
614:
565:
485:
412:
401:Two-dimensional form
389:, and in studies of
61:
18:Van der Pol equation
4686:Mitchell Feigenbaum
4628:Population dynamics
4613:HĂ©nonâHeiles system
4473:Irrational rotation
4426:Dynamical billiards
4411:Coupled map lattice
4271:Liouville's theorem
4203:Hausdorff dimension
4188:Conservative system
4173:Bifurcation diagram
3863:2013PhRvL.111w4101L
3794:2014PhRvL.112i4102W
3725:2020PhRvE.102d2213C
3661:2018PhRvE..97f2209C
3596:2015PhRvE..92f2927S
3243:2003ZaMM...83..611P
3211:, 153â163, (1993),
3121:1999IJBC....9.2197C
3004:1961BpJ.....1..445F
2992:Biophysical Journal
2926:1927Natur.120..363V
2729:there is a current
1619:Anatoly Dorodnitsyn
1343:
349:deterministic chaos
329:, i.e. the driving
315:electrical circuits
295:electrical engineer
4864:Santa Fe Institute
4731:Aleksandr Lyapunov
4561:Three-body problem
4448:Gingerbreadman map
4335:Bifurcation theory
4213:Lyapunov stability
4049:2017-02-14 at the
3461:Dorodnicyn, A. A.
2976:(1), 2202, (2007).
2783:. According to a
2767:
2687:Electrical circuit
2637:
2493:
2429:quantum mechanical
2423:Quantum oscillator
2389:
2347:
2271:
2116:
2034:
1951:
1927:
1843:
1796:
1761:Knowing that in a
1728:
1570:
1409:
1329:
1231:
1226:
1177:
1049:
914:
903:
840:
775:
740:
673:
635:
596:
550:
527:
468:
286:
262:
254:
233:
161:
4905:Dynamical systems
4872:
4871:
4736:BenoĂźt Mandelbrot
4701:Martin Gutzwiller
4691:Peter Grassberger
4574:
4573:
4556:Rössler attractor
4304:
4303:
4208:Invariant measure
4130:Lyapunov exponent
3985:Littlewood, J. E.
3981:Cartwright, M. L.
3703:Physical Review E
3639:Physical Review E
3574:Physical Review E
3532:978-0-367-09206-1
3444:Prikl. Mat. Mech.
3367:978-1-4757-3069-2
3301:978-3-540-60934-6
3170:10.1121/1.4798467
3105:(11): 2197â2202.
2725:. In the serial
2720:mutual inductance
2596:
2548:
2468:Chaotic behaviour
2433:Lindblad equation
2403:corresponding to
2401:conjugate momenta
2386:
2313:
2101:
2061:
2016:
1976:
1918:
1900:
1869:
1679:
1645:
1532:
1505:
1377:
1362:
1318:
1225:
1169:
1155:
1118:
1090:
1044:
902:
826:
700:
664:
632:
591:
577:
526:
497:
457:
406:Liénard's theorem
375:action potentials
268:Evolution of the
250:Liénard transform
245:
212:
144:
96:
32:dynamical systems
16:(Redirected from
4912:
4890:Dutch inventions
4844:Butterfly effect
4756:Itamar Procaccia
4706:Brosl Hasslacher
4603:Elastic pendulum
4531:Duffing equation
4478:KaplanâYorke map
4396:Arnold's cat map
4384:
4360:Stability theory
4345:Dynamical system
4340:Control of chaos
4320:
4312:
4296:Takens's theorem
4228:Poincaré section
4098:
4080:
4073:
4066:
4057:
4035:
4009:
4008:
3977:
3971:
3970:
3968:
3966:
3951:
3945:
3944:
3926:
3920:
3905:
3899:
3898:
3856:
3836:
3830:
3829:
3787:
3767:
3761:
3760:
3718:
3698:
3689:
3688:
3654:
3630:
3624:
3623:
3589:
3565:
3559:
3558:
3552:
3544:
3516:
3510:
3509:
3499:
3475:
3469:
3468:
3458:
3452:
3451:
3435:
3429:
3428:
3400:
3394:
3393:
3387:
3379:
3351:
3345:
3344:
3312:
3306:
3305:
3277:
3271:
3270:
3226:
3220:
3201:
3195:
3188:
3182:
3181:
3155:
3149:
3148:
3114:
3112:chao-dyn/9905035
3091:
3085:
3084:
3048:
3042:
3041:
3031:
2983:
2977:
2960:
2954:
2953:
2934:10.1038/120363a0
2909:
2903:
2902:
2874:
2868:
2865:
2859:
2858:
2830:
2824:
2815:
2672:angular velocity
2669:
2653:
2646:
2644:
2643:
2638:
2597:
2595:
2587:
2579:
2574:
2573:
2549:
2547:
2546:
2545:
2532:
2528:
2527:
2517:
2505:
2490:
2483:
2476:
2398:
2396:
2395:
2390:
2388:
2387:
2379:
2373:
2372:
2356:
2354:
2353:
2348:
2340:
2339:
2315:
2314:
2306:
2300:
2299:
2280:
2278:
2277:
2272:
2267:
2266:
2251:
2250:
2217:
2216:
2207:
2206:
2191:
2190:
2178:
2177:
2125:
2123:
2122:
2117:
2103:
2102:
2094:
2088:
2087:
2063:
2062:
2054:
2043:
2041:
2040:
2035:
2018:
2017:
2009:
2003:
2002:
1978:
1977:
1969:
1936:
1934:
1933:
1928:
1920:
1919:
1911:
1902:
1901:
1893:
1890:
1889:
1871:
1870:
1862:
1852:
1850:
1849:
1844:
1836:
1835:
1831:
1805:
1803:
1802:
1797:
1792:
1791:
1787:
1763:Hopf bifurcation
1757:
1749:
1742:Hopf bifurcation
1737:
1735:
1734:
1729:
1724:
1723:
1693:
1692:
1680:
1672:
1667:
1666:
1662:
1646:
1638:
1616:
1609:
1598:
1594:
1586:
1579:
1577:
1576:
1571:
1563:
1562:
1558:
1533:
1528:
1520:
1506:
1504:
1493:
1488:
1487:
1483:
1418:
1416:
1415:
1410:
1378:
1373:
1365:
1363:
1361:
1353:
1345:
1342:
1337:
1319:
1314:
1306:
1259:
1255:
1248:
1244:
1240:
1238:
1237:
1232:
1227:
1221:
1220:
1211:
1186:
1184:
1183:
1178:
1170:
1162:
1157:
1156:
1148:
1140:
1136:
1129:
1128:
1119:
1111:
1092:
1091:
1083:
1069:
1058:
1056:
1055:
1050:
1045:
1043:
1039:
1038:
1017:
1012:
1011:
993:
988:
987:
971:
963:
948:
940:Lienard equation
937:
923:
921:
920:
915:
904:
901:
893:
885:
869:
849:
847:
846:
841:
827:
825:
824:
823:
810:
806:
805:
795:
786:
772:
749:
747:
746:
741:
727:
726:
702:
701:
693:
682:
680:
679:
674:
666:
665:
657:
644:
642:
641:
636:
634:
633:
625:
605:
603:
602:
597:
592:
584:
579:
578:
570:
559:
557:
556:
551:
549:
545:
538:
537:
528:
519:
499:
498:
490:
477:
475:
474:
469:
464:
459:
458:
450:
441:
436:
435:
387:geological fault
279:
246:
226:
222:
213:
190:
186:
175:is the position
174:
170:
168:
167:
162:
145:
143:
135:
127:
122:
121:
97:
95:
94:
93:
80:
76:
75:
65:
30:In the study of
21:
4920:
4919:
4915:
4914:
4913:
4911:
4910:
4909:
4875:
4874:
4873:
4868:
4836:
4830:
4776:Caroline Series
4671:Mary Cartwright
4653:
4647:
4598:Double pendulum
4580:
4570:
4519:
4512:
4438:Exponential map
4389:
4375:
4369:
4327:
4321:
4314:
4300:
4266:Ergodic theorem
4259:
4252:
4243:Stable manifold
4233:Recurrence plot
4149:
4103:
4089:
4084:
4051:Wayback Machine
4020:
4017:
4012:
3979:
3978:
3974:
3964:
3962:
3953:
3952:
3948:
3941:
3928:
3927:
3923:
3906:
3902:
3838:
3837:
3833:
3769:
3768:
3764:
3700:
3699:
3692:
3632:
3631:
3627:
3567:
3566:
3562:
3545:
3533:
3518:
3517:
3513:
3477:
3476:
3472:
3460:
3459:
3455:
3437:
3436:
3432:
3417:
3402:
3401:
3397:
3380:
3368:
3353:
3352:
3348:
3333:10.1137/0142047
3314:
3313:
3309:
3302:
3279:
3278:
3274:
3228:
3227:
3223:
3202:
3198:
3189:
3185:
3157:
3156:
3152:
3093:
3092:
3088:
3050:
3049:
3045:
2985:
2984:
2980:
2961:
2957:
2911:
2910:
2906:
2876:
2875:
2871:
2866:
2862:
2832:
2831:
2827:
2816:
2812:
2808:
2799:Mary Cartwright
2795:
2763:
2749:
2742:
2713:self inductance
2680:
2678:Popular culture
2667:
2651:
2588:
2580:
2565:
2537:
2533:
2519:
2518:
2511:
2510:
2496:
2485:
2478:
2471:
2461:
2425:
2413:geometric phase
2364:
2359:
2358:
2331:
2291:
2286:
2285:
2258:
2242:
2208:
2198:
2182:
2169:
2146:
2145:
2079:
2047:
2046:
1994:
1962:
1961:
1943:
1881:
1855:
1854:
1819:
1808:
1807:
1775:
1767:
1766:
1752:
1747:
1744:
1712:
1681:
1647:
1626:
1625:
1611:
1604:
1596:
1588:
1581:
1543:
1521:
1497:
1468:
1421:
1420:
1366:
1354:
1346:
1307:
1270:
1269:
1257:
1253:
1246:
1242:
1212:
1192:
1191:
1120:
1103:
1099:
1076:
1075:
1067:
1030:
1003:
979:
972:
964:
951:
950:
946:
928:
894:
886:
872:
871:
864:
815:
811:
797:
796:
789:
788:
781:
767:
757:
718:
686:
685:
650:
649:
612:
611:
563:
562:
529:
510:
506:
483:
482:
427:
410:
409:
403:
291:
277:
235:
224:
220:
202:
188:
184:
172:
136:
128:
113:
85:
81:
67:
66:
59:
58:
28:
23:
22:
15:
12:
11:
5:
4918:
4916:
4908:
4907:
4902:
4897:
4892:
4887:
4877:
4876:
4870:
4869:
4867:
4866:
4861:
4859:Predictability
4856:
4851:
4846:
4840:
4838:
4832:
4831:
4829:
4828:
4826:Lai-Sang Young
4823:
4821:James A. Yorke
4818:
4816:Amie Wilkinson
4813:
4808:
4803:
4798:
4793:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4751:Henri Poincaré
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4698:
4693:
4688:
4683:
4678:
4673:
4668:
4663:
4657:
4655:
4649:
4648:
4646:
4645:
4640:
4635:
4630:
4625:
4620:
4618:Kicked rotator
4615:
4610:
4605:
4600:
4595:
4590:
4588:Chua's circuit
4584:
4582:
4576:
4575:
4572:
4571:
4569:
4568:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4522:
4520:
4517:
4514:
4513:
4511:
4510:
4508:Zaslavskii map
4505:
4503:Tinkerbell map
4500:
4495:
4490:
4485:
4480:
4475:
4470:
4465:
4460:
4455:
4450:
4445:
4440:
4435:
4434:
4433:
4423:
4418:
4413:
4408:
4403:
4398:
4392:
4390:
4387:
4381:
4371:
4370:
4368:
4367:
4362:
4357:
4352:
4350:Ergodic theory
4347:
4342:
4337:
4331:
4329:
4323:
4322:
4307:
4305:
4302:
4301:
4299:
4298:
4293:
4288:
4283:
4278:
4273:
4268:
4262:
4260:
4257:
4254:
4253:
4251:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4210:
4205:
4200:
4195:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4154:
4151:
4150:
4148:
4147:
4142:
4140:Periodic point
4137:
4132:
4127:
4122:
4117:
4112:
4106:
4104:
4101:
4095:
4091:
4090:
4085:
4083:
4082:
4075:
4068:
4060:
4054:
4053:
4041:
4036:
4016:
4015:External links
4013:
4011:
4010:
3972:
3960:New York Times
3946:
3939:
3921:
3919:, pp. 213â214.
3900:
3847:(23): 234101.
3831:
3762:
3690:
3625:
3560:
3531:
3511:
3470:
3453:
3446:(in Russian).
3430:
3415:
3395:
3366:
3346:
3327:(3): 678â693.
3307:
3300:
3272:
3221:
3203:Grimshaw, R.,
3196:
3183:
3150:
3086:
3043:
2978:
2962:Kanamaru, T.,
2955:
2904:
2869:
2860:
2825:
2809:
2807:
2804:
2803:
2802:
2794:
2791:
2786:New York Times
2761:
2757:voltage source
2751:on the triode
2747:
2740:
2679:
2676:
2648:
2647:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2594:
2591:
2586:
2583:
2577:
2572:
2568:
2564:
2561:
2558:
2555:
2552:
2544:
2540:
2536:
2531:
2526:
2522:
2460:
2457:
2424:
2421:
2385:
2382:
2376:
2371:
2367:
2346:
2343:
2338:
2334:
2330:
2327:
2324:
2321:
2318:
2312:
2309:
2303:
2298:
2294:
2282:
2281:
2270:
2265:
2261:
2257:
2254:
2249:
2245:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2215:
2211:
2205:
2201:
2197:
2194:
2189:
2185:
2181:
2176:
2172:
2168:
2165:
2162:
2159:
2156:
2153:
2127:
2126:
2115:
2112:
2109:
2106:
2100:
2097:
2091:
2086:
2082:
2078:
2075:
2072:
2069:
2066:
2060:
2057:
2044:
2033:
2030:
2027:
2024:
2021:
2015:
2012:
2006:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1975:
1972:
1942:
1939:
1926:
1923:
1917:
1914:
1908:
1905:
1899:
1896:
1888:
1884:
1880:
1877:
1874:
1868:
1865:
1842:
1839:
1834:
1830:
1826:
1822:
1818:
1815:
1795:
1790:
1786:
1782:
1778:
1774:
1743:
1740:
1739:
1738:
1727:
1722:
1719:
1715:
1711:
1708:
1705:
1702:
1699:
1696:
1691:
1688:
1684:
1678:
1675:
1670:
1665:
1661:
1657:
1654:
1650:
1644:
1641:
1636:
1633:
1622:
1569:
1566:
1561:
1557:
1553:
1550:
1546:
1542:
1539:
1536:
1531:
1527:
1524:
1518:
1515:
1512:
1509:
1503:
1500:
1496:
1491:
1486:
1482:
1478:
1475:
1471:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1376:
1372:
1369:
1360:
1357:
1352:
1349:
1341:
1336:
1332:
1328:
1325:
1322:
1317:
1313:
1310:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1266:
1265:
1264:
1261:
1250:
1230:
1224:
1219:
1215:
1208:
1205:
1202:
1199:
1176:
1173:
1168:
1165:
1160:
1154:
1151:
1144:
1139:
1135:
1132:
1127:
1123:
1117:
1114:
1109:
1106:
1102:
1098:
1095:
1089:
1086:
1064:
1048:
1042:
1037:
1033:
1029:
1026:
1023:
1020:
1016:
1010:
1006:
1002:
999:
996:
992:
986:
982:
978:
975:
970:
967:
961:
958:
943:
925:
913:
910:
907:
900:
897:
892:
889:
882:
879:
860:
859:
839:
836:
833:
830:
822:
818:
814:
809:
804:
800:
756:
753:
752:
751:
739:
736:
733:
730:
725:
721:
717:
714:
711:
708:
705:
699:
696:
683:
672:
669:
663:
660:
631:
628:
622:
619:
608:
607:
595:
590:
587:
582:
576:
573:
560:
548:
544:
541:
536:
532:
525:
522:
516:
513:
509:
505:
502:
496:
493:
467:
463:
456:
453:
447:
444:
440:
434:
430:
426:
423:
420:
417:
402:
399:
325:, they become
297:and physicist
290:
287:
160:
157:
154:
151:
148:
142:
139:
134:
131:
125:
120:
116:
112:
109:
106:
103:
100:
92:
88:
84:
79:
74:
70:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4917:
4906:
4903:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4882:
4880:
4865:
4862:
4860:
4857:
4855:
4854:Edge of chaos
4852:
4850:
4847:
4845:
4842:
4841:
4839:
4833:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4811:Marcelo Viana
4809:
4807:
4804:
4802:
4801:Audrey Terras
4799:
4797:
4796:Floris Takens
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4697:
4696:Celso Grebogi
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4676:Chen Guanrong
4674:
4672:
4669:
4667:
4664:
4662:
4661:Michael Berry
4659:
4658:
4656:
4650:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4585:
4583:
4577:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4536:Lorenz system
4534:
4532:
4529:
4527:
4524:
4523:
4521:
4515:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4484:
4483:Langton's ant
4481:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4458:Horseshoe map
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4432:
4429:
4428:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4393:
4391:
4385:
4382:
4379:
4372:
4366:
4363:
4361:
4358:
4356:
4355:Quantum chaos
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4336:
4333:
4332:
4330:
4324:
4319:
4315:
4311:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4263:
4261:
4255:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4163:Arnold tongue
4161:
4159:
4156:
4155:
4152:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4107:
4105:
4099:
4096:
4092:
4088:
4081:
4076:
4074:
4069:
4067:
4062:
4061:
4058:
4052:
4048:
4045:
4042:
4040:
4037:
4033:
4029:
4028:
4023:
4019:
4018:
4014:
4006:
4002:
3998:
3994:
3990:
3986:
3982:
3976:
3973:
3961:
3957:
3950:
3947:
3942:
3940:0-14-009250-1
3936:
3932:
3925:
3922:
3918:
3914:
3910:
3904:
3901:
3896:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3855:
3850:
3846:
3842:
3835:
3832:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3791:
3786:
3781:
3778:(9): 094102.
3777:
3773:
3766:
3763:
3758:
3754:
3750:
3746:
3742:
3738:
3734:
3730:
3726:
3722:
3717:
3712:
3709:(4): 042213.
3708:
3704:
3697:
3695:
3691:
3686:
3682:
3678:
3674:
3670:
3666:
3662:
3658:
3653:
3648:
3645:(6): 062209.
3644:
3640:
3636:
3629:
3626:
3621:
3617:
3613:
3609:
3605:
3601:
3597:
3593:
3588:
3583:
3580:(6): 062927.
3579:
3575:
3571:
3564:
3561:
3556:
3550:
3542:
3538:
3534:
3528:
3524:
3523:
3515:
3512:
3507:
3503:
3498:
3493:
3489:
3485:
3481:
3474:
3471:
3466:
3465:
3457:
3454:
3449:
3445:
3441:
3434:
3431:
3426:
3422:
3418:
3416:0-8493-8607-1
3412:
3408:
3407:
3399:
3396:
3391:
3385:
3377:
3373:
3369:
3363:
3359:
3358:
3350:
3347:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3311:
3308:
3303:
3297:
3293:
3289:
3285:
3284:
3276:
3273:
3268:
3264:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3225:
3222:
3218:
3217:0-8493-8607-1
3214:
3210:
3206:
3200:
3197:
3193:
3187:
3184:
3179:
3175:
3171:
3167:
3163:
3162:
3154:
3151:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3113:
3108:
3104:
3100:
3096:
3090:
3087:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3047:
3044:
3039:
3035:
3030:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2997:
2993:
2989:
2982:
2979:
2975:
2971:
2970:
2965:
2959:
2956:
2951:
2947:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2915:
2908:
2905:
2900:
2896:
2892:
2888:
2884:
2880:
2873:
2870:
2864:
2861:
2856:
2852:
2848:
2844:
2840:
2836:
2829:
2826:
2823:
2820:
2819:bdtoolbox.org
2814:
2811:
2805:
2800:
2797:
2796:
2792:
2790:
2788:
2787:
2782:
2781:
2776:
2772:
2764:
2758:
2754:
2750:
2743:
2736:
2732:
2728:
2724:
2721:
2717:
2714:
2710:
2706:
2703:
2699:
2696:
2692:
2688:
2684:
2677:
2675:
2673:
2665:
2664:wave function
2661:
2657:
2634:
2631:
2628:
2622:
2619:
2613:
2610:
2607:
2604:
2601:
2598:
2592:
2589:
2584:
2581:
2570:
2566:
2562:
2559:
2553:
2550:
2542:
2538:
2534:
2529:
2524:
2520:
2509:
2508:
2507:
2503:
2499:
2488:
2481:
2474:
2469:
2465:
2458:
2456:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2422:
2420:
2418:
2414:
2410:
2406:
2402:
2383:
2380:
2374:
2369:
2365:
2344:
2336:
2332:
2328:
2325:
2319:
2316:
2310:
2307:
2301:
2296:
2292:
2268:
2263:
2259:
2255:
2247:
2243:
2239:
2236:
2230:
2227:
2224:
2221:
2218:
2213:
2209:
2203:
2199:
2195:
2187:
2183:
2179:
2174:
2170:
2166:
2163:
2160:
2157:
2151:
2144:
2143:
2142:
2140:
2136:
2132:
2113:
2110:
2107:
2104:
2098:
2095:
2084:
2080:
2076:
2073:
2067:
2064:
2058:
2055:
2045:
2031:
2028:
2025:
2022:
2019:
2013:
2010:
1999:
1995:
1991:
1988:
1982:
1979:
1973:
1970:
1960:
1959:
1958:
1956:
1947:
1940:
1938:
1924:
1921:
1915:
1912:
1906:
1903:
1897:
1894:
1886:
1882:
1878:
1875:
1872:
1866:
1863:
1840:
1837:
1832:
1828:
1824:
1820:
1816:
1813:
1793:
1788:
1784:
1780:
1776:
1772:
1764:
1759:
1755:
1741:
1720:
1717:
1713:
1706:
1703:
1700:
1697:
1694:
1689:
1686:
1682:
1676:
1673:
1668:
1663:
1659:
1655:
1652:
1648:
1642:
1639:
1634:
1631:
1623:
1620:
1615:
1608:
1602:
1601:Airy function
1592:
1584:
1567:
1559:
1555:
1551:
1548:
1544:
1537:
1534:
1529:
1525:
1522:
1516:
1513:
1510:
1507:
1501:
1498:
1494:
1489:
1484:
1480:
1476:
1473:
1469:
1465:
1462:
1459:
1456:
1450:
1447:
1444:
1441:
1438:
1435:
1429:
1426:
1406:
1400:
1397:
1394:
1391:
1388:
1385:
1379:
1374:
1370:
1367:
1358:
1355:
1350:
1347:
1339:
1334:
1330:
1326:
1323:
1320:
1315:
1311:
1308:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1267:
1262:
1251:
1228:
1222:
1217:
1213:
1206:
1203:
1200:
1197:
1189:
1188:
1174:
1171:
1166:
1163:
1158:
1152:
1149:
1142:
1137:
1133:
1130:
1125:
1121:
1115:
1112:
1107:
1104:
1100:
1096:
1093:
1087:
1084:
1073:
1065:
1062:
1046:
1035:
1031:
1024:
1021:
1018:
1014:
1008:
1004:
1000:
997:
994:
990:
984:
980:
976:
973:
968:
965:
959:
956:
944:
941:
935:
931:
926:
911:
908:
905:
898:
895:
890:
887:
880:
877:
867:
862:
861:
857:
853:
837:
834:
831:
828:
820:
816:
812:
807:
802:
798:
784:
779:
778:
777:
770:
765:
761:
754:
737:
734:
731:
723:
719:
715:
712:
706:
703:
697:
694:
684:
670:
667:
661:
658:
648:
647:
646:
629:
626:
620:
617:
593:
588:
585:
580:
574:
571:
561:
546:
542:
539:
534:
530:
523:
520:
514:
511:
507:
503:
500:
494:
491:
481:
480:
479:
465:
461:
454:
451:
445:
442:
438:
432:
428:
424:
421:
418:
415:
407:
400:
398:
397:oscillators.
396:
392:
388:
384:
380:
376:
372:
368:
364:
361:
357:
352:
350:
346:
343:an irregular
342:
338:
337:
332:
328:
324:
320:
316:
312:
308:
304:
300:
296:
288:
283:
275:
271:
266:
258:
251:
230:
218:
200:
198:
194:
182:
178:
158:
155:
152:
149:
146:
140:
137:
132:
129:
118:
114:
110:
107:
101:
98:
90:
86:
82:
77:
72:
68:
57:
53:
49:
45:
41:
37:
33:
19:
4885:Chaotic maps
4806:Mary Tsingou
4771:David Ruelle
4766:Otto Rössler
4711:Michel HĂ©non
4681:Leon O. Chua
4638:Tilt-A-Whirl
4608:FPUT problem
4565:
4493:Standard map
4488:Logistic map
4313:
4087:Chaos theory
4025:
3988:
3975:
3963:. Retrieved
3959:
3949:
3930:
3924:
3908:
3903:
3844:
3840:
3834:
3775:
3771:
3765:
3706:
3702:
3642:
3638:
3628:
3577:
3573:
3563:
3521:
3514:
3487:
3483:
3473:
3463:
3456:
3447:
3443:
3433:
3405:
3398:
3356:
3349:
3324:
3320:
3310:
3282:
3275:
3234:
3230:
3224:
3204:
3199:
3191:
3186:
3160:
3153:
3102:
3098:
3089:
3056:
3052:
3046:
2995:
2991:
2981:
2973:
2969:Scholarpedia
2967:
2958:
2917:
2913:
2907:
2882:
2878:
2872:
2863:
2838:
2828:
2813:
2784:
2778:
2773:described a
2771:James Gleick
2768:
2759:
2753:control grid
2745:
2738:
2730:
2722:
2715:
2707:, a coupled
2704:
2697:
2689:involving a
2660:displacement
2649:
2501:
2497:
2494:
2486:
2479:
2472:
2452:
2448:
2440:
2426:
2417:Hannay angle
2408:
2404:
2283:
2138:
2134:
2130:
2128:
1952:
1760:
1753:
1745:
1613:
1606:
1590:
1582:
1071:
933:
929:
865:
782:
776:
768:
609:
404:
367:planar field
353:
334:
319:vacuum tubes
292:
197:nonlinearity
183:of the time
179:âwhich is a
44:conservative
35:
29:
4791:Nina Snaith
4781:Yakov Sinai
4666:Rufus Bowen
4416:Duffing map
4401:Baker's map
4326:Theoretical
4238:SRB measure
4145:Phase space
4115:Bifurcation
3490:: 620â622.
2775:vacuum tube
2727:RLC circuit
1955:Hamiltonian
1853:which gives
341:frequencies
323:limit cycle
311:limit cycle
274:phase plane
270:limit cycle
48:oscillating
42:) is a non-
4879:Categories
4849:Complexity
4746:Edward Ott
4593:Convection
4518:Continuous
4193:Ergodicity
3917:0719018110
3716:1711.07376
3652:1610.05218
3587:1512.06758
3541:1112373147
3450:: 313â328.
2806:References
2711:-set with
1066:For large
645:leads to:
395:vocal fold
383:seismology
360:biological
317:employing
229:nullclines
217:phase plot
177:coordinate
4761:Mary Rees
4721:Bryna Kra
4654:theorists
4463:Ikeda map
4453:HĂ©non map
4443:Gauss map
4125:Limit set
4110:Attractor
4032:EMS Press
4005:0024-6107
3879:0031-9007
3854:1306.6359
3810:0031-9007
3785:1307.7044
3757:224801468
3741:2470-0045
3549:cite book
3506:1385-7258
3384:cite book
3376:851704808
3341:0036-1399
3267:120504403
3259:0044-2267
3209:CRC Press
3178:1939-800X
3137:0218-1274
3073:0096-8390
3020:0006-3495
2950:186244992
2942:0028-0836
2899:1941-5982
2855:0024-6107
2702:capacitor
2662:, of the
2656:amplitude
2620:ω
2614:
2605:−
2563:−
2554:μ
2551:−
2384:˙
2329:−
2320:μ
2311:˙
2240:−
2231:μ
2228:−
2099:˙
2077:−
2068:μ
2059:¨
2014:˙
1992:−
1983:μ
1980:−
1974:¨
1916:˙
1907:ε
1904:−
1898:˙
1867:¨
1821:ε
1777:ε
1773:∝
1718:−
1714:μ
1701:μ
1698:
1687:−
1683:μ
1669:−
1653:−
1649:μ
1640:α
1549:−
1545:μ
1530:μ
1526:⋯
1514:μ
1511:
1502:μ
1490:−
1474:−
1470:μ
1466:α
1457:μ
1448:
1439:−
1407:μ
1398:
1389:−
1331:∫
1327:μ
1303:μ
1300:∫
1285:∫
1258:(â2, 2/3)
1243:(2, â2/3)
1207:−
1167:μ
1153:˙
1131:−
1108:−
1097:μ
1088:˙
1032:μ
1005:μ
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969:π
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698:˙
662:˙
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455:˙
446:−
425:−
391:phonation
327:entrained
111:−
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4837:articles
4579:Physical
4498:Tent map
4388:Discrete
4328:branches
4258:Theorems
4094:Concepts
4047:Archived
3895:33622111
3887:24476274
3818:24655255
3749:33212685
3685:51635019
3677:30011548
3620:14930486
3612:26764794
3425:28275539
3081:51648050
3038:19431309
2793:See also
2709:inductor
2695:resistor
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1254:(1, 2/3)
1247:(1, 2/3)
363:sciences
356:physical
181:function
4835:Related
4643:Weather
4581:systems
4374:Chaotic
4120:Fractal
4034:, 2001
3965:11 July
3859:Bibcode
3826:7950471
3790:Bibcode
3721:Bibcode
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3592:Bibcode
3239:Bibcode
3145:9120223
3117:Bibcode
3029:1366333
3000:Bibcode
2922:Bibcode
2769:Author
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379:neurons
303:Philips
289:History
272:in the
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336:Nature
331:signal
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