4903:
5402:
5324:
4668:
4590:
4512:
4434:
5137:
5059:
4981:
4356:
4278:
4200:
4122:
2719:
1494:
7248:
2321:
1113:
31:
1048:
1843:
7240:
2714:{\displaystyle {\begin{aligned}{\dot {r}}&={\frac {\partial {\mathcal {H}}}{\partial {p_{r}}}}={\frac {p_{r}}{M+m}}\\{\dot {p_{r}}}&=-{\frac {\partial {\mathcal {H}}}{\partial {r}}}={\frac {p_{\theta }^{2}}{mr^{3}}}-Mg+mg\cos {\theta }\\{\dot {\theta }}&={\frac {\partial {\mathcal {H}}}{\partial {p_{\theta }}}}={\frac {p_{\theta }}{mr^{2}}}\\{\dot {p_{\theta }}}&=-{\frac {\partial {\mathcal {H}}}{\partial {\theta }}}=-mgr\sin {\theta }\end{aligned}}}
3330:
1489:{\displaystyle {\begin{aligned}p_{r}&={\frac {\partial {\mathcal {L}}}{\partial {\dot {r}}}}={\frac {\partial T}{\partial {\dot {r}}}}=(M+m){\dot {r}}\\p_{\theta }&={\frac {\partial {\mathcal {L}}}{\partial {\dot {\theta }}}}={\frac {\partial T}{\partial {\dot {\theta }}}}=mr^{2}{\dot {\theta }}\\\therefore {\mathcal {H}}&={\frac {p_{r}^{2}}{2(M+m)}}+{\frac {p_{\theta }^{2}}{2mr^{2}}}+Mgr-mgr\cos {\theta }\end{aligned}}}
670:
5487:
4753:
1565:
2073:
3675:
3031:
474:
1043:{\displaystyle {\begin{aligned}{\mathcal {L}}&=T-U\\&={\frac {1}{2}}M{\dot {r}}^{2}+{\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-Mgr+mgr\cos {\theta }\\{\mathcal {H}}&=T+U\\&={\frac {1}{2}}M{\dot {r}}^{2}+{\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)+Mgr-mgr\cos {\theta }\end{aligned}}}
1838:{\displaystyle {\begin{aligned}{\frac {\partial {\mathcal {L}}}{\partial \theta }}&={\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\theta }}}}\right)\\-mgr\sin {\theta }&=2mr{\dot {r}}{\dot {\theta }}+mr^{2}{\ddot {\theta }}\\r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}+g\sin {\theta }&=0\end{aligned}}}
4902:
5401:
5323:
4667:
4589:
4511:
4433:
5944:
5136:
5058:
4980:
4355:
4277:
4199:
4121:
1874:
6218:
3453:
3325:{\displaystyle {\mathcal {H}}\left(r,\theta ,{\dot {r}},{\dot {\theta }}\right)=\underbrace {{\frac {1}{2}}M_{t}\left(R{\dot {\theta }}-{\dot {r}}\right)^{2}+{\frac {1}{2}}mr^{2}{\dot {\theta }}^{2}} _{T}+\underbrace {gr\left(M-m\cos {\theta }\right)+gR\left(m\sin {\theta }-M\theta \right)} _{U},}
266:
2068:{\displaystyle {\begin{aligned}{\frac {\partial {\mathcal {L}}}{\partial r}}&={\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {r}}}}\right)\\mr{\dot {\theta }}^{2}-Mg+mg\cos {\theta }&=(M+m){\ddot {r}}\end{aligned}}}
5726:
6086:
3670:{\displaystyle {\begin{aligned}\mu _{t}({\ddot {r}}-R{\ddot {\theta }})&=r{\dot {\theta }}^{2}+g(\cos {\theta }-\mu )\\r{\ddot {\theta }}&=-2{\dot {r}}{\dot {\theta }}+R{\dot {\theta }}^{2}-g\sin {\theta }\\\end{aligned}}}
4094:
The swinging mass of the swinging Atwood's machine undergoes interesting trajectories or orbits when subject to different initial conditions, and for different mass ratios. These include periodic orbits and collision orbits.
2216:
5521:
Collision (or terminating singular) orbits are subset of singular orbits formed when the swinging mass is ejected from its pivot with an initial velocity, such that it returns to the pivot (i.e. it collides with the pivot):
4085:
is small (near vertical), the trajectory describes a "teardrop", when it is large, it describes a "heart". These trajectories can be exactly solved algebraically, which is unusual for a system with a non-linear
Hamiltonian.
611:
2871:
6055:
4029:
469:{\displaystyle {\begin{aligned}T&={\frac {1}{2}}Mv_{M}^{2}+{\frac {1}{2}}mv_{m}^{2}\\&={\frac {1}{2}}M{\dot {r}}^{2}+{\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)\end{aligned}}}
4879:
185:
6091:
3458:
2326:
1879:
1570:
1118:
675:
554:
271:
5471:
5393:
5206:
5128:
5050:
4972:
4737:
4659:
4581:
4503:
4425:
4347:
4269:
4191:
3935:
5586:
3402:
6946:
4787:
When the different harmonic components in the system are in phase, the resulting trajectory is simple and periodic, such as the "smile" trajectory, which resembles that of an ordinary
3853:
5591:
The simplest case of collision orbits are the ones with a mass ratio of 3, which will always return symmetrically to the origin after being ejected from the origin, and were termed
6960:
5718:
2109:
6272:
6223:
Note that the numerator is a constant dependent only on the initial position in this case, as we have assumed the initial condition to be at rest. However, the energy constant
3726:
2771:
34:
The
Swinging Atwood's machine. The smaller mass, labelled m, is allowed to swing freely whereas the larger mass, M, can only move up and down. Assume the pivots to be points.
662:
638:
5939:{\displaystyle E={\frac {1}{2}}M{\dot {r}}^{2}+{\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)+Mgr-mgr\cos {\theta }=Mgr_{0}-mgr_{0}\cos {\theta _{0}}}
5515:
5088:
4781:
4083:
4056:
2972:
2313:
1105:
5166:
5010:
4932:
3781:
6350:
6298:
6380:
A new integrable case for the problem of three dimensional
Swinging Atwood Machine (3D-SAM) was announced in 2016. Like the 2D version, the problem is integrable when
5685:
5431:
5353:
4697:
4619:
4541:
4463:
2918:
2742:
2259:
1557:
1537:
517:
6324:
5261:
4385:
4307:
4229:
4151:
3879:
142:
2945:
2286:
1078:
6407:
6370:
116:
6928:
Olivier, Pujol, JP Perez, JP Ramis, C. Simo, S. Simon, JA Weil (2010), "Swinging Atwood's
Machine: Experimental and numerical results, and a theoretical study,"
6213:{\displaystyle {\begin{aligned}r&={\frac {h}{1-{\frac {\cos {\theta }}{\mu }}}}\\h&=r_{0}\left(1-{\frac {\cos {\theta _{0}}}{\mu }}\right)\end{aligned}}}
6241:
6078:
5652:
5634:
of this bounding curve. The equation for this curve can be derived by analyzing the energy of the system, and using conservation of energy. Let us suppose that
5616:
5304:
5284:
3445:
3425:
3023:
3003:
2898:
2239:
1866:
1517:
537:
497:
258:
238:
218:
6552:
Pujol, Olivier; Perez, J.P.; Simo, C.; Simon, S.; Weil, J.A. (2010). "Swinging Atwood's
Machine: Experimental and numerical results, and a theoretical study".
165:
5618:
must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time.
2117:
6852:
549:
2783:
5955:
3947:
17:
7537:
7367:
6887:
6835:
Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems",
5595:
orbits in
Tufillaro's initial paper. They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance.
6919:
Ouazzani-T.H., A. and
Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine,"
4797:
6372:
gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region.
7247:
7007:
6944:
Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine",
7200:
5225:
under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards:
192:
The swinging Atwood's machine is a system with two degrees of freedom. We may derive its equations of motion using either
7147:
7480:
6761:
7715:
7522:
7210:
6910:
Nunes, A., J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine,"
7397:
6975:
5436:
5358:
5171:
5093:
5015:
4937:
4702:
4624:
4546:
4468:
4390:
4312:
4234:
4156:
3884:
85:
of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight.
5528:
3348:
6796:
Elmandouh, A.A. (2016). "On the integrability of the motion of 3D-Swinging Atwood machine and related problems".
7814:
3786:
160:
counteracting the counterweight's weight. Research on the SAM started as part of a 1982 senior thesis entitled
7215:
7205:
6876:
Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics,"
6488:
Tufillaro, Nicholas B.; Nunes, A.; Casasayas, J. (1988). "Unbounded orbits of a swinging Atwood's machine".
7470:
7335:
6837:
6300:, this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For
5221:
The motion is singular if at some point, the swinging mass passes through the origin. Since the system is
148:
of conditions that lead to a variety of motions that can be classified as terminating or non-terminating,
6937:
6912:
6896:
6869:
6680:
6601:
6490:
6444:
5690:
4031:), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of
2081:
6867:
Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine,"
7475:
7000:
6250:
6244:
1499:
Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in
157:
7542:
3881:
case is the only known integrable mass ratio. It has been shown that the system is not integrable for
3683:
2747:
7590:
5949:
However, notice that in the boundary case, the velocity of the swinging mass is zero. Hence we have:
6243:
can also be calculated for nonzero initial velocity, and the equation still holds in all cases. The
643:
619:
7294:
6965:
6885:
Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood
Machine. Far- and near-resonance region",
6851:
Babelon, O, M. Talon, MC Peyranere (2010), "Kowalevski's analysis of a swinging Atwood's machine,"
6442:
Tufillaro, Nicholas B.; Abbott, Tyler A.; Griffiths, David J. (1984). "Swinging Atwood's
Machine".
7239:
7455:
7220:
7127:
5626:
For any initial position, it can be shown that the swinging mass is bounded by a curve that is a
5493:
5067:
4759:
4104:
4061:
4034:
2950:
2291:
1083:
7195:
5145:
4989:
4911:
3746:
519:
is the angle of the swinging mass relative to pointing straight downwards. The potential energy
7495:
7107:
6637:
6329:
6277:
6894:
Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,""
7645:
7552:
7350:
7177:
7112:
7087:
6993:
5657:
5410:
5332:
4676:
4598:
4520:
4442:
2903:
2727:
2244:
1542:
1522:
502:
193:
6903:
Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the
Swinging Atwood Machine",
6303:
5231:
4364:
4286:
4208:
4130:
3858:
121:
7655:
7407:
7307:
7152:
6807:
6699:
6678:
Tufillaro, Nicholas B. (1994). "Teardrop and heart orbits of a swinging Atwoods machine,".
6610:
6573:
6499:
6453:
2974:. However, due to energy conservation, the phase space is constrained to three dimensions.
2923:
2264:
1056:
197:
7485:
6383:
6355:
8:
7615:
7572:
7557:
7402:
7355:
7340:
7325:
7225:
7132:
7117:
7102:
6878:
6769:
6730:
5222:
4791:, and various loops. In general a periodic orbit exists when the following is satisfied:
95:
78:
7157:
6980:
6811:
6782:
6703:
6653:
6614:
6577:
6503:
6457:
2221:
Hamiltonian analysis may also be applied to determine four first order ODEs in terms of
7793:
7660:
7490:
7377:
7372:
7264:
7142:
7044:
6921:
6858:
Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators,"
6743:
6689:
6563:
6226:
6063:
5637:
5601:
5289:
5269:
3736:
3430:
3410:
3008:
2988:
2883:
2224:
1851:
1502:
522:
482:
243:
223:
203:
173:
59:
92:
either the pendulum or counterweight eventually collides with its pulley), except for
7665:
7630:
7620:
7517:
7137:
7059:
6798:
6657:
3740:
2983:
149:
7680:
7773:
7685:
7635:
7532:
7460:
7412:
7289:
7274:
7269:
7064:
6819:
6815:
6778:
6739:
6707:
6649:
6618:
6599:
Tufillaro, Nicholas B. (1986). "Integrable motion of a swinging Atwood's machine".
6581:
6507:
6461:
5631:
540:
51:
4884:
The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed
4107:. The orbit is called nonsingular if the swinging mass does not touch the pulley.
2211:{\displaystyle (\mu +1){\ddot {r}}-r{\dot {\theta }}^{2}+g(\mu -\cos {\theta })=0}
184:
7705:
7600:
7527:
7360:
7172:
7162:
6860:
156:
or chaotic, bounded or unbounded, singular or non-singular due to the pendulum's
145:
7695:
7640:
6970:
6585:
3944:
Numerical studies indicate that when the orbit is singular (initial conditions:
3937:. For many other values of the mass ratio (and initial conditions) SAM displays
7788:
7755:
7750:
7745:
7547:
7437:
7432:
7330:
7279:
7069:
2774:
43:
5306:. As such, special techniques must be used to rigorously analyze these cases.
7808:
7783:
7740:
7730:
7725:
7625:
7605:
7465:
7387:
7284:
7092:
6848:, B.S. Thesis, National Institute of Physics, University of the Philippines.
6661:
5627:
153:
7735:
7700:
7610:
7567:
7422:
7417:
7016:
6535:
3938:
169:
55:
30:
7382:
606:{\displaystyle {\begin{aligned}U&=Mgr-mgr\cos {\theta }\end{aligned}}}
7720:
7710:
7595:
7345:
7167:
7074:
6728:
Tufillaro, Nicholas B. (1985). "Motions of a swinging Atwood's machine".
6694:
2877:
2866:{\displaystyle {\ddot {r}}=g{\frac {1-\mu }{1+\mu }}=g{\frac {m-M}{m+M}}}
47:
7778:
7675:
7122:
6050:{\displaystyle Mgr-mgr\cos {\theta }=Mgr_{0}-mgr_{0}\cos {\theta _{0}}}
4024:{\displaystyle r=0,{\dot {r}}=v,\theta =\theta _{0},{\dot {\theta }}=0}
1053:
We can then express the Hamiltonian in terms of the canonical momenta,
5598:
When the swinging mass returns to the origin, the counterweight mass,
7690:
7650:
7392:
7054:
7039:
6935:
Sears, R. (1995) "Comment on "A surprising mechanics demonstration,"
6929:
6554:
6711:
6622:
6511:
6465:
7427:
6352:
it is a hyperbola; in either of these cases, it is not bounded. As
4788:
88:
The conventional Atwood's machine allows only "runaway" solutions (
66:
6568:
6060:
To see that it is the equation of a conic section, we isolate for
7097:
7049:
5309:
The following are plots of arbitrarily selected singular orbits.
5286:
is close to zero and the equations of motion require dividing by
82:
2773:
to zero, the resulting special case is the regular non-swinging
7670:
6985:
6981:
Open source Java code for running the Swinging Atwood's Machine
164:(referring to the shape of some trajectories of the system) by
6638:"Swinging Atwood's Machine : integrability and dynamics"
5486:
4874:{\displaystyle r(t+\tau )=r(t),\,\theta (t+\tau )=\theta (t)}
4752:
3743:
and nonintegrable. SAM is integrable when the mass ratio
6976:
2010 Videos of an experimental Swinging Atwood's Machine
6441:
6635:
6487:
499:
is the distance of the swinging mass to its pivot, and
6525:
6523:
6521:
2724:
Notice that in both of these derivations, if one sets
27:
Variation of Atwood's machine incorporating a pendulum
6386:
6375:
6358:
6332:
6306:
6280:
6253:
6229:
6089:
6066:
5958:
5729:
5693:
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5640:
5604:
5531:
5496:
5439:
5413:
5361:
5335:
5292:
5272:
5234:
5174:
5148:
5096:
5070:
5018:
4992:
4940:
4914:
4908:
A "smile" orbit of the swinging Atwood's machine for
4800:
4762:
4705:
4679:
4627:
4601:
4549:
4523:
4471:
4445:
4393:
4367:
4315:
4289:
4237:
4211:
4159:
4133:
4064:
4037:
3950:
3887:
3861:
3789:
3749:
3686:
3456:
3433:
3413:
3351:
3034:
3011:
2991:
2953:
2926:
2906:
2886:
2876:
The swinging Atwood's machine has a four-dimensional
2786:
2750:
2730:
2324:
2294:
2267:
2247:
2227:
2120:
2084:
2078:
We simplify the equations by defining the mass ratio
1877:
1854:
1568:
1545:
1525:
1505:
1116:
1086:
1059:
673:
646:
622:
552:
525:
505:
485:
269:
246:
226:
206:
124:
98:
6673:
6671:
6551:
6518:
6853:Journal of Physics A: Mathematical and Theoretical
6401:
6364:
6344:
6318:
6292:
6266:
6235:
6212:
6072:
6049:
5938:
5712:
5679:
5646:
5610:
5580:
5509:
5465:
5425:
5387:
5347:
5298:
5278:
5255:
5200:
5160:
5122:
5082:
5044:
5004:
4966:
4926:
4873:
4775:
4731:
4691:
4653:
4613:
4575:
4535:
4497:
4457:
4419:
4379:
4341:
4301:
4263:
4223:
4185:
4145:
4077:
4050:
4023:
3929:
3873:
3847:
3775:
3720:
3669:
3439:
3419:
3396:
3324:
3017:
2997:
2966:
2939:
2912:
2892:
2865:
2765:
2736:
2713:
2307:
2280:
2253:
2233:
2210:
2103:
2067:
1860:
1837:
1551:
1531:
1511:
1488:
1099:
1072:
1042:
656:
632:
605:
531:
511:
491:
468:
252:
232:
212:
136:
110:
6762:"Collision orbits of a swinging Atwood's machine"
5266:The region close to the pivot is singular, since
188:Motion of Swinging Atwood's Machine for M/m = 4.5
7806:
6668:
6636:Casasayas, J.; Nunes, A.; Tufillaro, N. (1990).
81:, massless string suspended on two frictionless
6437:
6435:
6433:
6431:
6429:
6427:
6425:
6423:
6421:
5720:. The total energy of the system is therefore:
2982:If the pulleys in the system are taken to have
6846:Dynamics of a Double-Swinging Atwood's machine
5407:An orbit of the swinging Atwood's machine for
5329:An orbit of the swinging Atwood's machine for
5142:An orbit of the swinging Atwood's machine for
5064:An orbit of the swinging Atwood's machine for
4986:An orbit of the swinging Atwood's machine for
4673:An orbit of the swinging Atwood's machine for
4595:An orbit of the swinging Atwood's machine for
4517:An orbit of the swinging Atwood's machine for
4439:An orbit of the swinging Atwood's machine for
4361:An orbit of the swinging Atwood's machine for
4283:An orbit of the swinging Atwood's machine for
4205:An orbit of the swinging Atwood's machine for
4127:An orbit of the swinging Atwood's machine for
3447:become zero. The equations of motion are now:
2977:
118:. However, the swinging Atwood's machine with
7001:
5466:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
5388:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
5201:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
5123:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
5045:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4967:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4732:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4654:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4576:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4498:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4420:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4342:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4264:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
4186:{\displaystyle \theta _{0}={\frac {\pi }{2}}}
3930:{\displaystyle \mu \in (0,1)\cup (3,\infty )}
42:(SAM) is a mechanism that resembles a simple
6888:International Journal of Theoretical Physics
6418:
5581:{\displaystyle r(\tau )=r(0)=0,\,\tau >0}
3397:{\displaystyle M_{t}=M+m+{\frac {I}{R^{2}}}}
46:except that one of the masses is allowed to
6755:
6753:
6547:
6545:
3783:. The system also looks pretty regular for
3342:is the effective total mass of the system,
65:Specifically, it comprises two masses (the
7008:
6994:
2920:and their corresponding canonical momenta
2261:and their corresponding canonical momenta
6795:
6759:
6727:
6723:
6721:
6693:
6677:
6598:
6592:
6567:
6529:
5568:
4837:
3848:{\displaystyle \mu =4n^{2}-1=3,15,35,...}
6750:
6542:
5485:
4751:
4103:For certain conditions, system exhibits
183:
50:in a two-dimensional plane, producing a
29:
6483:
6481:
6479:
6477:
6475:
3407:This reduces to the version above when
616:We may then write down the Lagrangian,
14:
7807:
6789:
6718:
3025:, the Hamiltonian of the SAM is then:
179:
6989:
4098:
6966:Oscilaciones en la máquina de Atwood
6472:
240:. The kinetic energy of the system,
7148:Measure-preserving dynamical system
7030:
5713:{\displaystyle \theta =\theta _{0}}
5481:
2104:{\displaystyle \mu ={\frac {M}{m}}}
24:
6829:
6376:Recent three dimensional extension
5216:
4747:
3921:
3037:
2670:
2663:
2658:
2572:
2565:
2560:
2457:
2450:
2445:
2363:
2356:
2351:
1947:
1940:
1935:
1897:
1890:
1885:
1638:
1631:
1626:
1588:
1581:
1576:
1356:
1301:
1293:
1269:
1262:
1257:
1185:
1177:
1153:
1146:
1141:
861:
680:
649:
625:
25:
7826:
7716:Oleksandr Mykolayovych Sharkovsky
6954:
6947:Mechanics Research Communications
6783:10.1051/jphys:0198500460120205300
6654:10.1051/jphys:0199000510160169300
6267:{\displaystyle {\frac {1}{\mu }}}
7246:
7238:
7015:
6744:10.1051/jphys:019850046090149500
5400:
5322:
5135:
5057:
4979:
4901:
4666:
4588:
4510:
4432:
4354:
4276:
4198:
4120:
3731:
3721:{\displaystyle \mu _{t}=M_{t}/m}
2766:{\displaystyle {\dot {\theta }}}
6760:Tufillaro, Nicholas B. (1985).
6530:Tufillaro, Nicholas B. (1982).
4112:Selection of nonsingular orbits
4089:
58:for some system parameters and
7481:Rabinovich–Fabrikant equations
6820:10.1016/j.physleta.2016.01.021
6629:
5621:
5556:
5550:
5541:
5535:
5244:
5238:
4868:
4862:
4853:
4841:
4831:
4825:
4816:
4804:
3924:
3912:
3906:
3894:
3562:
3542:
3504:
3471:
2199:
2179:
2133:
2121:
2046:
2034:
1401:
1389:
1218:
1206:
657:{\displaystyle {\mathcal {H}}}
633:{\displaystyle {\mathcal {L}}}
18:Swinging Atwood's Machine
13:
1:
6971:"Smiles and Teardrops" (1982)
6844:Barrera, Emmanuel Jan (2003)
6412:
220:and the non-swinging mass be
6838:Brazilian Journal of Physics
5473:, and zero initial velocity.
5395:, and zero initial velocity.
5314:Selection of singular orbits
5208:, and zero initial velocity.
5130:, and zero initial velocity.
5052:, and zero initial velocity.
4974:, and zero initial velocity.
4893:Selection of periodic orbits
4739:, and zero initial velocity.
4661:, and zero initial velocity.
4583:, and zero initial velocity.
4505:, and zero initial velocity.
4427:, and zero initial velocity.
4349:, and zero initial velocity.
4271:, and zero initial velocity.
4193:, and zero initial velocity.
7:
7216:Poincaré recurrence theorem
6938:American Journal of Physics
6913:American Journal of Physics
6897:American Journal of Physics
6870:European Journal of Physics
6681:American Journal of Physics
6602:American Journal of Physics
6586:10.1016/j.physd.2010.02.017
6491:American Journal of Physics
6445:American Journal of Physics
6326:, it is a parabola and for
5510:{\displaystyle \theta _{0}}
5083:{\displaystyle \mu =1.1727}
4776:{\displaystyle \theta _{0}}
4078:{\displaystyle \theta _{0}}
4051:{\displaystyle \theta _{0}}
2978:System with massive pulleys
2967:{\displaystyle p_{\theta }}
2308:{\displaystyle p_{\theta }}
1100:{\displaystyle p_{\theta }}
541:acceleration due to gravity
200:. Let the swinging mass be
10:
7831:
7211:Poincaré–Bendixson theorem
6891:, Vol. 27(11), p1405-1410.
6841:Vol.28 n.4 SĂŁo Paulo Dec.
5161:{\displaystyle \mu =1.555}
5005:{\displaystyle \mu =2.394}
4927:{\displaystyle \mu =1.665}
4888:orbits in his 1984 paper.
3776:{\displaystyle \mu =M/m=3}
2111:. The above then becomes:
158:reactive centrifugal force
7764:
7581:
7563:Swinging Atwood's machine
7508:
7446:
7316:
7303:
7255:
7236:
7206:Krylov–Bogolyubov theorem
7186:
7083:
7023:
6345:{\displaystyle \mu <1}
6293:{\displaystyle \mu >1}
5654:is released from rest at
40:swinging Atwood's machine
7471:Lotka–Volterra equations
7295:Synchronization of chaos
7098:axiom A dynamical system
6932:239, pp. 1067–1081.
6247:of the conic section is
5630:. The pivot is always a
5517:ranging from 0.1 to 3.1.
4783:ranging from 0.1 to 3.1.
73:and counterweight, mass
7456:Double scroll attractor
7221:Stable manifold theorem
7128:False nearest neighbors
6961:Imperial College Course
6941:, Vol. 63(9), p854-855.
5680:{\displaystyle r=r_{0}}
5426:{\displaystyle \mu =25}
5348:{\displaystyle \mu =10}
4692:{\displaystyle \mu =24}
4614:{\displaystyle \mu =21}
4536:{\displaystyle \mu =19}
4458:{\displaystyle \mu =16}
4105:complex harmonic motion
2913:{\displaystyle \theta }
2737:{\displaystyle \theta }
2254:{\displaystyle \theta }
1552:{\displaystyle \theta }
1532:{\displaystyle \theta }
640:, and the Hamiltonian,
512:{\displaystyle \theta }
7496:Van der Pol oscillator
7476:Mackey–Glass equations
7108:Box-counting dimension
6950:Vol. 33 (5), p711–716.
6907:II France 1, p711-715.
6403:
6366:
6346:
6320:
6319:{\displaystyle \mu =1}
6294:
6268:
6237:
6214:
6074:
6051:
5940:
5714:
5681:
5648:
5612:
5582:
5518:
5511:
5467:
5427:
5389:
5349:
5300:
5280:
5257:
5256:{\displaystyle r(0)=0}
5202:
5162:
5124:
5084:
5046:
5006:
4968:
4928:
4875:
4784:
4777:
4733:
4693:
4655:
4615:
4577:
4537:
4499:
4459:
4421:
4381:
4380:{\displaystyle \mu =6}
4343:
4303:
4302:{\displaystyle \mu =5}
4265:
4225:
4224:{\displaystyle \mu =3}
4187:
4147:
4146:{\displaystyle \mu =2}
4079:
4052:
4025:
3931:
3875:
3874:{\displaystyle \mu =3}
3849:
3777:
3722:
3671:
3441:
3421:
3398:
3326:
3019:
2999:
2968:
2941:
2914:
2894:
2867:
2767:
2738:
2715:
2309:
2282:
2255:
2235:
2212:
2105:
2069:
1862:
1839:
1553:
1533:
1513:
1490:
1101:
1074:
1044:
658:
634:
607:
533:
513:
493:
470:
254:
234:
214:
189:
138:
137:{\displaystyle M>m}
112:
35:
7815:Hamiltonian mechanics
7646:Svetlana Jitomirskaya
7553:Multiscroll attractor
7398:Interval exchange map
7351:Dyadic transformation
7336:Complex quadratic map
7178:Topological conjugacy
7113:Correlation dimension
7088:Anosov diffeomorphism
6900:Vol.60(10), p951-953.
6404:
6367:
6347:
6321:
6295:
6269:
6238:
6215:
6075:
6052:
5941:
5715:
5682:
5649:
5613:
5583:
5512:
5489:
5468:
5428:
5390:
5350:
5301:
5281:
5258:
5203:
5163:
5125:
5085:
5047:
5007:
4969:
4929:
4876:
4778:
4755:
4734:
4694:
4656:
4616:
4578:
4538:
4500:
4460:
4422:
4382:
4344:
4304:
4266:
4226:
4188:
4148:
4080:
4053:
4026:
3932:
3876:
3850:
3778:
3739:can be classified as
3723:
3672:
3442:
3422:
3399:
3327:
3020:
3000:
2969:
2942:
2940:{\displaystyle p_{r}}
2915:
2895:
2868:
2768:
2744:and angular velocity
2739:
2716:
2310:
2283:
2281:{\displaystyle p_{r}}
2256:
2236:
2213:
2106:
2070:
1863:
1840:
1554:
1534:
1514:
1491:
1102:
1075:
1073:{\displaystyle p_{r}}
1045:
659:
635:
608:
539:is solely due to the
534:
514:
494:
471:
255:
235:
215:
194:Hamiltonian mechanics
187:
139:
113:
33:
7656:Edward Norton Lorenz
6916:Vol.63(2), p121-126.
6532:Smiles and Teardrops
6402:{\displaystyle M=3m}
6384:
6365:{\displaystyle \mu }
6356:
6330:
6304:
6278:
6251:
6227:
6087:
6064:
5956:
5727:
5691:
5658:
5638:
5602:
5529:
5494:
5437:
5411:
5359:
5333:
5290:
5270:
5232:
5172:
5146:
5094:
5068:
5016:
4990:
4938:
4912:
4798:
4760:
4703:
4677:
4625:
4599:
4547:
4521:
4469:
4443:
4391:
4365:
4313:
4287:
4235:
4209:
4157:
4131:
4062:
4035:
3948:
3885:
3859:
3787:
3747:
3684:
3454:
3431:
3411:
3349:
3032:
3009:
2989:
2951:
2924:
2904:
2884:
2784:
2748:
2728:
2322:
2292:
2265:
2245:
2225:
2118:
2082:
1875:
1852:
1566:
1543:
1523:
1503:
1114:
1084:
1057:
671:
644:
620:
550:
523:
503:
483:
267:
244:
224:
204:
198:Lagrangian mechanics
162:Smiles and Teardrops
122:
96:
7616:Mitchell Feigenbaum
7558:Population dynamics
7543:Hénon–Heiles system
7403:Irrational rotation
7356:Dynamical billiards
7341:Coupled map lattice
7201:Liouville's theorem
7133:Hausdorff dimension
7118:Conservative system
7103:Bifurcation diagram
6879:Journal de Physique
6812:2016PhLA..380..989E
6770:Journal de Physique
6731:Journal de Physique
6704:1994AmJPh..62..231T
6642:Journal de Physique
6615:1986AmJPh..54..142T
6578:2010PhyD..239.1067P
6504:1988AmJPh..56.1117T
6458:1984AmJPh..52..895T
3737:Hamiltonian systems
2487:
1426:
1384:
342:
311:
180:Equations of motion
111:{\displaystyle M=m}
77:) connected by an
7794:Santa Fe Institute
7661:Aleksandr Lyapunov
7491:Three-body problem
7378:Gingerbreadman map
7265:Bifurcation theory
7143:Lyapunov stability
6922:Il Nuovo Cimento B
6905:Journal of Physics
6399:
6362:
6342:
6316:
6290:
6264:
6233:
6210:
6208:
6070:
6047:
5936:
5710:
5677:
5644:
5608:
5578:
5519:
5507:
5490:Type B orbits for
5463:
5423:
5385:
5345:
5296:
5276:
5253:
5198:
5158:
5120:
5080:
5042:
5002:
4964:
4924:
4871:
4785:
4773:
4756:Type A orbits for
4729:
4689:
4651:
4611:
4573:
4533:
4495:
4455:
4417:
4377:
4339:
4299:
4261:
4221:
4183:
4143:
4099:Nonsingular orbits
4075:
4048:
4021:
3927:
3871:
3845:
3773:
3718:
3667:
3665:
3437:
3417:
3394:
3322:
3318:
3311:
3221:
3214:
3015:
2995:
2964:
2937:
2910:
2890:
2863:
2763:
2734:
2711:
2709:
2473:
2305:
2278:
2251:
2231:
2208:
2101:
2065:
2063:
1858:
1835:
1833:
1549:
1529:
1509:
1486:
1484:
1412:
1370:
1097:
1070:
1040:
1038:
654:
630:
603:
601:
529:
509:
489:
466:
464:
328:
297:
250:
230:
210:
190:
174:David J. Griffiths
166:Nicholas Tufillaro
134:
108:
60:initial conditions
36:
7802:
7801:
7666:Benoît Mandelbrot
7631:Martin Gutzwiller
7621:Peter Grassberger
7504:
7503:
7486:Rössler attractor
7234:
7233:
7138:Invariant measure
7060:Lyapunov exponent
6873:Vol.10(10), p173.
6799:Physics Letters A
6648:(16): 1693–1702.
6562:(12): 1067–1081.
6262:
6236:{\displaystyle h}
6199:
6137:
6134:
6073:{\displaystyle r}
5831:
5799:
5779:
5759:
5744:
5647:{\displaystyle m}
5611:{\displaystyle M}
5461:
5383:
5299:{\displaystyle r}
5279:{\displaystyle r}
5196:
5118:
5040:
4962:
4727:
4649:
4571:
4493:
4415:
4337:
4259:
4181:
4012:
3972:
3637:
3618:
3606:
3581:
3527:
3501:
3483:
3440:{\displaystyle I}
3420:{\displaystyle R}
3392:
3227:
3225:
3201:
3176:
3151:
3136:
3106:
3096:
3094:
3083:
3068:
3018:{\displaystyle R}
2998:{\displaystyle I}
2984:moment of inertia
2893:{\displaystyle r}
2861:
2829:
2796:
2760:
2679:
2642:
2620:
2588:
2547:
2503:
2466:
2429:
2407:
2379:
2338:
2234:{\displaystyle r}
2164:
2145:
2099:
2058:
1989:
1963:
1959:
1926:
1904:
1861:{\displaystyle r}
1801:
1789:
1771:
1752:
1724:
1712:
1654:
1650:
1617:
1595:
1512:{\displaystyle r}
1445:
1405:
1344:
1317:
1313:
1285:
1281:
1230:
1201:
1197:
1169:
1165:
987:
955:
935:
915:
900:
806:
774:
754:
734:
719:
532:{\displaystyle U}
492:{\displaystyle r}
448:
416:
396:
376:
361:
323:
292:
253:{\displaystyle T}
233:{\displaystyle M}
213:{\displaystyle m}
16:(Redirected from
7822:
7774:Butterfly effect
7686:Itamar Procaccia
7636:Brosl Hasslacher
7533:Elastic pendulum
7461:Duffing equation
7408:Kaplan–Yorke map
7326:Arnold's cat map
7314:
7313:
7290:Stability theory
7275:Dynamical system
7270:Control of chaos
7250:
7242:
7226:Takens's theorem
7158:Poincaré section
7028:
7027:
7010:
7003:
6996:
6987:
6986:
6824:
6823:
6793:
6787:
6786:
6766:
6757:
6748:
6747:
6738:(9): 1495–1500.
6725:
6716:
6715:
6697:
6695:chao-dyn/9302006
6675:
6666:
6665:
6633:
6627:
6626:
6596:
6590:
6589:
6571:
6549:
6540:
6539:
6527:
6516:
6515:
6485:
6470:
6469:
6439:
6408:
6406:
6405:
6400:
6371:
6369:
6368:
6363:
6351:
6349:
6348:
6343:
6325:
6323:
6322:
6317:
6299:
6297:
6296:
6291:
6273:
6271:
6270:
6265:
6263:
6255:
6242:
6240:
6239:
6234:
6219:
6217:
6216:
6211:
6209:
6205:
6201:
6200:
6195:
6194:
6193:
6192:
6175:
6162:
6161:
6138:
6136:
6135:
6130:
6129:
6117:
6105:
6079:
6077:
6076:
6071:
6056:
6054:
6053:
6048:
6046:
6045:
6044:
6028:
6027:
6009:
6008:
5990:
5945:
5943:
5942:
5937:
5935:
5934:
5933:
5917:
5916:
5898:
5897:
5879:
5844:
5840:
5839:
5838:
5833:
5832:
5824:
5820:
5819:
5807:
5806:
5801:
5800:
5792:
5780:
5772:
5767:
5766:
5761:
5760:
5752:
5745:
5737:
5719:
5717:
5716:
5711:
5709:
5708:
5686:
5684:
5683:
5678:
5676:
5675:
5653:
5651:
5650:
5645:
5617:
5615:
5614:
5609:
5587:
5585:
5584:
5579:
5516:
5514:
5513:
5508:
5506:
5505:
5482:Collision orbits
5472:
5470:
5469:
5464:
5462:
5454:
5449:
5448:
5432:
5430:
5429:
5424:
5404:
5394:
5392:
5391:
5386:
5384:
5376:
5371:
5370:
5354:
5352:
5351:
5346:
5326:
5305:
5303:
5302:
5297:
5285:
5283:
5282:
5277:
5262:
5260:
5259:
5254:
5207:
5205:
5204:
5199:
5197:
5189:
5184:
5183:
5167:
5165:
5164:
5159:
5139:
5129:
5127:
5126:
5121:
5119:
5111:
5106:
5105:
5089:
5087:
5086:
5081:
5061:
5051:
5049:
5048:
5043:
5041:
5033:
5028:
5027:
5011:
5009:
5008:
5003:
4983:
4973:
4971:
4970:
4965:
4963:
4955:
4950:
4949:
4933:
4931:
4930:
4925:
4905:
4880:
4878:
4877:
4872:
4782:
4780:
4779:
4774:
4772:
4771:
4738:
4736:
4735:
4730:
4728:
4720:
4715:
4714:
4698:
4696:
4695:
4690:
4670:
4660:
4658:
4657:
4652:
4650:
4642:
4637:
4636:
4620:
4618:
4617:
4612:
4592:
4582:
4580:
4579:
4574:
4572:
4564:
4559:
4558:
4542:
4540:
4539:
4534:
4514:
4504:
4502:
4501:
4496:
4494:
4486:
4481:
4480:
4464:
4462:
4461:
4456:
4436:
4426:
4424:
4423:
4418:
4416:
4408:
4403:
4402:
4386:
4384:
4383:
4378:
4358:
4348:
4346:
4345:
4340:
4338:
4330:
4325:
4324:
4308:
4306:
4305:
4300:
4280:
4270:
4268:
4267:
4262:
4260:
4252:
4247:
4246:
4230:
4228:
4227:
4222:
4202:
4192:
4190:
4189:
4184:
4182:
4174:
4169:
4168:
4152:
4150:
4149:
4144:
4124:
4084:
4082:
4081:
4076:
4074:
4073:
4057:
4055:
4054:
4049:
4047:
4046:
4030:
4028:
4027:
4022:
4014:
4013:
4005:
3999:
3998:
3974:
3973:
3965:
3936:
3934:
3933:
3928:
3880:
3878:
3877:
3872:
3854:
3852:
3851:
3846:
3808:
3807:
3782:
3780:
3779:
3774:
3763:
3727:
3725:
3724:
3719:
3714:
3709:
3708:
3696:
3695:
3676:
3674:
3673:
3668:
3666:
3662:
3645:
3644:
3639:
3638:
3630:
3620:
3619:
3611:
3608:
3607:
3599:
3583:
3582:
3574:
3555:
3535:
3534:
3529:
3528:
3520:
3503:
3502:
3494:
3485:
3484:
3476:
3470:
3469:
3446:
3444:
3443:
3438:
3426:
3424:
3423:
3418:
3403:
3401:
3400:
3395:
3393:
3391:
3390:
3378:
3361:
3360:
3331:
3329:
3328:
3323:
3317:
3312:
3307:
3306:
3302:
3292:
3264:
3260:
3259:
3220:
3215:
3210:
3209:
3208:
3203:
3202:
3194:
3190:
3189:
3177:
3169:
3164:
3163:
3158:
3154:
3153:
3152:
3144:
3138:
3137:
3129:
3117:
3116:
3107:
3099:
3090:
3086:
3085:
3084:
3076:
3070:
3069:
3061:
3041:
3040:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2973:
2971:
2970:
2965:
2963:
2962:
2946:
2944:
2943:
2938:
2936:
2935:
2919:
2917:
2916:
2911:
2899:
2897:
2896:
2891:
2872:
2870:
2869:
2864:
2862:
2860:
2849:
2838:
2830:
2828:
2817:
2806:
2798:
2797:
2789:
2772:
2770:
2769:
2764:
2762:
2761:
2753:
2743:
2741:
2740:
2735:
2720:
2718:
2717:
2712:
2710:
2706:
2680:
2678:
2677:
2668:
2667:
2666:
2656:
2644:
2643:
2638:
2637:
2628:
2621:
2619:
2618:
2617:
2604:
2603:
2594:
2589:
2587:
2586:
2585:
2584:
2570:
2569:
2568:
2558:
2549:
2548:
2540:
2533:
2504:
2502:
2501:
2500:
2486:
2481:
2472:
2467:
2465:
2464:
2455:
2454:
2453:
2443:
2431:
2430:
2425:
2424:
2415:
2408:
2406:
2395:
2394:
2385:
2380:
2378:
2377:
2376:
2375:
2361:
2360:
2359:
2349:
2340:
2339:
2331:
2314:
2312:
2311:
2306:
2304:
2303:
2287:
2285:
2284:
2279:
2277:
2276:
2260:
2258:
2257:
2252:
2240:
2238:
2237:
2232:
2217:
2215:
2214:
2209:
2198:
2172:
2171:
2166:
2165:
2157:
2147:
2146:
2138:
2110:
2108:
2107:
2102:
2100:
2092:
2074:
2072:
2071:
2066:
2064:
2060:
2059:
2051:
2026:
1997:
1996:
1991:
1990:
1982:
1968:
1964:
1962:
1961:
1960:
1952:
1945:
1944:
1943:
1933:
1927:
1925:
1914:
1905:
1903:
1895:
1894:
1893:
1883:
1867:
1865:
1864:
1859:
1844:
1842:
1841:
1836:
1834:
1820:
1803:
1802:
1794:
1791:
1790:
1782:
1773:
1772:
1764:
1754:
1753:
1745:
1742:
1741:
1726:
1725:
1717:
1714:
1713:
1705:
1686:
1659:
1655:
1653:
1652:
1651:
1643:
1636:
1635:
1634:
1624:
1618:
1616:
1605:
1596:
1594:
1586:
1585:
1584:
1574:
1558:
1556:
1555:
1550:
1538:
1536:
1535:
1530:
1518:
1516:
1515:
1510:
1495:
1493:
1492:
1487:
1485:
1481:
1446:
1444:
1443:
1442:
1425:
1420:
1411:
1406:
1404:
1383:
1378:
1369:
1360:
1359:
1346:
1345:
1337:
1334:
1333:
1318:
1316:
1315:
1314:
1306:
1299:
1291:
1286:
1284:
1283:
1282:
1274:
1267:
1266:
1265:
1255:
1246:
1245:
1232:
1231:
1223:
1202:
1200:
1199:
1198:
1190:
1183:
1175:
1170:
1168:
1167:
1166:
1158:
1151:
1150:
1149:
1139:
1130:
1129:
1106:
1104:
1103:
1098:
1096:
1095:
1079:
1077:
1076:
1071:
1069:
1068:
1049:
1047:
1046:
1041:
1039:
1035:
1000:
996:
995:
994:
989:
988:
980:
976:
975:
963:
962:
957:
956:
948:
936:
928:
923:
922:
917:
916:
908:
901:
893:
885:
865:
864:
854:
819:
815:
814:
813:
808:
807:
799:
795:
794:
782:
781:
776:
775:
767:
755:
747:
742:
741:
736:
735:
727:
720:
712:
704:
684:
683:
663:
661:
660:
655:
653:
652:
639:
637:
636:
631:
629:
628:
612:
610:
609:
604:
602:
598:
538:
536:
535:
530:
518:
516:
515:
510:
498:
496:
495:
490:
475:
473:
472:
467:
465:
461:
457:
456:
455:
450:
449:
441:
437:
436:
424:
423:
418:
417:
409:
397:
389:
384:
383:
378:
377:
369:
362:
354:
346:
341:
336:
324:
316:
310:
305:
293:
285:
259:
257:
256:
251:
239:
237:
236:
231:
219:
217:
216:
211:
143:
141:
140:
135:
117:
115:
114:
109:
76:
72:
52:dynamical system
44:Atwood's machine
21:
7830:
7829:
7825:
7824:
7823:
7821:
7820:
7819:
7805:
7804:
7803:
7798:
7766:
7760:
7706:Caroline Series
7601:Mary Cartwright
7583:
7577:
7528:Double pendulum
7510:
7500:
7449:
7442:
7368:Exponential map
7319:
7305:
7299:
7257:
7251:
7244:
7230:
7196:Ergodic theorem
7189:
7182:
7173:Stable manifold
7163:Recurrence plot
7079:
7033:
7019:
7014:
6957:
6861:Physica Scripta
6832:
6830:Further reading
6827:
6794:
6790:
6764:
6758:
6751:
6726:
6719:
6712:10.1119/1.17602
6676:
6669:
6634:
6630:
6623:10.1119/1.14710
6597:
6593:
6550:
6543:
6528:
6519:
6512:10.1119/1.15774
6486:
6473:
6466:10.1119/1.13791
6452:(10): 895–903.
6440:
6419:
6415:
6385:
6382:
6381:
6378:
6357:
6354:
6353:
6331:
6328:
6327:
6305:
6302:
6301:
6279:
6276:
6275:
6254:
6252:
6249:
6248:
6228:
6225:
6224:
6207:
6206:
6188:
6184:
6183:
6176:
6174:
6167:
6163:
6157:
6153:
6146:
6140:
6139:
6125:
6118:
6116:
6109:
6104:
6097:
6090:
6088:
6085:
6084:
6065:
6062:
6061:
6040:
6036:
6035:
6023:
6019:
6004:
6000:
5986:
5957:
5954:
5953:
5929:
5925:
5924:
5912:
5908:
5893:
5889:
5875:
5834:
5823:
5822:
5821:
5815:
5811:
5802:
5791:
5790:
5789:
5788:
5784:
5771:
5762:
5751:
5750:
5749:
5736:
5728:
5725:
5724:
5704:
5700:
5692:
5689:
5688:
5671:
5667:
5659:
5656:
5655:
5639:
5636:
5635:
5624:
5603:
5600:
5599:
5530:
5527:
5526:
5501:
5497:
5495:
5492:
5491:
5484:
5479:
5478:
5477:
5474:
5453:
5444:
5440:
5438:
5435:
5434:
5412:
5409:
5408:
5405:
5396:
5375:
5366:
5362:
5360:
5357:
5356:
5334:
5331:
5330:
5327:
5316:
5315:
5291:
5288:
5287:
5271:
5268:
5267:
5233:
5230:
5229:
5219:
5217:Singular orbits
5214:
5213:
5212:
5209:
5188:
5179:
5175:
5173:
5170:
5169:
5147:
5144:
5143:
5140:
5131:
5110:
5101:
5097:
5095:
5092:
5091:
5069:
5066:
5065:
5062:
5053:
5032:
5023:
5019:
5017:
5014:
5013:
4991:
4988:
4987:
4984:
4975:
4954:
4945:
4941:
4939:
4936:
4935:
4913:
4910:
4909:
4906:
4895:
4894:
4799:
4796:
4795:
4767:
4763:
4761:
4758:
4757:
4750:
4748:Periodic orbits
4745:
4744:
4743:
4740:
4719:
4710:
4706:
4704:
4701:
4700:
4678:
4675:
4674:
4671:
4662:
4641:
4632:
4628:
4626:
4623:
4622:
4600:
4597:
4596:
4593:
4584:
4563:
4554:
4550:
4548:
4545:
4544:
4522:
4519:
4518:
4515:
4506:
4485:
4476:
4472:
4470:
4467:
4466:
4444:
4441:
4440:
4437:
4428:
4407:
4398:
4394:
4392:
4389:
4388:
4366:
4363:
4362:
4359:
4350:
4329:
4320:
4316:
4314:
4311:
4310:
4288:
4285:
4284:
4281:
4272:
4251:
4242:
4238:
4236:
4233:
4232:
4210:
4207:
4206:
4203:
4194:
4173:
4164:
4160:
4158:
4155:
4154:
4132:
4129:
4128:
4125:
4114:
4113:
4101:
4092:
4069:
4065:
4063:
4060:
4059:
4042:
4038:
4036:
4033:
4032:
4004:
4003:
3994:
3990:
3964:
3963:
3949:
3946:
3945:
3886:
3883:
3882:
3860:
3857:
3856:
3803:
3799:
3788:
3785:
3784:
3759:
3748:
3745:
3744:
3734:
3710:
3704:
3700:
3691:
3687:
3685:
3682:
3681:
3664:
3663:
3658:
3640:
3629:
3628:
3627:
3610:
3609:
3598:
3597:
3584:
3573:
3572:
3566:
3565:
3551:
3530:
3519:
3518:
3517:
3507:
3493:
3492:
3475:
3474:
3465:
3461:
3457:
3455:
3452:
3451:
3432:
3429:
3428:
3412:
3409:
3408:
3386:
3382:
3377:
3356:
3352:
3350:
3347:
3346:
3341:
3338:
3313:
3288:
3278:
3274:
3255:
3239:
3235:
3228:
3226:
3216:
3204:
3193:
3192:
3191:
3185:
3181:
3168:
3159:
3143:
3142:
3128:
3127:
3123:
3119:
3118:
3112:
3108:
3098:
3097:
3095:
3075:
3074:
3060:
3059:
3046:
3042:
3036:
3035:
3033:
3030:
3029:
3010:
3007:
3006:
2990:
2987:
2986:
2980:
2958:
2954:
2952:
2949:
2948:
2931:
2927:
2925:
2922:
2921:
2905:
2902:
2901:
2885:
2882:
2881:
2850:
2839:
2837:
2818:
2807:
2805:
2788:
2787:
2785:
2782:
2781:
2752:
2751:
2749:
2746:
2745:
2729:
2726:
2725:
2708:
2707:
2702:
2673:
2669:
2662:
2661:
2657:
2655:
2645:
2633:
2629:
2627:
2626:
2623:
2622:
2613:
2609:
2605:
2599:
2595:
2593:
2580:
2576:
2575:
2571:
2564:
2563:
2559:
2557:
2550:
2539:
2538:
2535:
2534:
2529:
2496:
2492:
2488:
2482:
2477:
2471:
2460:
2456:
2449:
2448:
2444:
2442:
2432:
2420:
2416:
2414:
2413:
2410:
2409:
2396:
2390:
2386:
2384:
2371:
2367:
2366:
2362:
2355:
2354:
2350:
2348:
2341:
2330:
2329:
2325:
2323:
2320:
2319:
2299:
2295:
2293:
2290:
2289:
2272:
2268:
2266:
2263:
2262:
2246:
2243:
2242:
2226:
2223:
2222:
2194:
2167:
2156:
2155:
2154:
2137:
2136:
2119:
2116:
2115:
2091:
2083:
2080:
2079:
2062:
2061:
2050:
2049:
2027:
2022:
1992:
1981:
1980:
1979:
1970:
1969:
1951:
1950:
1946:
1939:
1938:
1934:
1932:
1928:
1918:
1913:
1906:
1896:
1889:
1888:
1884:
1882:
1878:
1876:
1873:
1872:
1853:
1850:
1849:
1832:
1831:
1821:
1816:
1793:
1792:
1781:
1780:
1763:
1762:
1756:
1755:
1744:
1743:
1737:
1733:
1716:
1715:
1704:
1703:
1687:
1682:
1661:
1660:
1642:
1641:
1637:
1630:
1629:
1625:
1623:
1619:
1609:
1604:
1597:
1587:
1580:
1579:
1575:
1573:
1569:
1567:
1564:
1563:
1544:
1541:
1540:
1524:
1521:
1520:
1504:
1501:
1500:
1483:
1482:
1477:
1438:
1434:
1427:
1421:
1416:
1410:
1385:
1379:
1374:
1368:
1361:
1355:
1354:
1348:
1347:
1336:
1335:
1329:
1325:
1305:
1304:
1300:
1292:
1290:
1273:
1272:
1268:
1261:
1260:
1256:
1254:
1247:
1241:
1237:
1234:
1233:
1222:
1221:
1189:
1188:
1184:
1176:
1174:
1157:
1156:
1152:
1145:
1144:
1140:
1138:
1131:
1125:
1121:
1117:
1115:
1112:
1111:
1091:
1087:
1085:
1082:
1081:
1064:
1060:
1058:
1055:
1054:
1037:
1036:
1031:
990:
979:
978:
977:
971:
967:
958:
947:
946:
945:
944:
940:
927:
918:
907:
906:
905:
892:
883:
882:
866:
860:
859:
856:
855:
850:
809:
798:
797:
796:
790:
786:
777:
766:
765:
764:
763:
759:
746:
737:
726:
725:
724:
711:
702:
701:
685:
679:
678:
674:
672:
669:
668:
664:of the system:
648:
647:
645:
642:
641:
624:
623:
621:
618:
617:
600:
599:
594:
560:
553:
551:
548:
547:
524:
521:
520:
504:
501:
500:
484:
481:
480:
463:
462:
451:
440:
439:
438:
432:
428:
419:
408:
407:
406:
405:
401:
388:
379:
368:
367:
366:
353:
344:
343:
337:
332:
315:
306:
301:
284:
277:
270:
268:
265:
264:
245:
242:
241:
225:
222:
221:
205:
202:
201:
182:
146:parameter space
123:
120:
119:
97:
94:
93:
74:
70:
28:
23:
22:
15:
12:
11:
5:
7828:
7818:
7817:
7800:
7799:
7797:
7796:
7791:
7789:Predictability
7786:
7781:
7776:
7770:
7768:
7762:
7761:
7759:
7758:
7756:Lai-Sang Young
7753:
7751:James A. Yorke
7748:
7746:Amie Wilkinson
7743:
7738:
7733:
7728:
7723:
7718:
7713:
7708:
7703:
7698:
7693:
7688:
7683:
7681:Henri Poincaré
7678:
7673:
7668:
7663:
7658:
7653:
7648:
7643:
7638:
7633:
7628:
7623:
7618:
7613:
7608:
7603:
7598:
7593:
7587:
7585:
7579:
7578:
7576:
7575:
7570:
7565:
7560:
7555:
7550:
7548:Kicked rotator
7545:
7540:
7535:
7530:
7525:
7520:
7518:Chua's circuit
7514:
7512:
7506:
7505:
7502:
7501:
7499:
7498:
7493:
7488:
7483:
7478:
7473:
7468:
7463:
7458:
7452:
7450:
7447:
7444:
7443:
7441:
7440:
7438:Zaslavskii map
7435:
7433:Tinkerbell map
7430:
7425:
7420:
7415:
7410:
7405:
7400:
7395:
7390:
7385:
7380:
7375:
7370:
7365:
7364:
7363:
7353:
7348:
7343:
7338:
7333:
7328:
7322:
7320:
7317:
7311:
7301:
7300:
7298:
7297:
7292:
7287:
7282:
7280:Ergodic theory
7277:
7272:
7267:
7261:
7259:
7253:
7252:
7237:
7235:
7232:
7231:
7229:
7228:
7223:
7218:
7213:
7208:
7203:
7198:
7192:
7190:
7187:
7184:
7183:
7181:
7180:
7175:
7170:
7165:
7160:
7155:
7150:
7145:
7140:
7135:
7130:
7125:
7120:
7115:
7110:
7105:
7100:
7095:
7090:
7084:
7081:
7080:
7078:
7077:
7072:
7070:Periodic point
7067:
7062:
7057:
7052:
7047:
7042:
7036:
7034:
7031:
7025:
7021:
7020:
7013:
7012:
7005:
6998:
6990:
6984:
6983:
6978:
6973:
6968:
6963:
6956:
6955:External links
6953:
6952:
6951:
6942:
6933:
6926:
6917:
6908:
6901:
6892:
6883:
6882:Vol.51, p1693.
6874:
6865:
6856:
6849:
6842:
6831:
6828:
6826:
6825:
6788:
6749:
6717:
6688:(3): 231–233.
6667:
6628:
6591:
6541:
6517:
6471:
6416:
6414:
6411:
6398:
6395:
6392:
6389:
6377:
6374:
6361:
6341:
6338:
6335:
6315:
6312:
6309:
6289:
6286:
6283:
6261:
6258:
6232:
6221:
6220:
6204:
6198:
6191:
6187:
6182:
6179:
6173:
6170:
6166:
6160:
6156:
6152:
6149:
6147:
6145:
6142:
6141:
6133:
6128:
6124:
6121:
6115:
6112:
6108:
6103:
6100:
6098:
6096:
6093:
6092:
6069:
6058:
6057:
6043:
6039:
6034:
6031:
6026:
6022:
6018:
6015:
6012:
6007:
6003:
5999:
5996:
5993:
5989:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5947:
5946:
5932:
5928:
5923:
5920:
5915:
5911:
5907:
5904:
5901:
5896:
5892:
5888:
5885:
5882:
5878:
5874:
5871:
5868:
5865:
5862:
5859:
5856:
5853:
5850:
5847:
5843:
5837:
5830:
5827:
5818:
5814:
5810:
5805:
5798:
5795:
5787:
5783:
5778:
5775:
5770:
5765:
5758:
5755:
5748:
5743:
5740:
5735:
5732:
5707:
5703:
5699:
5696:
5674:
5670:
5666:
5663:
5643:
5623:
5620:
5607:
5589:
5588:
5577:
5574:
5571:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5546:
5543:
5540:
5537:
5534:
5504:
5500:
5483:
5480:
5476:
5475:
5460:
5457:
5452:
5447:
5443:
5422:
5419:
5416:
5406:
5399:
5397:
5382:
5379:
5374:
5369:
5365:
5344:
5341:
5338:
5328:
5321:
5318:
5317:
5313:
5312:
5311:
5295:
5275:
5264:
5263:
5252:
5249:
5246:
5243:
5240:
5237:
5218:
5215:
5211:
5210:
5195:
5192:
5187:
5182:
5178:
5157:
5154:
5151:
5141:
5134:
5132:
5117:
5114:
5109:
5104:
5100:
5079:
5076:
5073:
5063:
5056:
5054:
5039:
5036:
5031:
5026:
5022:
5001:
4998:
4995:
4985:
4978:
4976:
4961:
4958:
4953:
4948:
4944:
4923:
4920:
4917:
4907:
4900:
4897:
4896:
4892:
4891:
4890:
4882:
4881:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4770:
4766:
4749:
4746:
4742:
4741:
4726:
4723:
4718:
4713:
4709:
4688:
4685:
4682:
4672:
4665:
4663:
4648:
4645:
4640:
4635:
4631:
4610:
4607:
4604:
4594:
4587:
4585:
4570:
4567:
4562:
4557:
4553:
4532:
4529:
4526:
4516:
4509:
4507:
4492:
4489:
4484:
4479:
4475:
4454:
4451:
4448:
4438:
4431:
4429:
4414:
4411:
4406:
4401:
4397:
4376:
4373:
4370:
4360:
4353:
4351:
4336:
4333:
4328:
4323:
4319:
4298:
4295:
4292:
4282:
4275:
4273:
4258:
4255:
4250:
4245:
4241:
4220:
4217:
4214:
4204:
4197:
4195:
4180:
4177:
4172:
4167:
4163:
4142:
4139:
4136:
4126:
4119:
4116:
4115:
4111:
4110:
4109:
4100:
4097:
4091:
4088:
4072:
4068:
4045:
4041:
4020:
4017:
4011:
4008:
4002:
3997:
3993:
3989:
3986:
3983:
3980:
3977:
3971:
3968:
3962:
3959:
3956:
3953:
3939:chaotic motion
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3870:
3867:
3864:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3806:
3802:
3798:
3795:
3792:
3772:
3769:
3766:
3762:
3758:
3755:
3752:
3733:
3730:
3717:
3713:
3707:
3703:
3699:
3694:
3690:
3678:
3677:
3661:
3657:
3654:
3651:
3648:
3643:
3636:
3633:
3626:
3623:
3617:
3614:
3605:
3602:
3596:
3593:
3590:
3587:
3585:
3580:
3577:
3571:
3568:
3567:
3564:
3561:
3558:
3554:
3550:
3547:
3544:
3541:
3538:
3533:
3526:
3523:
3516:
3513:
3510:
3508:
3506:
3500:
3497:
3491:
3488:
3482:
3479:
3473:
3468:
3464:
3460:
3459:
3436:
3416:
3405:
3404:
3389:
3385:
3381:
3376:
3373:
3370:
3367:
3364:
3359:
3355:
3339:
3336:
3333:
3332:
3321:
3316:
3310:
3305:
3301:
3298:
3295:
3291:
3287:
3284:
3281:
3277:
3273:
3270:
3267:
3263:
3258:
3254:
3251:
3248:
3245:
3242:
3238:
3234:
3231:
3224:
3219:
3213:
3207:
3200:
3197:
3188:
3184:
3180:
3175:
3172:
3167:
3162:
3157:
3150:
3147:
3141:
3135:
3132:
3126:
3122:
3115:
3111:
3105:
3102:
3093:
3089:
3082:
3079:
3073:
3067:
3064:
3058:
3055:
3052:
3049:
3045:
3039:
3014:
2994:
2979:
2976:
2961:
2957:
2934:
2930:
2909:
2889:
2874:
2873:
2859:
2856:
2853:
2848:
2845:
2842:
2836:
2833:
2827:
2824:
2821:
2816:
2813:
2810:
2804:
2801:
2795:
2792:
2775:Atwood machine
2759:
2756:
2733:
2722:
2721:
2705:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2676:
2672:
2665:
2660:
2654:
2651:
2648:
2646:
2641:
2636:
2632:
2625:
2624:
2616:
2612:
2608:
2602:
2598:
2592:
2583:
2579:
2574:
2567:
2562:
2556:
2553:
2551:
2546:
2543:
2537:
2536:
2532:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2499:
2495:
2491:
2485:
2480:
2476:
2470:
2463:
2459:
2452:
2447:
2441:
2438:
2435:
2433:
2428:
2423:
2419:
2412:
2411:
2405:
2402:
2399:
2393:
2389:
2383:
2374:
2370:
2365:
2358:
2353:
2347:
2344:
2342:
2337:
2334:
2328:
2327:
2302:
2298:
2275:
2271:
2250:
2230:
2219:
2218:
2207:
2204:
2201:
2197:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2170:
2163:
2160:
2153:
2150:
2144:
2141:
2135:
2132:
2129:
2126:
2123:
2098:
2095:
2090:
2087:
2076:
2075:
2057:
2054:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2028:
2025:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1995:
1988:
1985:
1978:
1975:
1972:
1971:
1967:
1958:
1955:
1949:
1942:
1937:
1931:
1924:
1921:
1917:
1912:
1909:
1907:
1902:
1899:
1892:
1887:
1881:
1880:
1857:
1846:
1845:
1830:
1827:
1824:
1822:
1819:
1815:
1812:
1809:
1806:
1800:
1797:
1788:
1785:
1779:
1776:
1770:
1767:
1761:
1758:
1757:
1751:
1748:
1740:
1736:
1732:
1729:
1723:
1720:
1711:
1708:
1702:
1699:
1696:
1693:
1690:
1688:
1685:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1662:
1658:
1649:
1646:
1640:
1633:
1628:
1622:
1615:
1612:
1608:
1603:
1600:
1598:
1593:
1590:
1583:
1578:
1572:
1571:
1548:
1528:
1508:
1497:
1496:
1480:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1441:
1437:
1433:
1430:
1424:
1419:
1415:
1409:
1403:
1400:
1397:
1394:
1391:
1388:
1382:
1377:
1373:
1367:
1364:
1362:
1358:
1353:
1350:
1349:
1343:
1340:
1332:
1328:
1324:
1321:
1312:
1309:
1303:
1298:
1295:
1289:
1280:
1277:
1271:
1264:
1259:
1253:
1250:
1248:
1244:
1240:
1236:
1235:
1229:
1226:
1220:
1217:
1214:
1211:
1208:
1205:
1196:
1193:
1187:
1182:
1179:
1173:
1164:
1161:
1155:
1148:
1143:
1137:
1134:
1132:
1128:
1124:
1120:
1119:
1094:
1090:
1067:
1063:
1051:
1050:
1034:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
999:
993:
986:
983:
974:
970:
966:
961:
954:
951:
943:
939:
934:
931:
926:
921:
914:
911:
904:
899:
896:
891:
888:
886:
884:
881:
878:
875:
872:
869:
867:
863:
858:
857:
853:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
818:
812:
805:
802:
793:
789:
785:
780:
773:
770:
762:
758:
753:
750:
745:
740:
733:
730:
723:
718:
715:
710:
707:
705:
703:
700:
697:
694:
691:
688:
686:
682:
677:
676:
651:
627:
614:
613:
597:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
561:
559:
556:
555:
528:
508:
488:
477:
476:
460:
454:
447:
444:
435:
431:
427:
422:
415:
412:
404:
400:
395:
392:
387:
382:
375:
372:
365:
360:
357:
352:
349:
347:
345:
340:
335:
331:
327:
322:
319:
314:
309:
304:
300:
296:
291:
288:
283:
280:
278:
276:
273:
272:
249:
229:
209:
181:
178:
172:, directed by
133:
130:
127:
107:
104:
101:
26:
9:
6:
4:
3:
2:
7827:
7816:
7813:
7812:
7810:
7795:
7792:
7790:
7787:
7785:
7784:Edge of chaos
7782:
7780:
7777:
7775:
7772:
7771:
7769:
7763:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7741:Marcelo Viana
7739:
7737:
7734:
7732:
7731:Audrey Terras
7729:
7727:
7726:Floris Takens
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7694:
7692:
7689:
7687:
7684:
7682:
7679:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7659:
7657:
7654:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7626:Celso Grebogi
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7606:Chen Guanrong
7604:
7602:
7599:
7597:
7594:
7592:
7591:Michael Berry
7589:
7588:
7586:
7580:
7574:
7571:
7569:
7566:
7564:
7561:
7559:
7556:
7554:
7551:
7549:
7546:
7544:
7541:
7539:
7536:
7534:
7531:
7529:
7526:
7524:
7521:
7519:
7516:
7515:
7513:
7507:
7497:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7477:
7474:
7472:
7469:
7467:
7466:Lorenz system
7464:
7462:
7459:
7457:
7454:
7453:
7451:
7445:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7419:
7416:
7414:
7413:Langton's ant
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7388:Horseshoe map
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7362:
7359:
7358:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7324:
7323:
7321:
7315:
7312:
7309:
7302:
7296:
7293:
7291:
7288:
7286:
7285:Quantum chaos
7283:
7281:
7278:
7276:
7273:
7271:
7268:
7266:
7263:
7262:
7260:
7254:
7249:
7245:
7241:
7227:
7224:
7222:
7219:
7217:
7214:
7212:
7209:
7207:
7204:
7202:
7199:
7197:
7194:
7193:
7191:
7185:
7179:
7176:
7174:
7171:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7151:
7149:
7146:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7099:
7096:
7094:
7093:Arnold tongue
7091:
7089:
7086:
7085:
7082:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7037:
7035:
7029:
7026:
7022:
7018:
7011:
7006:
7004:
6999:
6997:
6992:
6991:
6988:
6982:
6979:
6977:
6974:
6972:
6969:
6967:
6964:
6962:
6959:
6958:
6949:
6948:
6943:
6940:
6939:
6934:
6931:
6927:
6925:Vol. 110 (9).
6924:
6923:
6918:
6915:
6914:
6909:
6906:
6902:
6899:
6898:
6893:
6890:
6889:
6884:
6881:
6880:
6875:
6872:
6871:
6866:
6863:
6862:
6857:
6854:
6850:
6847:
6843:
6840:
6839:
6834:
6833:
6821:
6817:
6813:
6809:
6805:
6801:
6800:
6792:
6784:
6780:
6777:: 2053–2056.
6776:
6772:
6771:
6763:
6756:
6754:
6745:
6741:
6737:
6733:
6732:
6724:
6722:
6713:
6709:
6705:
6701:
6696:
6691:
6687:
6683:
6682:
6674:
6672:
6663:
6659:
6655:
6651:
6647:
6643:
6639:
6632:
6624:
6620:
6616:
6612:
6608:
6604:
6603:
6595:
6587:
6583:
6579:
6575:
6570:
6565:
6561:
6557:
6556:
6548:
6546:
6537:
6533:
6526:
6524:
6522:
6513:
6509:
6505:
6501:
6497:
6493:
6492:
6484:
6482:
6480:
6478:
6476:
6467:
6463:
6459:
6455:
6451:
6447:
6446:
6438:
6436:
6434:
6432:
6430:
6428:
6426:
6424:
6422:
6417:
6410:
6396:
6393:
6390:
6387:
6373:
6359:
6339:
6336:
6333:
6313:
6310:
6307:
6287:
6284:
6281:
6259:
6256:
6246:
6230:
6202:
6196:
6189:
6185:
6180:
6177:
6171:
6168:
6164:
6158:
6154:
6150:
6148:
6143:
6131:
6126:
6122:
6119:
6113:
6110:
6106:
6101:
6099:
6094:
6083:
6082:
6081:
6067:
6041:
6037:
6032:
6029:
6024:
6020:
6016:
6013:
6010:
6005:
6001:
5997:
5994:
5991:
5987:
5983:
5980:
5977:
5974:
5971:
5968:
5965:
5962:
5959:
5952:
5951:
5950:
5930:
5926:
5921:
5918:
5913:
5909:
5905:
5902:
5899:
5894:
5890:
5886:
5883:
5880:
5876:
5872:
5869:
5866:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5841:
5835:
5828:
5825:
5816:
5812:
5808:
5803:
5796:
5793:
5785:
5781:
5776:
5773:
5768:
5763:
5756:
5753:
5746:
5741:
5738:
5733:
5730:
5723:
5722:
5721:
5705:
5701:
5697:
5694:
5672:
5668:
5664:
5661:
5641:
5633:
5629:
5628:conic section
5619:
5605:
5596:
5594:
5575:
5572:
5569:
5565:
5562:
5559:
5553:
5547:
5544:
5538:
5532:
5525:
5524:
5523:
5502:
5498:
5488:
5458:
5455:
5450:
5445:
5441:
5420:
5417:
5414:
5403:
5398:
5380:
5377:
5372:
5367:
5363:
5342:
5339:
5336:
5325:
5320:
5319:
5310:
5307:
5293:
5273:
5250:
5247:
5241:
5235:
5228:
5227:
5226:
5224:
5193:
5190:
5185:
5180:
5176:
5155:
5152:
5149:
5138:
5133:
5115:
5112:
5107:
5102:
5098:
5077:
5074:
5071:
5060:
5055:
5037:
5034:
5029:
5024:
5020:
4999:
4996:
4993:
4982:
4977:
4959:
4956:
4951:
4946:
4942:
4921:
4918:
4915:
4904:
4899:
4898:
4889:
4887:
4865:
4859:
4856:
4850:
4847:
4844:
4838:
4834:
4828:
4822:
4819:
4813:
4810:
4807:
4801:
4794:
4793:
4792:
4790:
4768:
4764:
4754:
4724:
4721:
4716:
4711:
4707:
4686:
4683:
4680:
4669:
4664:
4646:
4643:
4638:
4633:
4629:
4608:
4605:
4602:
4591:
4586:
4568:
4565:
4560:
4555:
4551:
4530:
4527:
4524:
4513:
4508:
4490:
4487:
4482:
4477:
4473:
4452:
4449:
4446:
4435:
4430:
4412:
4409:
4404:
4399:
4395:
4374:
4371:
4368:
4357:
4352:
4334:
4331:
4326:
4321:
4317:
4296:
4293:
4290:
4279:
4274:
4256:
4253:
4248:
4243:
4239:
4218:
4215:
4212:
4201:
4196:
4178:
4175:
4170:
4165:
4161:
4140:
4137:
4134:
4123:
4118:
4117:
4108:
4106:
4096:
4087:
4070:
4066:
4043:
4039:
4018:
4015:
4009:
4006:
4000:
3995:
3991:
3987:
3984:
3981:
3978:
3975:
3969:
3966:
3960:
3957:
3954:
3951:
3942:
3940:
3918:
3915:
3909:
3903:
3900:
3897:
3891:
3888:
3868:
3865:
3862:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3804:
3800:
3796:
3793:
3790:
3770:
3767:
3764:
3760:
3756:
3753:
3750:
3742:
3738:
3732:Integrability
3729:
3715:
3711:
3705:
3701:
3697:
3692:
3688:
3659:
3655:
3652:
3649:
3646:
3641:
3634:
3631:
3624:
3621:
3615:
3612:
3603:
3600:
3594:
3591:
3588:
3586:
3578:
3575:
3569:
3559:
3556:
3552:
3548:
3545:
3539:
3536:
3531:
3524:
3521:
3514:
3511:
3509:
3498:
3495:
3489:
3486:
3480:
3477:
3466:
3462:
3450:
3449:
3448:
3434:
3414:
3387:
3383:
3379:
3374:
3371:
3368:
3365:
3362:
3357:
3353:
3345:
3344:
3343:
3319:
3314:
3308:
3303:
3299:
3296:
3293:
3289:
3285:
3282:
3279:
3275:
3271:
3268:
3265:
3261:
3256:
3252:
3249:
3246:
3243:
3240:
3236:
3232:
3229:
3222:
3217:
3211:
3205:
3198:
3195:
3186:
3182:
3178:
3173:
3170:
3165:
3160:
3155:
3148:
3145:
3139:
3133:
3130:
3124:
3120:
3113:
3109:
3103:
3100:
3091:
3087:
3080:
3077:
3071:
3065:
3062:
3056:
3053:
3050:
3047:
3043:
3028:
3027:
3026:
3012:
2992:
2985:
2975:
2959:
2955:
2932:
2928:
2907:
2887:
2879:
2857:
2854:
2851:
2846:
2843:
2840:
2834:
2831:
2825:
2822:
2819:
2814:
2811:
2808:
2802:
2799:
2793:
2790:
2780:
2779:
2778:
2776:
2757:
2754:
2731:
2703:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2674:
2652:
2649:
2647:
2639:
2634:
2630:
2614:
2610:
2606:
2600:
2596:
2590:
2581:
2577:
2554:
2552:
2544:
2541:
2530:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2497:
2493:
2489:
2483:
2478:
2474:
2468:
2461:
2439:
2436:
2434:
2426:
2421:
2417:
2403:
2400:
2397:
2391:
2387:
2381:
2372:
2368:
2345:
2343:
2335:
2332:
2318:
2317:
2316:
2300:
2296:
2273:
2269:
2248:
2228:
2205:
2202:
2195:
2191:
2188:
2185:
2182:
2176:
2173:
2168:
2161:
2158:
2151:
2148:
2142:
2139:
2130:
2127:
2124:
2114:
2113:
2112:
2096:
2093:
2088:
2085:
2055:
2052:
2043:
2040:
2037:
2031:
2029:
2023:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1993:
1986:
1983:
1976:
1973:
1965:
1956:
1953:
1929:
1922:
1919:
1915:
1910:
1908:
1900:
1871:
1870:
1869:
1855:
1828:
1825:
1823:
1817:
1813:
1810:
1807:
1804:
1798:
1795:
1786:
1783:
1777:
1774:
1768:
1765:
1759:
1749:
1746:
1738:
1734:
1730:
1727:
1721:
1718:
1709:
1706:
1700:
1697:
1694:
1691:
1689:
1683:
1679:
1676:
1673:
1670:
1667:
1664:
1656:
1647:
1644:
1620:
1613:
1610:
1606:
1601:
1599:
1591:
1562:
1561:
1560:
1546:
1539:. First, the
1526:
1506:
1478:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1439:
1435:
1431:
1428:
1422:
1417:
1413:
1407:
1398:
1395:
1392:
1386:
1380:
1375:
1371:
1365:
1363:
1351:
1341:
1338:
1330:
1326:
1322:
1319:
1310:
1307:
1296:
1287:
1278:
1275:
1251:
1249:
1242:
1238:
1227:
1224:
1215:
1212:
1209:
1203:
1194:
1191:
1180:
1171:
1162:
1159:
1135:
1133:
1126:
1122:
1110:
1109:
1108:
1092:
1088:
1065:
1061:
1032:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
997:
991:
984:
981:
972:
968:
964:
959:
952:
949:
941:
937:
932:
929:
924:
919:
912:
909:
902:
897:
894:
889:
887:
879:
876:
873:
870:
868:
851:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
816:
810:
803:
800:
791:
787:
783:
778:
771:
768:
760:
756:
751:
748:
743:
738:
731:
728:
721:
716:
713:
708:
706:
698:
695:
692:
689:
687:
667:
666:
665:
595:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
562:
557:
546:
545:
544:
542:
526:
506:
486:
458:
452:
445:
442:
433:
429:
425:
420:
413:
410:
402:
398:
393:
390:
385:
380:
373:
370:
363:
358:
355:
350:
348:
338:
333:
329:
325:
320:
317:
312:
307:
302:
298:
294:
289:
286:
281:
279:
274:
263:
262:
261:
247:
227:
207:
199:
195:
186:
177:
175:
171:
167:
163:
159:
155:
154:quasiperiodic
151:
147:
131:
128:
125:
105:
102:
99:
91:
86:
84:
80:
68:
63:
61:
57:
53:
49:
45:
41:
32:
19:
7736:Mary Tsingou
7701:David Ruelle
7696:Otto Rössler
7641:Michel HĂ©non
7611:Leon O. Chua
7568:Tilt-A-Whirl
7562:
7538:FPUT problem
7423:Standard map
7418:Logistic map
7243:
7017:Chaos theory
6945:
6936:
6920:
6911:
6904:
6895:
6886:
6877:
6868:
6859:
6855:Vol. 43 (8).
6845:
6836:
6803:
6797:
6791:
6774:
6768:
6735:
6729:
6685:
6679:
6645:
6641:
6631:
6606:
6600:
6594:
6559:
6553:
6536:Reed College
6531:
6495:
6489:
6449:
6443:
6379:
6245:eccentricity
6222:
6059:
5948:
5625:
5597:
5592:
5590:
5520:
5308:
5265:
5220:
4885:
4883:
4786:
4102:
4093:
4090:Trajectories
3943:
3735:
3679:
3406:
3334:
2981:
2875:
2723:
2220:
2077:
1847:
1498:
1052:
615:
478:
191:
170:Reed College
161:
144:has a large
89:
87:
79:inextensible
64:
39:
37:
7721:Nina Snaith
7711:Yakov Sinai
7596:Rufus Bowen
7346:Duffing map
7331:Baker's map
7256:Theoretical
7168:SRB measure
7075:Phase space
7045:Bifurcation
5622:Boundedness
3005:and radius
2880:defined by
2878:phase space
7779:Complexity
7676:Edward Ott
7523:Convection
7448:Continuous
7123:Ergodicity
6864:Vol.35(1).
6609:(2): 142.
6534:(Thesis).
6413:References
3855:, but the
3741:integrable
1868:equation:
1559:equation:
7691:Mary Rees
7651:Bryna Kra
7584:theorists
7393:Ikeda map
7383:HĂ©non map
7373:Gauss map
7055:Limit set
7040:Attractor
6930:Physica D
6662:0302-0738
6569:0912.5168
6555:Physica D
6360:μ
6334:μ
6308:μ
6282:μ
6260:μ
6197:μ
6186:θ
6181:
6172:−
6132:μ
6127:θ
6123:
6114:−
6038:θ
6033:
6011:−
5988:θ
5984:
5969:−
5927:θ
5922:
5900:−
5877:θ
5873:
5858:−
5829:˙
5826:θ
5797:˙
5757:˙
5702:θ
5695:θ
5570:τ
5539:τ
5499:θ
5456:π
5442:θ
5415:μ
5378:π
5364:θ
5337:μ
5223:invariant
5191:π
5177:θ
5150:μ
5113:π
5099:θ
5072:μ
5035:π
5021:θ
4994:μ
4957:π
4943:θ
4916:μ
4860:θ
4851:τ
4839:θ
4814:τ
4765:θ
4722:π
4708:θ
4681:μ
4644:π
4630:θ
4603:μ
4566:π
4552:θ
4525:μ
4488:π
4474:θ
4447:μ
4410:π
4396:θ
4369:μ
4332:π
4318:θ
4291:μ
4254:π
4240:θ
4213:μ
4176:π
4162:θ
4135:μ
4067:θ
4040:θ
4010:˙
4007:θ
3992:θ
3985:θ
3970:˙
3922:∞
3910:∪
3892:∈
3889:μ
3863:μ
3810:−
3791:μ
3751:μ
3689:μ
3660:θ
3656:
3647:−
3635:˙
3632:θ
3616:˙
3613:θ
3604:˙
3592:−
3579:¨
3576:θ
3560:μ
3557:−
3553:θ
3549:
3525:˙
3522:θ
3499:¨
3496:θ
3487:−
3481:¨
3463:μ
3309:⏟
3300:θ
3294:−
3290:θ
3286:
3257:θ
3253:
3244:−
3212:⏟
3199:˙
3196:θ
3149:˙
3140:−
3134:˙
3131:θ
3081:˙
3078:θ
3066:˙
3054:θ
2960:θ
2908:θ
2844:−
2826:μ
2815:μ
2812:−
2794:¨
2758:˙
2755:θ
2732:θ
2704:θ
2700:
2685:−
2675:θ
2671:∂
2659:∂
2653:−
2640:˙
2635:θ
2601:θ
2582:θ
2573:∂
2561:∂
2545:˙
2542:θ
2531:θ
2527:
2506:−
2479:θ
2458:∂
2446:∂
2440:−
2427:˙
2364:∂
2352:∂
2336:˙
2301:θ
2249:θ
2196:θ
2192:
2186:−
2183:μ
2162:˙
2159:θ
2149:−
2143:¨
2125:μ
2086:μ
2056:¨
2024:θ
2020:
1999:−
1987:˙
1984:θ
1957:˙
1948:∂
1936:∂
1898:∂
1886:∂
1818:θ
1814:
1799:˙
1796:θ
1787:˙
1769:¨
1766:θ
1750:¨
1747:θ
1722:˙
1719:θ
1710:˙
1684:θ
1680:
1665:−
1648:˙
1645:θ
1639:∂
1627:∂
1592:θ
1589:∂
1577:∂
1547:θ
1527:θ
1479:θ
1475:
1460:−
1418:θ
1352:∴
1342:˙
1339:θ
1311:˙
1308:θ
1302:∂
1294:∂
1279:˙
1276:θ
1270:∂
1258:∂
1243:θ
1228:˙
1195:˙
1186:∂
1178:∂
1163:˙
1154:∂
1142:∂
1093:θ
1033:θ
1029:
1014:−
985:˙
982:θ
953:˙
913:˙
852:θ
848:
821:−
804:˙
801:θ
772:˙
732:˙
696:−
596:θ
592:
577:−
507:θ
446:˙
443:θ
414:˙
374:˙
7809:Category
7767:articles
7509:Physical
7428:Tent map
7318:Discrete
7258:branches
7188:Theorems
7024:Concepts
6498:: 1117.
4789:pendulum
1848:And the
150:periodic
67:pendulum
54:that is
7765:Related
7573:Weather
7511:systems
7304:Chaotic
7050:Fractal
6808:Bibcode
6806:: 989.
6700:Bibcode
6611:Bibcode
6574:Bibcode
6500:Bibcode
6454:Bibcode
4058:. When
83:pulleys
69:, mass
56:chaotic
7671:Hee Oh
7306:maps (
7153:Mixing
6660:
6274:. For
5593:Type B
5078:1.1727
4886:Type A
3680:where
3335:Where
479:where
260:, is:
7582:Chaos
7361:outer
7065:Orbit
6765:(PDF)
6690:arXiv
6564:arXiv
5632:focus
5156:1.555
5000:2.394
4922:1.665
48:swing
7308:list
7032:Core
6658:ISSN
6337:<
6285:>
5687:and
5573:>
3427:and
2947:and
2288:and
1519:and
129:>
90:i.e.
38:The
6816:doi
6804:380
6779:doi
6740:doi
6708:doi
6650:doi
6619:doi
6582:doi
6560:239
6508:doi
6462:doi
6178:cos
6120:cos
6030:cos
5981:cos
5919:cos
5870:cos
3653:sin
3546:cos
3283:sin
3250:cos
2697:sin
2524:cos
2189:cos
2017:cos
1811:sin
1677:sin
1472:cos
1026:cos
845:cos
589:cos
196:or
168:at
7811::
6814:.
6802:.
6775:46
6773:.
6767:.
6752:^
6736:46
6734:.
6720:^
6706:.
6698:.
6686:62
6684:.
6670:^
6656:.
6646:51
6644:.
6640:.
6617:.
6607:54
6605:.
6580:.
6572:.
6558:.
6544:^
6520:^
6506:.
6496:56
6494:.
6474:^
6460:.
6450:52
6448:.
6420:^
6409:.
6080::
5433:,
5421:25
5355:,
5343:10
5168:,
5090:,
5012:,
4934:,
4699:,
4687:24
4621:,
4609:21
4543:,
4531:19
4465:,
4453:16
4387:,
4309:,
4231:,
4153:,
3941:.
3831:35
3825:15
3728:.
2900:,
2777::
2315::
2241:,
1107::
1080:,
543::
176:.
152:,
62:.
7310:)
7009:e
7002:t
6995:v
6822:.
6818::
6810::
6785:.
6781::
6746:.
6742::
6714:.
6710::
6702::
6692::
6664:.
6652::
6625:.
6621::
6613::
6588:.
6584::
6576::
6566::
6538:.
6514:.
6510::
6502::
6468:.
6464::
6456::
6397:m
6394:3
6391:=
6388:M
6340:1
6314:1
6311:=
6288:1
6257:1
6231:h
6203:)
6190:0
6169:1
6165:(
6159:0
6155:r
6151:=
6144:h
6111:1
6107:h
6102:=
6095:r
6068:r
6042:0
6025:0
6021:r
6017:g
6014:m
6006:0
6002:r
5998:g
5995:M
5992:=
5978:r
5975:g
5972:m
5966:r
5963:g
5960:M
5931:0
5914:0
5910:r
5906:g
5903:m
5895:0
5891:r
5887:g
5884:M
5881:=
5867:r
5864:g
5861:m
5855:r
5852:g
5849:M
5846:+
5842:)
5836:2
5817:2
5813:r
5809:+
5804:2
5794:r
5786:(
5782:m
5777:2
5774:1
5769:+
5764:2
5754:r
5747:M
5742:2
5739:1
5734:=
5731:E
5706:0
5698:=
5673:0
5669:r
5665:=
5662:r
5642:m
5606:M
5576:0
5566:,
5563:0
5560:=
5557:)
5554:0
5551:(
5548:r
5545:=
5542:)
5536:(
5533:r
5503:0
5459:2
5451:=
5446:0
5418:=
5381:2
5373:=
5368:0
5340:=
5294:r
5274:r
5251:0
5248:=
5245:)
5242:0
5239:(
5236:r
5194:2
5186:=
5181:0
5153:=
5116:2
5108:=
5103:0
5075:=
5038:2
5030:=
5025:0
4997:=
4960:2
4952:=
4947:0
4919:=
4869:)
4866:t
4863:(
4857:=
4854:)
4848:+
4845:t
4842:(
4835:,
4832:)
4829:t
4826:(
4823:r
4820:=
4817:)
4811:+
4808:t
4805:(
4802:r
4769:0
4725:2
4717:=
4712:0
4684:=
4647:2
4639:=
4634:0
4606:=
4569:2
4561:=
4556:0
4528:=
4491:2
4483:=
4478:0
4450:=
4413:2
4405:=
4400:0
4375:6
4372:=
4335:2
4327:=
4322:0
4297:5
4294:=
4257:2
4249:=
4244:0
4219:3
4216:=
4179:2
4171:=
4166:0
4141:2
4138:=
4071:0
4044:0
4019:0
4016:=
4001:,
3996:0
3988:=
3982:,
3979:v
3976:=
3967:r
3961:,
3958:0
3955:=
3952:r
3925:)
3919:,
3916:3
3913:(
3907:)
3904:1
3901:,
3898:0
3895:(
3869:3
3866:=
3843:.
3840:.
3837:.
3834:,
3828:,
3822:,
3819:3
3816:=
3813:1
3805:2
3801:n
3797:4
3794:=
3771:3
3768:=
3765:m
3761:/
3757:M
3754:=
3716:m
3712:/
3706:t
3702:M
3698:=
3693:t
3650:g
3642:2
3625:R
3622:+
3601:r
3595:2
3589:=
3570:r
3563:)
3543:(
3540:g
3537:+
3532:2
3515:r
3512:=
3505:)
3490:R
3478:r
3472:(
3467:t
3435:I
3415:R
3388:2
3384:R
3380:I
3375:+
3372:m
3369:+
3366:M
3363:=
3358:t
3354:M
3340:t
3337:M
3320:,
3315:U
3304:)
3297:M
3280:m
3276:(
3272:R
3269:g
3266:+
3262:)
3247:m
3241:M
3237:(
3233:r
3230:g
3223:+
3218:T
3206:2
3187:2
3183:r
3179:m
3174:2
3171:1
3166:+
3161:2
3156:)
3146:r
3125:R
3121:(
3114:t
3110:M
3104:2
3101:1
3092:=
3088:)
3072:,
3063:r
3057:,
3051:,
3048:r
3044:(
3038:H
3013:R
2993:I
2956:p
2933:r
2929:p
2888:r
2858:M
2855:+
2852:m
2847:M
2841:m
2835:g
2832:=
2823:+
2820:1
2809:1
2803:g
2800:=
2791:r
2694:r
2691:g
2688:m
2682:=
2664:H
2650:=
2631:p
2615:2
2611:r
2607:m
2597:p
2591:=
2578:p
2566:H
2555:=
2521:g
2518:m
2515:+
2512:g
2509:M
2498:3
2494:r
2490:m
2484:2
2475:p
2469:=
2462:r
2451:H
2437:=
2422:r
2418:p
2404:m
2401:+
2398:M
2392:r
2388:p
2382:=
2373:r
2369:p
2357:H
2346:=
2333:r
2297:p
2274:r
2270:p
2229:r
2206:0
2203:=
2200:)
2180:(
2177:g
2174:+
2169:2
2152:r
2140:r
2134:)
2131:1
2128:+
2122:(
2097:m
2094:M
2089:=
2053:r
2047:)
2044:m
2041:+
2038:M
2035:(
2032:=
2014:g
2011:m
2008:+
2005:g
2002:M
1994:2
1977:r
1974:m
1966:)
1954:r
1941:L
1930:(
1923:t
1920:d
1916:d
1911:=
1901:r
1891:L
1856:r
1829:0
1826:=
1808:g
1805:+
1784:r
1778:2
1775:+
1760:r
1739:2
1735:r
1731:m
1728:+
1707:r
1701:r
1698:m
1695:2
1692:=
1674:r
1671:g
1668:m
1657:)
1632:L
1621:(
1614:t
1611:d
1607:d
1602:=
1582:L
1507:r
1469:r
1466:g
1463:m
1457:r
1454:g
1451:M
1448:+
1440:2
1436:r
1432:m
1429:2
1423:2
1414:p
1408:+
1402:)
1399:m
1396:+
1393:M
1390:(
1387:2
1381:2
1376:r
1372:p
1366:=
1357:H
1331:2
1327:r
1323:m
1320:=
1297:T
1288:=
1263:L
1252:=
1239:p
1225:r
1219:)
1216:m
1213:+
1210:M
1207:(
1204:=
1192:r
1181:T
1172:=
1160:r
1147:L
1136:=
1127:r
1123:p
1089:p
1066:r
1062:p
1023:r
1020:g
1017:m
1011:r
1008:g
1005:M
1002:+
998:)
992:2
973:2
969:r
965:+
960:2
950:r
942:(
938:m
933:2
930:1
925:+
920:2
910:r
903:M
898:2
895:1
890:=
880:U
877:+
874:T
871:=
862:H
842:r
839:g
836:m
833:+
830:r
827:g
824:M
817:)
811:2
792:2
788:r
784:+
779:2
769:r
761:(
757:m
752:2
749:1
744:+
739:2
729:r
722:M
717:2
714:1
709:=
699:U
693:T
690:=
681:L
650:H
626:L
586:r
583:g
580:m
574:r
571:g
568:M
565:=
558:U
527:U
487:r
459:)
453:2
434:2
430:r
426:+
421:2
411:r
403:(
399:m
394:2
391:1
386:+
381:2
371:r
364:M
359:2
356:1
351:=
339:2
334:m
330:v
326:m
321:2
318:1
313:+
308:2
303:M
299:v
295:M
290:2
287:1
282:=
275:T
248:T
228:M
208:m
132:m
126:M
106:m
103:=
100:M
75:M
71:m
20:)
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