1977:
5185:
52:
1918:, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.
1774:, and the heartbeat while resting. The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting. For a physical pendulum under friction, the resting state will be a fixed-point attractor. The difference with the clock pendulum is that there, energy is injected by the
109:
741:
5177:
3400:
1871:
1782:
1664:, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between
1641:
1838:
1656:
of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical
1802:
There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If
307:
is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit
3322:
root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simpleâit is not simply that the points nearest one root all map there, giving a
1719:
during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are
2841:, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.
3096:
is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire
1648:. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor.
3440:; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are
768:. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points.
4859:
140:
is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.
4032:
1744:, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2 points, 3 points, 3Ă2 points, 4 points, 5 points, or any given positive integer number of points.
1949:, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of SinaiâRuelleâBowen type.
288:, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the
1671:
In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the
265:. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to
3394:
1594:. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g.
2227:
1668:. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).
850:
1485:
625:
3047:
2896:
2627:
2601:
2443:
2728:
2122:
2096:
1174:
2575:
2397:
2034:
1331:
440:
229:
of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be
2154:
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied
3280:
3165:
2968:
2932:
2148:
521:
735:
3237:
3211:
2365:
2319:
1357:
1028:
1002:
576:
2273:
2066:
657:
1521:
1203:
1108:
2970:
all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
3300:
3185:
3115:
3094:
3070:
2998:
2808:
2788:
2768:
2748:
2691:
2671:
2647:
2543:
2523:
2503:
2483:
2463:
2417:
2339:
2293:
2247:
2002:
1452:
1432:
1412:
1384:
1290:
1270:
1250:
1223:
1148:
1128:
1079:
1055:
967:
947:
927:
903:
876:
697:
677:
596:
541:
480:
460:
405:
3545:
The image and video show the attractor of a second order 3-D Sprott-type polynomial, originally computed by
Nicholas Desprez using the Chaoscope freeware (cf.
368:
would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.
2810:, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.
3323:
basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function
237:. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a
3455:
may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The
3781:
1657:
system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a
2068:
is visited over the course of 10 iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A
4277:
218:, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of
3302:
is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.
5474:
5304:
4106:
308:
set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.
598:
units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane
3411: â 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.
1915:
4291:
1736:
system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a
292:
of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.
3730:
Grebogi Celso, Ott Edward, Yorke James A (1987). "Chaos, Strange
Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics".
4877:
3452:
1490:
Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive
5184:
4944:
4084:
5137:
5084:
190:
in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a
5417:
4323:
38:
5652:
4356:
4232:
4216:
1733:
1458:
on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of
749:
95:
73:
5459:
5147:
3835:
66:
5334:
171:
dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in
4296:
4270:
117:
4883:
1808:
186:
If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented
5152:
3326:
5142:
4739:
4696:
2161:
1875:
1665:
1536:
775:
5499:
5407:
5272:
4612:
1911:
1721:
1645:
4817:
4899:
4383:
4260:
2694:
2069:
1034:
5412:
4937:
3464:
2974:
1858:
waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its
31:
17:
5479:
1461:
601:
5527:
4549:
4274:
3282:
gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix
2934:
from all initial points except 0; there is no attractor and therefore no basin of attraction. But if
1688:
1653:
1595:
1587:
744:
Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of the
4189:
3868:
3003:
2852:
2606:
2580:
2422:
5231:
4405:
2700:
375:), in which case the evolution of any two distinct points of the attractor result in exponentially
60:
5176:
2101:
2075:
1153:
5392:
5157:
5064:
2831:
2548:
2370:
2007:
1953:
1938:
1560:
1295:
410:
5132:
4167:(2011). "Stochastic climate dynamics: Random attractors and time-dependent invariant measures".
3846:(2011). "Stochastic climate dynamics: Random attractors and time-dependent invariant measures".
5432:
5044:
4891:
4842:
4450:
4316:
4184:
3863:
3667:
Greenwood, J. A.; J. B. P. Williamson (6 December 1966). "Contact of
Nominally Flat Surfaces".
1976:
1786:
258:
77:
3242:
3127:
2937:
2901:
2127:
485:
5582:
5489:
5287:
5114:
5049:
5024:
4930:
4676:
4368:
3775:
3121:
1716:
1712:
1491:
702:
303:
is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A
280:: if it were not for some driving force, the motion would cease. (Dissipation may come from
3216:
3190:
2344:
2298:
1336:
1007:
972:
546:
5592:
5344:
5244:
5089:
4837:
4832:
4622:
4554:
4176:
4143:
4047:
4003:
3935:
3900:
3855:
3808:
3739:
3676:
3633:
3526:
2252:
2039:
1576:
1563:. More complex attractors that cannot be categorized as simple geometric subsets, such as
630:
262:
215:
5422:
4244:
3516:
1961:
1497:
1179:
1084:
8:
5751:
5552:
5509:
5494:
5339:
5292:
5277:
5262:
5162:
5069:
5054:
5039:
4595:
4572:
4455:
4440:
4373:
3521:
2827:
1903:
1692:
1603:
1544:
157:
5094:
4507:
4180:
4147:
4098:
4051:
4007:
3948:
3904:
3859:
3812:
3743:
3680:
3637:
5730:
5597:
5427:
5314:
5309:
5201:
5079:
4981:
4822:
4802:
4766:
4761:
4524:
4063:
4019:
3824:
3763:
3711:
3692:
3649:
3511:
3471:
3315:
3285:
3170:
3100:
3079:
3055:
2983:
2793:
2773:
2753:
2733:
2676:
2656:
2632:
2528:
2508:
2488:
2468:
2448:
2402:
2324:
2278:
2232:
1987:
1934:
1711:
shape, are both much more complex surfaces when examined under a microscope, and their
1703:. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly
1672:
1611:
1607:
1599:
1556:
1552:
1437:
1417:
1397:
1369:
1275:
1255:
1235:
1208:
1133:
1113:
1064:
1040:
952:
932:
912:
888:
861:
682:
662:
581:
526:
465:
445:
390:
277:
3318:
of iterating to a root of a nonlinear expression. If the expression has more than one
3314:
can give rise to a richer variety of behavior than can linear systems. One example is
379:, which complicates prediction when even the smallest noise is present in the system.
121:
5602:
5567:
5557:
5454:
5074:
4996:
4865:
4827:
4751:
4659:
4564:
4470:
4445:
4435:
4378:
4361:
4351:
4346:
4309:
4228:
4212:
4155:
4080:
3916:
3755:
3696:
3653:
3560:
3407:
showing basins of attraction in the complex plane for using Newton's method to solve
3239:, making the entire number line the basin of attraction for 0. And the matrix system
2978:
2823:
1804:
1767:
1700:
1684:
230:
180:
153:
5617:
4067:
4023:
3828:
5710:
5622:
5572:
5469:
5397:
5349:
5226:
5211:
5206:
5001:
4782:
4649:
4632:
4460:
4194:
4151:
4055:
4011:
3908:
3873:
3816:
3767:
3747:
3684:
3641:
3433:
3311:
2838:
1965:
1548:
1540:
254:
191:
133:
4249:
4033:"Small random perturbations of dynamical systems and the definition of attractors"
3912:
3474:, if it has a global attractor, then this attractor will be of finite dimensions.
5642:
5537:
5464:
5297:
5109:
5099:
4797:
4734:
4395:
4281:
3796:
3751:
3719:. U.S. Department of Commerce, National Institute of Standards (NIST). p. 5.
3496:
3486:
352:
is not a limit set. Because of the dissipation due to air resistance, the point
5632:
5577:
4492:
4198:
3973:
3877:
3072:
will have all elements of the dynamic vector diverge to infinity if the largest
1567:
wild sets, were known of at the time but were thought to be fragile anomalies.
1494:(preventing a point from being an attractor), others relax the requirement that
5725:
5692:
5687:
5682:
5484:
5374:
5369:
5267:
5216:
5006:
4812:
4744:
4715:
4671:
4654:
4637:
4590:
4534:
4519:
4487:
4425:
3563:
3491:
3470:
For the three-dimensional, incompressible NavierâStokes equation with periodic
3404:
1859:
1789:
1771:
1763:
1737:
285:
176:
4756:
5745:
5720:
5677:
5667:
5662:
5562:
5542:
5402:
5324:
5221:
5029:
4666:
4642:
4512:
4482:
4465:
4430:
4415:
4205:
4127:
3991:
3437:
3050:
2834:
1930:
1844:
1658:
1572:
1568:
482:-dimensional phase space, representing the initial state of the system, then
312:
4286:
3117:-dimensional space of potential initial vectors is the basin of attraction.
2837:, every point in the phase space is in the basin of attraction. However, in
2321:
will rapidly lead to function values that go to negative infinity; starting
740:
108:
5672:
5637:
5547:
5504:
5359:
5354:
4953:
4911:
4906:
4807:
4787:
4544:
4477:
4221:
4164:
4094:
3987:
3978:
3920:
3843:
3759:
3688:
3501:
3396:, the following initial conditions are in successive basins of attraction:
2155:
1981:
1942:
1926:
1907:
1811:), the trajectory is no longer closed, and the limit cycle becomes a limit
1741:
1644:
Weakly attracting fixed point for a complex number evolving according to a
1455:
376:
234:
219:
5319:
1957:
1704:
5657:
5647:
5532:
5282:
5104:
5011:
4872:
4792:
4502:
4497:
3969:
3621:
3319:
2819:
1946:
1759:
1753:
1591:
879:
289:
129:
3729:
3399:
144:
In finite-dimensional systems, the evolving variable may be represented
5715:
5612:
5059:
4725:
4710:
4705:
4686:
4420:
4131:
4059:
4015:
3820:
3645:
3546:
3506:
3073:
1775:
1580:
442:
be a function which specifies the dynamics of the system. That is, if
226:
4254:
1870:
1781:
5627:
5587:
5329:
4991:
4681:
4627:
4539:
4390:
3891:
Strelioff, C.; HĂŒbler, A. (2006). "Medium-Term
Prediction of Chaos".
3568:
745:
296:
270:
172:
4134:; Pelikan; Yorke (1984). "Strange attractors that are not chaotic".
3593:
Attractors for infinite-dimensional non-autonomous dynamical systems
1971:
1640:
5364:
4582:
4126:
3590:
1680:
1676:
1661:
1623:
1564:
1391:
315:
281:
266:
199:
187:
2485:
is an attractor for the function's behaviour. For other values of
5034:
4986:
4529:
4332:
4162:
3660:
3441:
2229:, whose basins of attraction for various values of the parameter
2150:, interspersed with regions of simpler behaviour (white stripes).
2098:, a second bifurcation (leading to four attractor values) around
1899:
1696:
203:
164:
145:
3841:
3436:
to find their roots. Each root has a basin of attraction in the
1937:
of a system describing fluid flow. Strange attractors are often
5607:
4922:
4600:
1708:
1619:
1615:
1532:
856:
343:
is also a limit set, as trajectories converge to it; the point
2822:, over which iterations are defined, such that any point (any
1724:
or fixed points, some of which are categorized as attractors.
3467:
are all known to have global attractors of finite dimension.
1812:
1762:
is a periodic orbit of a continuous dynamical system that is
195:
3951:, Global Attractors in Partial Differential Equations,
3666:
1833:
incommensurate frequencies. For example, here is a 2-torus:
4301:
4265:
1914:
also exist. If a strange attractor is chaotic, exhibiting
1855:
276:
Dynamical systems in the physical world tend to arise from
4077:
269:
the equations, either through analytical means or through
3986:
2818:
An attractor's basin of attraction is the region of the
1837:
1543:. Until the 1960s, attractors were thought of as being
37:"Strange attractor" redirects here. For other uses, see
3605:
2036:. The colour of a point indicates how often the point
3329:
3288:
3245:
3219:
3193:
3173:
3130:
3103:
3082:
3058:
3006:
2986:
2940:
2904:
2855:
2849:
An univariate linear homogeneous difference equation
2796:
2776:
2756:
2736:
2703:
2679:
2659:
2635:
2609:
2583:
2551:
2531:
2511:
2491:
2471:
2451:
2425:
2405:
2373:
2347:
2327:
2301:
2281:
2255:
2235:
2164:
2130:
2104:
2078:
2042:
2010:
1990:
1933:
to describe the attractor resulting from a series of
1906:. This is often the case when the dynamics on it are
1500:
1464:
1440:
1420:
1400:
1372:
1339:
1298:
1278:
1258:
1238:
1211:
1182:
1156:
1136:
1116:
1087:
1067:
1043:
1010:
975:
955:
935:
915:
891:
864:
778:
705:
685:
665:
633:
604:
584:
549:
529:
488:
468:
448:
413:
393:
4266:
Chaoscope, a 3D Strange
Attractor rendering freeware
3933:
Dence, Thomas, "Cubics, chaos and Newton's method",
2577:
will lead to function values that alternate between
359:
is also an attractor. If there was no dissipation,
116:. Another visualization of the same 3D attractor is
2697:), or, as a result of further doubling, any number
1843:A time series corresponding to this attractor is a
578:is the result of the evolution of this state after
3558:
3388:
3294:
3274:
3231:
3213:but to converge to an attractor at the value 0 if
3205:
3179:
3159:
3109:
3088:
3064:
3041:
2992:
2962:
2926:
2890:
2802:
2782:
2762:
2742:
2722:
2685:
2665:
2641:
2621:
2595:
2569:
2537:
2517:
2497:
2477:
2457:
2437:
2411:
2391:
2359:
2333:
2313:
2287:
2267:
2241:
2221:
2142:
2116:
2090:
2060:
2028:
1996:
1984:. The attractor(s) for any value of the parameter
1515:
1479:
1446:
1426:
1406:
1378:
1351:
1325:
1284:
1264:
1244:
1217:
1197:
1168:
1142:
1122:
1102:
1073:
1049:
1022:
996:
961:
941:
921:
897:
870:
844:
729:
691:
671:
651:
619:
590:
570:
535:
515:
474:
454:
434:
399:
3447:
2790:are visited in turn; finally, for some values of
1972:Attractors characterize the evolution of a system
882:characterized by the following three conditions:
5743:
3591:Carvalho, A.; Langa, J.A.; Robinson, J. (2012).
2124:. The behaviour is increasingly complicated for
3890:
3305:
1225:in the phase space with the following property:
4292:Interactive trigonometric attractors generator
1579:and that its attractor had the structure of a
4938:
4317:
4250:A gallery of trigonometric strange attractors
3703:
3549:and the linked project files for parameters).
371:Some attractors are known to be chaotic (see
175:, they may be separate variables such as the
4107:Notices of the American Mathematical Society
4074:
3794:
3780:: CS1 maint: multiple names: authors list (
2844:
1414:, every point that is sufficiently close to
3620:
2367:will also go to negative infinity. But for
1952:Examples of strange attractors include the
248:
4945:
4931:
4324:
4310:
4261:A gallery of polynomial strange attractors
1916:sensitive dependence on initial conditions
1727:
1390:Since the basin of attraction contains an
382:
4188:
4030:
3867:
3709:
1902:structure, that is if it has non-integer
1467:
1454:. The definition of an attractor uses a
1366:There is no proper (non-empty) subset of
607:
299:are similar to the attractor concept. An
96:Learn how and when to remove this message
3453:Parabolic partial differential equations
3398:
3389:{\displaystyle f(x)=x^{3}-2x^{2}-11x+12}
2004:are shown on the ordinate in the domain
1975:
1869:
1780:
1639:
1487:, the Euclidean norm is typically used.
739:
107:
59:This article includes a list of general
4878:List of fractals by Hausdorff dimension
3994:(1971). "On the nature of turbulence".
3968:
3624:(1985). "On the concept of attractor".
3595:. Vol. 182. Springer. p. 109.
3432:Newton's method can also be applied to
2813:
2222:{\displaystyle x_{n+1}=rx_{n}(1-x_{n})}
14:
5744:
4093:
4040:Communications in Mathematical Physics
3996:Communications in Mathematical Physics
3801:Communications in Mathematical Physics
3795:Ruelle, David; Takens, Floris (1971).
3626:Communications in Mathematical Physics
3187:except zero to diverge to infinity if
2830:be iterated into the attractor. For a
1815:. This kind of attractor is called an
1628:
1526:
845:{\displaystyle f(t,(x,v))=(x+tv,v).\ }
372:
257:is generally described by one or more
211:
113:
4926:
4305:
3559:
2649:, the attractor is a single point (a
3547:http://www.chaoscope.org/gallery.htm
1865:
1862:still consists only of sharp lines.
1854:periodic functions (not necessarily
1847:series: A discretely sampled sum of
1770:. Examples include the swings of a
318:has two invariant points: the point
120:. Code capable of rendering this is
45:
5085:Measure-preserving dynamical system
4967:
4287:Online strange attractors generator
4245:Basin of attraction on Scholarpedia
3955:, Elsevier, 2002, pp. 885â982.
273:, often with the aid of computers.
202:, or even a complicated set with a
24:
3962:
3842:Chekroun M. D.; Simonnet E. &
3463:, and the two-dimensional, forced
1941:in a few directions, but some are
1626:), then the attractor is called a
1163:
65:it lacks sufficient corresponding
39:Strange Attractor (disambiguation)
25:
5763:
5653:Oleksandr Mykolayovych Sharkovsky
4860:How Long Is the Coast of Britain?
4238:
3713:Surface Finish Metrology Tutorial
3606:Kantz, H.; Schreiber, T. (2004).
3120:Similar features apply to linear
1803:two of these frequencies form an
1778:mechanism to maintain the cycle.
679:is the position of the particle,
5183:
5175:
4952:
4099:"What is...a Strange Attractor?"
3669:Proceedings of the Royal Society
2770:, any given number of values of
2650:
1836:
1480:{\displaystyle \mathbb {R} ^{n}}
1386:having the first two properties.
737:, and the evolution is given by
620:{\displaystyle \mathbb {R} ^{2}}
327:of minimum height and the point
50:
3942:
3927:
3884:
1272:, there is a positive constant
1110:, which consists of all points
156:. The attractor is a region in
5418:RabinovichâFabrikant equations
4884:The Fractal Geometry of Nature
4163:Chekroun, M. D.; E. Simonnet;
3788:
3723:
3614:
3608:Nonlinear time series analysis
3599:
3584:
3552:
3539:
3448:Partial differential equations
3339:
3333:
3310:Equations or systems that are
3042:{\displaystyle X_{t}=AX_{t-1}}
2950:
2942:
2914:
2906:
2891:{\displaystyle x_{t}=ax_{t-1}}
2622:{\displaystyle x\approx 0.799}
2596:{\displaystyle x\approx 0.513}
2438:{\displaystyle x\approx 0.615}
2216:
2197:
2055:
2043:
1797:
1747:
1666:stable and unstable equilibria
1635:
1510:
1504:
1314:
1302:
1192:
1186:
1160:
1097:
1091:
991:
979:
833:
812:
806:
803:
791:
782:
724:
712:
646:
634:
565:
553:
504:
492:
429:
417:
336:of maximum height. The point
13:
1:
3953:Handbook of Dynamical Systems
3913:10.1103/PhysRevLett.96.044101
3797:"On the nature of turbulence"
3610:. Cambridge university press.
3532:
3167:causes all initial values of
2723:{\displaystyle k\times 2^{n}}
1912:strange nonchaotic attractors
1740:. This is illustrated by the
1604:fundamental geometric objects
523:and, for a positive value of
214:below). If the variable is a
4331:
4156:10.1016/0167-2789(84)90282-3
3752:10.1126/science.238.4827.632
3425:2.352836327 converges to â3;
3306:Nonlinear equation or system
2249:are shown in the figure. If
2117:{\displaystyle r\approx 3.5}
2091:{\displaystyle r\approx 3.0}
1646:complex quadratic polynomial
1586:Two simple attractors are a
1169:{\displaystyle t\to \infty }
194:, a finite set of points, a
27:Concept in dynamical systems
7:
5153:Poincaré recurrence theorem
4900:Chaos: Making a New Science
4226:Chaos: Making a New Science
4199:10.1016/j.physd.2011.06.005
3939:81, November 1997, 403â408.
3878:10.1016/j.physd.2011.06.005
3480:
3428:2.352836323 converges to 1.
3419:2.35284172 converges to â3;
2695:period-doubling bifurcation
2570:{\displaystyle 0<x<1}
2545:is 3.2, starting values of
2419:values rapidly converge to
2392:{\displaystyle 0<x<1}
2029:{\displaystyle 0<x<1}
1980:Bifurcation diagram of the
1792:: an attracting limit cycle
1531:Attractors are portions or
1326:{\displaystyle f(t,b)\in N}
435:{\displaystyle f(t,\cdot )}
112:Visual representation of a
10:
5768:
5148:PoincarĂ©âBendixson theorem
3422:2.35283735 converges to 4;
3416:2.35287527 converges to 4;
3000:, of the homogeneous form
2975:matrix difference equation
1876:Lorenz's strange attractor
1751:
1571:was able to show that his
1232:For any open neighborhood
36:
32:Attractor (disambiguation)
29:
5701:
5518:
5500:Swinging Atwood's machine
5445:
5383:
5253:
5240:
5192:
5173:
5143:KrylovâBogolyubov theorem
5123:
5020:
4960:
4851:
4775:
4724:
4695:
4611:
4581:
4563:
4404:
4339:
4257:Chua's circuit simulation
3710:Vorberger, T. V. (1990).
2845:Linear equation or system
2750:; at yet other values of
2505:, more than one value of
1547:of the phase space, like
1205:is the set of all points
5408:LotkaâVolterra equations
5232:Synchronization of chaos
5035:axiom A dynamical system
3275:{\displaystyle dX/dt=AX}
3160:{\displaystyle dx/dt=ax}
2963:{\displaystyle |a|<1}
2927:{\displaystyle |a|>1}
2898:diverges to infinity if
2445:, i.e. at this value of
2143:{\displaystyle r>3.6}
1559:, and simple regions of
1545:simple geometric subsets
516:{\displaystyle f(0,a)=a}
249:Motivation of attractors
5393:Double scroll attractor
5158:Stable manifold theorem
5065:False nearest neighbors
4255:Double scroll attractor
3465:NavierâStokes equations
2693:are visited in turn (a
1954:double-scroll attractor
1894:An attractor is called
1728:Finite number of points
1561:three-dimensional space
730:{\displaystyle a=(x,v)}
407:represent time and let
383:Mathematical definition
80:more precise citations.
5433:Van der Pol oscillator
5413:MackeyâGlass equations
5045:Box-counting dimension
4892:The Beauty of Fractals
3689:10.1098/rspa.1966.0242
3412:
3390:
3296:
3276:
3233:
3232:{\displaystyle a<0}
3207:
3206:{\displaystyle a>0}
3181:
3161:
3124:. The scalar equation
3122:differential equations
3111:
3090:
3066:
3043:
2994:
2964:
2928:
2892:
2826:) in that region will
2804:
2784:
2764:
2744:
2724:
2687:
2667:
2653:), at other values of
2643:
2623:
2597:
2571:
2539:
2519:
2499:
2479:
2459:
2439:
2413:
2393:
2361:
2360:{\displaystyle x>1}
2335:
2315:
2314:{\displaystyle x<0}
2289:
2269:
2243:
2223:
2151:
2144:
2118:
2092:
2062:
2030:
1998:
1891:
1886: = 10,
1882: = 28,
1794:
1649:
1517:
1481:
1448:
1428:
1408:
1380:
1353:
1352:{\displaystyle t>T}
1327:
1286:
1266:
1246:
1219:
1199:
1170:
1144:
1124:
1104:
1075:
1051:
1024:
1023:{\displaystyle t>0}
998:
997:{\displaystyle f(t,a)}
963:
943:
923:
899:
872:
846:
769:
731:
693:
673:
653:
621:
592:
572:
571:{\displaystyle f(t,a)}
537:
517:
476:
456:
436:
401:
377:diverging trajectories
125:
5583:Svetlana Jitomirskaya
5490:Multiscroll attractor
5335:Interval exchange map
5288:Dyadic transformation
5273:Complex quadratic map
5115:Topological conjugacy
5050:Correlation dimension
5025:Anosov diffeomorphism
4075:David Ruelle (1989).
3402:
3391:
3297:
3277:
3234:
3208:
3182:
3162:
3112:
3091:
3067:
3044:
2995:
2965:
2929:
2893:
2805:
2785:
2765:
2745:
2725:
2688:
2668:
2644:
2624:
2598:
2572:
2540:
2520:
2500:
2480:
2460:
2440:
2414:
2394:
2362:
2336:
2316:
2290:
2270:
2268:{\displaystyle r=2.6}
2244:
2224:
2145:
2119:
2093:
2063:
2061:{\displaystyle (r,x)}
2031:
1999:
1979:
1873:
1784:
1643:
1518:
1482:
1449:
1429:
1409:
1381:
1354:
1328:
1287:
1267:
1247:
1220:
1200:
1171:
1145:
1125:
1105:
1076:
1052:
1025:
999:
964:
944:
924:
900:
873:
847:
743:
732:
694:
674:
654:
652:{\displaystyle (x,v)}
622:
593:
573:
538:
518:
477:
457:
437:
402:
206:structure known as a
111:
5593:Edward Norton Lorenz
4838:Lewis Fry Richardson
4833:Hamid Naderi Yeganeh
4623:Burning Ship fractal
4555:Weierstrass function
4280:28 June 2022 at the
4210:The Essence of Chaos
3936:Mathematical Gazette
3527:Convergent evolution
3461:KuramotoâSivashinsky
3327:
3286:
3243:
3217:
3191:
3171:
3128:
3101:
3080:
3056:
3004:
2984:
2938:
2902:
2853:
2814:Basins of attraction
2794:
2774:
2754:
2734:
2701:
2677:
2657:
2633:
2629:. At some values of
2607:
2581:
2549:
2529:
2509:
2489:
2469:
2465:, a single value of
2449:
2423:
2403:
2371:
2345:
2325:
2299:
2279:
2253:
2233:
2162:
2128:
2102:
2076:
2040:
2008:
1988:
1826:-torus if there are
1516:{\displaystyle B(A)}
1498:
1462:
1438:
1418:
1398:
1370:
1337:
1296:
1276:
1256:
1236:
1209:
1198:{\displaystyle B(A)}
1180:
1154:
1134:
1114:
1103:{\displaystyle B(A)}
1085:
1065:
1041:
1008:
973:
953:
933:
913:
889:
862:
776:
703:
683:
663:
631:
602:
582:
547:
527:
486:
466:
446:
411:
391:
286:thermodynamic losses
263:difference equations
30:For other uses, see
5553:Mitchell Feigenbaum
5495:Population dynamics
5480:HĂ©nonâHeiles system
5340:Irrational rotation
5293:Dynamical billiards
5278:Coupled map lattice
5138:Liouville's theorem
5070:Hausdorff dimension
5055:Conservative system
5040:Bifurcation diagram
4596:Space-filling curve
4573:Multifractal system
4456:Space-filling curve
4441:Sierpinski triangle
4275:software laboratory
4181:2011PhyD..240.1685C
4148:1984PhyD...13..261G
4052:1981CMaPh..82..137R
4008:1971CMaPh..20..167R
3905:2006PhRvL..96d4101S
3860:2011PhyD..240.1685C
3813:1971CMaPh..20..167R
3744:1987Sci...238..632G
3681:1966RSPSA.295..300G
3638:1985CMaPh..99..177M
3522:Stable distribution
3472:boundary conditions
2973:Likewise, a linear
2525:may be visited: if
1904:Hausdorff dimension
1805:irrational fraction
1707:, and the marble's
1527:Types of attractors
1523:be a neighborhood.
1059:basin of attraction
295:Invariant sets and
278:dissipative systems
5731:Santa Fe Institute
5598:Aleksandr Lyapunov
5428:Three-body problem
5315:Gingerbreadman map
5202:Bifurcation theory
5080:Lyapunov stability
4823:Aleksandr Lyapunov
4803:Desmond Paul Henry
4767:Self-avoiding walk
4762:Percolation theory
4406:Iterated function
4347:Fractal dimensions
4297:Economic attractor
4079:. Academic Press.
4060:10.1007/BF01206949
4031:D. Ruelle (1981).
4016:10.1007/BF01646553
3821:10.1007/bf01646553
3646:10.1007/BF01212280
3561:Weisstein, Eric W.
3512:Hidden oscillation
3413:
3386:
3292:
3272:
3229:
3203:
3177:
3157:
3107:
3086:
3062:
3039:
2990:
2960:
2924:
2888:
2800:
2780:
2760:
2740:
2720:
2683:
2663:
2639:
2619:
2593:
2567:
2535:
2515:
2495:
2475:
2455:
2435:
2409:
2389:
2357:
2331:
2311:
2285:
2265:
2239:
2219:
2152:
2140:
2114:
2088:
2058:
2026:
1994:
1892:
1795:
1673:nonlinear dynamics
1650:
1513:
1477:
1444:
1424:
1404:
1376:
1349:
1323:
1282:
1262:
1242:
1215:
1195:
1166:
1140:
1120:
1100:
1071:
1047:
1020:
994:
959:
939:
919:
895:
868:
855:An attractor is a
842:
770:
727:
689:
669:
649:
617:
588:
568:
533:
513:
472:
452:
432:
397:
161:-dimensional space
126:
5739:
5738:
5603:BenoĂźt Mandelbrot
5568:Martin Gutzwiller
5558:Peter Grassberger
5441:
5440:
5423:Rössler attractor
5171:
5170:
5075:Invariant measure
4997:Lyapunov exponent
4920:
4919:
4866:Coastline paradox
4843:WacĆaw SierpiĆski
4828:Benoit Mandelbrot
4752:Fractal landscape
4660:Misiurewicz point
4565:Strange attractor
4446:Apollonian gasket
4436:Sierpinski carpet
4271:Research abstract
4175:(21): 1685â1700.
4086:978-0-12-601710-6
3854:(21): 1685â1700.
3738:(4827): 632â638.
3675:(1442): 300â319.
3517:Rössler attractor
3434:complex functions
3295:{\displaystyle A}
3180:{\displaystyle x}
3110:{\displaystyle n}
3089:{\displaystyle A}
3065:{\displaystyle A}
2993:{\displaystyle X}
2839:nonlinear systems
2824:initial condition
2803:{\displaystyle r}
2783:{\displaystyle x}
2763:{\displaystyle r}
2743:{\displaystyle x}
2686:{\displaystyle x}
2666:{\displaystyle r}
2642:{\displaystyle r}
2538:{\displaystyle r}
2518:{\displaystyle x}
2498:{\displaystyle r}
2478:{\displaystyle x}
2458:{\displaystyle r}
2412:{\displaystyle x}
2334:{\displaystyle x}
2288:{\displaystyle x}
2242:{\displaystyle r}
1997:{\displaystyle r}
1962:Rössler attractor
1923:strange attractor
1866:Strange attractor
1701:quantum mechanics
1685:surface roughness
1629:strange attractor
1447:{\displaystyle A}
1427:{\displaystyle A}
1407:{\displaystyle A}
1379:{\displaystyle A}
1285:{\displaystyle T}
1265:{\displaystyle A}
1245:{\displaystyle N}
1218:{\displaystyle b}
1176:. More formally,
1143:{\displaystyle A}
1123:{\displaystyle b}
1074:{\displaystyle A}
1050:{\displaystyle A}
962:{\displaystyle A}
949:is an element of
942:{\displaystyle a}
922:{\displaystyle f}
907:forward invariant
898:{\displaystyle A}
871:{\displaystyle A}
841:
699:is its velocity,
692:{\displaystyle v}
672:{\displaystyle x}
627:with coordinates
591:{\displaystyle t}
536:{\displaystyle t}
475:{\displaystyle n}
462:is a point in an
455:{\displaystyle a}
400:{\displaystyle t}
373:strange attractor
311:For example, the
282:internal friction
212:strange attractor
208:strange attractor
181:unemployment rate
134:dynamical systems
114:strange attractor
106:
105:
98:
16:(Redirected from
5759:
5711:Butterfly effect
5623:Itamar Procaccia
5573:Brosl Hasslacher
5470:Elastic pendulum
5398:Duffing equation
5345:KaplanâYorke map
5263:Arnold's cat map
5251:
5250:
5227:Stability theory
5212:Dynamical system
5207:Control of chaos
5187:
5179:
5163:Takens's theorem
5095:Poincaré section
4965:
4964:
4947:
4940:
4933:
4924:
4923:
4783:Michael Barnsley
4650:Lyapunov fractal
4508:SierpiĆski curve
4461:Blancmange curve
4326:
4319:
4312:
4303:
4302:
4206:Edward N. Lorenz
4202:
4192:
4159:
4142:(1â2): 261â268.
4123:
4121:
4119:
4103:
4090:
4071:
4037:
4027:
3983:
3956:
3949:GeneviĂšve Raugel
3946:
3940:
3931:
3925:
3924:
3888:
3882:
3881:
3871:
3839:
3833:
3832:
3792:
3786:
3785:
3779:
3771:
3727:
3721:
3720:
3718:
3707:
3701:
3700:
3664:
3658:
3657:
3618:
3612:
3611:
3603:
3597:
3596:
3588:
3582:
3581:
3580:
3578:
3576:
3556:
3550:
3543:
3395:
3393:
3392:
3387:
3370:
3369:
3354:
3353:
3301:
3299:
3298:
3293:
3281:
3279:
3278:
3273:
3256:
3238:
3236:
3235:
3230:
3212:
3210:
3209:
3204:
3186:
3184:
3183:
3178:
3166:
3164:
3163:
3158:
3141:
3116:
3114:
3113:
3108:
3095:
3093:
3092:
3087:
3071:
3069:
3068:
3063:
3048:
3046:
3045:
3040:
3038:
3037:
3016:
3015:
2999:
2997:
2996:
2991:
2969:
2967:
2966:
2961:
2953:
2945:
2933:
2931:
2930:
2925:
2917:
2909:
2897:
2895:
2894:
2889:
2887:
2886:
2865:
2864:
2809:
2807:
2806:
2801:
2789:
2787:
2786:
2781:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2729:
2727:
2726:
2721:
2719:
2718:
2692:
2690:
2689:
2684:
2672:
2670:
2669:
2664:
2648:
2646:
2645:
2640:
2628:
2626:
2625:
2620:
2602:
2600:
2599:
2594:
2576:
2574:
2573:
2568:
2544:
2542:
2541:
2536:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2484:
2482:
2481:
2476:
2464:
2462:
2461:
2456:
2444:
2442:
2441:
2436:
2418:
2416:
2415:
2410:
2398:
2396:
2395:
2390:
2366:
2364:
2363:
2358:
2340:
2338:
2337:
2332:
2320:
2318:
2317:
2312:
2294:
2292:
2291:
2286:
2274:
2272:
2271:
2266:
2248:
2246:
2245:
2240:
2228:
2226:
2225:
2220:
2215:
2214:
2196:
2195:
2180:
2179:
2149:
2147:
2146:
2141:
2123:
2121:
2120:
2115:
2097:
2095:
2094:
2089:
2067:
2065:
2064:
2059:
2035:
2033:
2032:
2027:
2003:
2001:
2000:
1995:
1966:Lorenz attractor
1890: = 8/3
1878:for values
1853:
1840:
1832:
1825:
1768:cyclic attractor
1766:. It concerns a
1541:dynamical system
1522:
1520:
1519:
1514:
1486:
1484:
1483:
1478:
1476:
1475:
1470:
1453:
1451:
1450:
1445:
1434:is attracted to
1433:
1431:
1430:
1425:
1413:
1411:
1410:
1405:
1385:
1383:
1382:
1377:
1358:
1356:
1355:
1350:
1332:
1330:
1329:
1324:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1251:
1249:
1248:
1243:
1224:
1222:
1221:
1216:
1204:
1202:
1201:
1196:
1175:
1173:
1172:
1167:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1109:
1107:
1106:
1101:
1080:
1078:
1077:
1072:
1056:
1054:
1053:
1048:
1029:
1027:
1026:
1021:
1003:
1001:
1000:
995:
968:
966:
965:
960:
948:
946:
945:
940:
928:
926:
925:
920:
904:
902:
901:
896:
877:
875:
874:
869:
851:
849:
848:
843:
839:
736:
734:
733:
728:
698:
696:
695:
690:
678:
676:
675:
670:
658:
656:
655:
650:
626:
624:
623:
618:
616:
615:
610:
597:
595:
594:
589:
577:
575:
574:
569:
542:
540:
539:
534:
522:
520:
519:
514:
481:
479:
478:
473:
461:
459:
458:
453:
441:
439:
438:
433:
406:
404:
403:
398:
367:
358:
351:
342:
335:
326:
255:dynamical system
173:economic systems
165:physical systems
101:
94:
90:
87:
81:
76:this article by
67:inline citations
54:
53:
46:
21:
5767:
5766:
5762:
5761:
5760:
5758:
5757:
5756:
5742:
5741:
5740:
5735:
5703:
5697:
5643:Caroline Series
5538:Mary Cartwright
5520:
5514:
5465:Double pendulum
5447:
5437:
5386:
5379:
5305:Exponential map
5256:
5242:
5236:
5194:
5188:
5181:
5167:
5133:Ergodic theorem
5126:
5119:
5110:Stable manifold
5100:Recurrence plot
5016:
4970:
4956:
4951:
4921:
4916:
4847:
4798:Felix Hausdorff
4771:
4735:Brownian motion
4720:
4691:
4614:
4607:
4577:
4559:
4550:Pythagoras tree
4407:
4400:
4396:Self-similarity
4340:Characteristics
4335:
4330:
4282:Wayback Machine
4241:
4190:10.1.1.156.5891
4117:
4115:
4101:
4097:(August 2006).
4087:
4035:
3965:
3963:Further reading
3960:
3959:
3947:
3943:
3932:
3928:
3893:Phys. Rev. Lett
3889:
3885:
3869:10.1.1.156.5891
3840:
3836:
3793:
3789:
3773:
3772:
3728:
3724:
3716:
3708:
3704:
3665:
3661:
3619:
3615:
3604:
3600:
3589:
3585:
3574:
3572:
3557:
3553:
3544:
3540:
3535:
3497:Stable manifold
3487:Cycle detection
3483:
3477:
3457:GinzburgâLandau
3450:
3365:
3361:
3349:
3345:
3328:
3325:
3324:
3316:Newton's method
3308:
3287:
3284:
3283:
3252:
3244:
3241:
3240:
3218:
3215:
3214:
3192:
3189:
3188:
3172:
3169:
3168:
3137:
3129:
3126:
3125:
3102:
3099:
3098:
3081:
3078:
3077:
3057:
3054:
3053:
3027:
3023:
3011:
3007:
3005:
3002:
3001:
2985:
2982:
2981:
2949:
2941:
2939:
2936:
2935:
2913:
2905:
2903:
2900:
2899:
2876:
2872:
2860:
2856:
2854:
2851:
2850:
2847:
2816:
2795:
2792:
2791:
2775:
2772:
2771:
2755:
2752:
2751:
2735:
2732:
2731:
2714:
2710:
2702:
2699:
2698:
2678:
2675:
2674:
2658:
2655:
2654:
2634:
2631:
2630:
2608:
2605:
2604:
2582:
2579:
2578:
2550:
2547:
2546:
2530:
2527:
2526:
2510:
2507:
2506:
2490:
2487:
2486:
2470:
2467:
2466:
2450:
2447:
2446:
2424:
2421:
2420:
2404:
2401:
2400:
2372:
2369:
2368:
2346:
2343:
2342:
2326:
2323:
2322:
2300:
2297:
2296:
2280:
2277:
2276:
2275:, all starting
2254:
2251:
2250:
2234:
2231:
2230:
2210:
2206:
2191:
2187:
2169:
2165:
2163:
2160:
2159:
2129:
2126:
2125:
2103:
2100:
2099:
2077:
2074:
2073:
2072:appears around
2041:
2038:
2037:
2009:
2006:
2005:
1989:
1986:
1985:
1974:
1958:HĂ©non attractor
1868:
1852:
1848:
1831:
1827:
1824:
1816:
1807:(i.e. they are
1800:
1793:
1756:
1750:
1730:
1638:
1529:
1499:
1496:
1495:
1471:
1466:
1465:
1463:
1460:
1459:
1439:
1436:
1435:
1419:
1416:
1415:
1399:
1396:
1395:
1371:
1368:
1367:
1338:
1335:
1334:
1297:
1294:
1293:
1277:
1274:
1273:
1257:
1254:
1253:
1237:
1234:
1233:
1210:
1207:
1206:
1181:
1178:
1177:
1155:
1152:
1151:
1135:
1132:
1131:
1115:
1112:
1111:
1086:
1083:
1082:
1066:
1063:
1062:
1042:
1039:
1038:
1033:There exists a
1009:
1006:
1005:
974:
971:
970:
954:
951:
950:
934:
931:
930:
914:
911:
910:
890:
887:
886:
863:
860:
859:
777:
774:
773:
704:
701:
700:
684:
681:
680:
664:
661:
660:
632:
629:
628:
611:
606:
605:
603:
600:
599:
583:
580:
579:
548:
545:
544:
528:
525:
524:
487:
484:
483:
467:
464:
463:
447:
444:
443:
412:
409:
408:
392:
389:
388:
385:
366:
360:
357:
353:
350:
344:
341:
337:
334:
328:
325:
319:
251:
102:
91:
85:
82:
72:Please help to
71:
55:
51:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5765:
5755:
5754:
5737:
5736:
5734:
5733:
5728:
5726:Predictability
5723:
5718:
5713:
5707:
5705:
5699:
5698:
5696:
5695:
5693:Lai-Sang Young
5690:
5688:James A. Yorke
5685:
5683:Amie Wilkinson
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5640:
5635:
5630:
5625:
5620:
5618:Henri Poincaré
5615:
5610:
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5565:
5560:
5555:
5550:
5545:
5540:
5535:
5530:
5524:
5522:
5516:
5515:
5513:
5512:
5507:
5502:
5497:
5492:
5487:
5485:Kicked rotator
5482:
5477:
5472:
5467:
5462:
5457:
5455:Chua's circuit
5451:
5449:
5443:
5442:
5439:
5438:
5436:
5435:
5430:
5425:
5420:
5415:
5410:
5405:
5400:
5395:
5389:
5387:
5384:
5381:
5380:
5378:
5377:
5375:Zaslavskii map
5372:
5370:Tinkerbell map
5367:
5362:
5357:
5352:
5347:
5342:
5337:
5332:
5327:
5322:
5317:
5312:
5307:
5302:
5301:
5300:
5290:
5285:
5280:
5275:
5270:
5265:
5259:
5257:
5254:
5248:
5238:
5237:
5235:
5234:
5229:
5224:
5219:
5217:Ergodic theory
5214:
5209:
5204:
5198:
5196:
5190:
5189:
5174:
5172:
5169:
5168:
5166:
5165:
5160:
5155:
5150:
5145:
5140:
5135:
5129:
5127:
5124:
5121:
5120:
5118:
5117:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5021:
5018:
5017:
5015:
5014:
5009:
5007:Periodic point
5004:
4999:
4994:
4989:
4984:
4979:
4973:
4971:
4968:
4962:
4958:
4957:
4950:
4949:
4942:
4935:
4927:
4918:
4917:
4915:
4914:
4909:
4904:
4896:
4888:
4880:
4875:
4870:
4869:
4868:
4855:
4853:
4849:
4848:
4846:
4845:
4840:
4835:
4830:
4825:
4820:
4815:
4813:Helge von Koch
4810:
4805:
4800:
4795:
4790:
4785:
4779:
4777:
4773:
4772:
4770:
4769:
4764:
4759:
4754:
4749:
4748:
4747:
4745:Brownian motor
4742:
4731:
4729:
4722:
4721:
4719:
4718:
4716:Pickover stalk
4713:
4708:
4702:
4700:
4693:
4692:
4690:
4689:
4684:
4679:
4674:
4672:Newton fractal
4669:
4664:
4663:
4662:
4655:Mandelbrot set
4652:
4647:
4646:
4645:
4640:
4638:Newton fractal
4635:
4625:
4619:
4617:
4609:
4608:
4606:
4605:
4604:
4603:
4593:
4591:Fractal canopy
4587:
4585:
4579:
4578:
4576:
4575:
4569:
4567:
4561:
4560:
4558:
4557:
4552:
4547:
4542:
4537:
4535:Vicsek fractal
4532:
4527:
4522:
4517:
4516:
4515:
4510:
4505:
4500:
4495:
4490:
4485:
4480:
4475:
4474:
4473:
4463:
4453:
4451:Fibonacci word
4448:
4443:
4438:
4433:
4428:
4426:Koch snowflake
4423:
4418:
4412:
4410:
4402:
4401:
4399:
4398:
4393:
4388:
4387:
4386:
4381:
4376:
4371:
4366:
4365:
4364:
4354:
4343:
4341:
4337:
4336:
4329:
4328:
4321:
4314:
4306:
4300:
4299:
4294:
4289:
4284:
4268:
4263:
4258:
4252:
4247:
4240:
4239:External links
4237:
4236:
4235:
4219:
4203:
4160:
4124:
4091:
4085:
4072:
4046:(1): 137â151.
4028:
4002:(3): 167â192.
3984:
3964:
3961:
3958:
3957:
3941:
3926:
3883:
3834:
3807:(3): 167â192.
3787:
3722:
3702:
3659:
3632:(2): 177â195.
3613:
3598:
3583:
3551:
3537:
3536:
3534:
3531:
3530:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3492:Hyperbolic set
3489:
3482:
3479:
3449:
3446:
3430:
3429:
3426:
3423:
3420:
3417:
3405:Newton fractal
3385:
3382:
3379:
3376:
3373:
3368:
3364:
3360:
3357:
3352:
3348:
3344:
3341:
3338:
3335:
3332:
3307:
3304:
3291:
3271:
3268:
3265:
3262:
3259:
3255:
3251:
3248:
3228:
3225:
3222:
3202:
3199:
3196:
3176:
3156:
3153:
3150:
3147:
3144:
3140:
3136:
3133:
3106:
3085:
3061:
3036:
3033:
3030:
3026:
3022:
3019:
3014:
3010:
2989:
2959:
2956:
2952:
2948:
2944:
2923:
2920:
2916:
2912:
2908:
2885:
2882:
2879:
2875:
2871:
2868:
2863:
2859:
2846:
2843:
2828:asymptotically
2815:
2812:
2799:
2779:
2759:
2739:
2717:
2713:
2709:
2706:
2682:
2673:two values of
2662:
2638:
2618:
2615:
2612:
2592:
2589:
2586:
2566:
2563:
2560:
2557:
2554:
2534:
2514:
2494:
2474:
2454:
2434:
2431:
2428:
2408:
2388:
2385:
2382:
2379:
2376:
2356:
2353:
2350:
2330:
2310:
2307:
2304:
2284:
2264:
2261:
2258:
2238:
2218:
2213:
2209:
2205:
2202:
2199:
2194:
2190:
2186:
2183:
2178:
2175:
2172:
2168:
2139:
2136:
2133:
2113:
2110:
2107:
2087:
2084:
2081:
2057:
2054:
2051:
2048:
2045:
2025:
2022:
2019:
2016:
2013:
1993:
1973:
1970:
1939:differentiable
1925:was coined by
1867:
1864:
1860:power spectrum
1850:
1829:
1820:
1809:incommensurate
1799:
1796:
1790:phase portrait
1785:
1772:pendulum clock
1752:Main article:
1749:
1746:
1738:periodic point
1729:
1726:
1637:
1634:
1528:
1525:
1512:
1509:
1506:
1503:
1474:
1469:
1443:
1423:
1403:
1388:
1387:
1375:
1363:
1362:
1361:
1360:
1348:
1345:
1342:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1281:
1261:
1241:
1227:
1226:
1214:
1194:
1191:
1188:
1185:
1165:
1162:
1159:
1139:
1119:
1099:
1096:
1093:
1090:
1070:
1046:
1031:
1019:
1016:
1013:
993:
990:
987:
984:
981:
978:
958:
938:
918:
894:
867:
853:
852:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
760:) =
726:
723:
720:
717:
714:
711:
708:
688:
668:
648:
645:
642:
639:
636:
614:
609:
587:
567:
564:
561:
558:
555:
552:
532:
512:
509:
506:
503:
500:
497:
494:
491:
471:
451:
431:
428:
425:
422:
419:
416:
396:
384:
381:
364:
355:
348:
339:
332:
323:
250:
247:
177:inflation rate
104:
103:
58:
56:
49:
26:
9:
6:
4:
3:
2:
5764:
5753:
5750:
5749:
5747:
5732:
5729:
5727:
5724:
5722:
5721:Edge of chaos
5719:
5717:
5714:
5712:
5709:
5708:
5706:
5700:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5678:Marcelo Viana
5676:
5674:
5671:
5669:
5668:Audrey Terras
5666:
5664:
5663:Floris Takens
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5563:Celso Grebogi
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5543:Chen Guanrong
5541:
5539:
5536:
5534:
5531:
5529:
5528:Michael Berry
5526:
5525:
5523:
5517:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5452:
5450:
5444:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5409:
5406:
5404:
5403:Lorenz system
5401:
5399:
5396:
5394:
5391:
5390:
5388:
5382:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5350:Langton's ant
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5325:Horseshoe map
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5299:
5296:
5295:
5294:
5291:
5289:
5286:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5264:
5261:
5260:
5258:
5252:
5249:
5246:
5239:
5233:
5230:
5228:
5225:
5223:
5222:Quantum chaos
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5199:
5197:
5191:
5186:
5182:
5178:
5164:
5161:
5159:
5156:
5154:
5151:
5149:
5146:
5144:
5141:
5139:
5136:
5134:
5131:
5130:
5128:
5122:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5030:Arnold tongue
5028:
5026:
5023:
5022:
5019:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4974:
4972:
4966:
4963:
4959:
4955:
4948:
4943:
4941:
4936:
4934:
4929:
4928:
4925:
4913:
4910:
4908:
4905:
4902:
4901:
4897:
4894:
4893:
4889:
4886:
4885:
4881:
4879:
4876:
4874:
4871:
4867:
4864:
4863:
4861:
4857:
4856:
4854:
4850:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4791:
4789:
4786:
4784:
4781:
4780:
4778:
4774:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4746:
4743:
4741:
4740:Brownian tree
4738:
4737:
4736:
4733:
4732:
4730:
4727:
4723:
4717:
4714:
4712:
4709:
4707:
4704:
4703:
4701:
4698:
4694:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4667:Multibrot set
4665:
4661:
4658:
4657:
4656:
4653:
4651:
4648:
4644:
4643:Douady rabbit
4641:
4639:
4636:
4634:
4631:
4630:
4629:
4626:
4624:
4621:
4620:
4618:
4616:
4610:
4602:
4599:
4598:
4597:
4594:
4592:
4589:
4588:
4586:
4584:
4580:
4574:
4571:
4570:
4568:
4566:
4562:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4514:
4513:Z-order curve
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4484:
4483:Hilbert curve
4481:
4479:
4476:
4472:
4469:
4468:
4467:
4466:De Rham curve
4464:
4462:
4459:
4458:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4431:Menger sponge
4429:
4427:
4424:
4422:
4419:
4417:
4416:Barnsley fern
4414:
4413:
4411:
4409:
4403:
4397:
4394:
4392:
4389:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4363:
4360:
4359:
4358:
4355:
4353:
4350:
4349:
4348:
4345:
4344:
4342:
4338:
4334:
4327:
4322:
4320:
4315:
4313:
4308:
4307:
4304:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4279:
4276:
4272:
4269:
4267:
4264:
4262:
4259:
4256:
4253:
4251:
4248:
4246:
4243:
4242:
4234:
4233:0-14-009250-1
4230:
4227:
4223:
4220:
4218:
4217:0-295-97514-8
4214:
4211:
4207:
4204:
4200:
4196:
4191:
4186:
4182:
4178:
4174:
4170:
4166:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4129:
4128:Celso Grebogi
4125:
4113:
4109:
4108:
4100:
4096:
4095:Ruelle, David
4092:
4088:
4082:
4078:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4034:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4001:
3997:
3993:
3992:Floris Takens
3989:
3985:
3981:
3980:
3975:
3971:
3967:
3966:
3954:
3950:
3945:
3938:
3937:
3930:
3922:
3918:
3914:
3910:
3906:
3902:
3899:(4): 044101.
3898:
3894:
3887:
3879:
3875:
3870:
3865:
3861:
3857:
3853:
3849:
3845:
3838:
3830:
3826:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3791:
3783:
3777:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3726:
3715:
3714:
3706:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3670:
3663:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3617:
3609:
3602:
3594:
3587:
3571:
3570:
3565:
3562:
3555:
3548:
3542:
3538:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3478:
3475:
3473:
3468:
3466:
3462:
3458:
3454:
3445:
3443:
3439:
3438:complex plane
3435:
3427:
3424:
3421:
3418:
3415:
3414:
3410:
3406:
3401:
3397:
3383:
3380:
3377:
3374:
3371:
3366:
3362:
3358:
3355:
3350:
3346:
3342:
3336:
3330:
3321:
3317:
3313:
3303:
3289:
3269:
3266:
3263:
3260:
3257:
3253:
3249:
3246:
3226:
3223:
3220:
3200:
3197:
3194:
3174:
3154:
3151:
3148:
3145:
3142:
3138:
3134:
3131:
3123:
3118:
3104:
3083:
3075:
3059:
3052:
3051:square matrix
3034:
3031:
3028:
3024:
3020:
3017:
3012:
3008:
2987:
2980:
2977:in a dynamic
2976:
2971:
2957:
2954:
2946:
2921:
2918:
2910:
2883:
2880:
2877:
2873:
2869:
2866:
2861:
2857:
2842:
2840:
2836:
2835:linear system
2833:
2829:
2825:
2821:
2811:
2797:
2777:
2757:
2737:
2715:
2711:
2707:
2704:
2696:
2680:
2660:
2652:
2651:"fixed point"
2636:
2616:
2613:
2610:
2590:
2587:
2584:
2564:
2561:
2558:
2555:
2552:
2532:
2512:
2492:
2472:
2452:
2432:
2429:
2426:
2406:
2386:
2383:
2380:
2377:
2374:
2354:
2351:
2348:
2328:
2308:
2305:
2302:
2282:
2262:
2259:
2256:
2236:
2211:
2207:
2203:
2200:
2192:
2188:
2184:
2181:
2176:
2173:
2170:
2166:
2157:
2137:
2134:
2131:
2111:
2108:
2105:
2085:
2082:
2079:
2071:
2052:
2049:
2046:
2023:
2020:
2017:
2014:
2011:
1991:
1983:
1978:
1969:
1967:
1963:
1959:
1955:
1950:
1948:
1944:
1940:
1936:
1932:
1931:Floris Takens
1928:
1924:
1919:
1917:
1913:
1909:
1905:
1901:
1897:
1889:
1885:
1881:
1877:
1872:
1863:
1861:
1857:
1846:
1845:quasiperiodic
1841:
1839:
1834:
1823:
1819:
1814:
1810:
1806:
1791:
1788:
1783:
1779:
1777:
1773:
1769:
1765:
1761:
1755:
1745:
1743:
1739:
1735:
1734:discrete-time
1725:
1723:
1718:
1714:
1713:shapes change
1710:
1706:
1705:hemispherical
1702:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1669:
1667:
1663:
1660:
1655:
1647:
1642:
1633:
1631:
1630:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1584:
1582:
1578:
1574:
1573:horseshoe map
1570:
1569:Stephen Smale
1566:
1565:topologically
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1524:
1507:
1501:
1493:
1488:
1472:
1457:
1441:
1421:
1401:
1393:
1373:
1365:
1364:
1346:
1343:
1340:
1333:for all real
1320:
1317:
1311:
1308:
1305:
1299:
1279:
1259:
1239:
1231:
1230:
1229:
1228:
1212:
1189:
1183:
1157:
1150:in the limit
1137:
1130:that "enter"
1117:
1094:
1088:
1068:
1060:
1057:, called the
1044:
1036:
1032:
1017:
1014:
1011:
988:
985:
982:
976:
956:
936:
916:
908:
892:
885:
884:
883:
881:
865:
858:
836:
830:
827:
824:
821:
818:
815:
809:
800:
797:
794:
788:
785:
779:
772:
771:
767:
764: +
763:
759:
755:
752:the function
751:
747:
742:
738:
721:
718:
715:
709:
706:
686:
666:
643:
640:
637:
612:
585:
562:
559:
556:
550:
530:
510:
507:
501:
498:
495:
489:
469:
449:
426:
423:
420:
414:
394:
380:
378:
374:
369:
363:
347:
331:
322:
317:
314:
309:
306:
302:
301:invariant set
298:
293:
291:
287:
283:
279:
274:
272:
268:
264:
260:
256:
246:
244:
240:
236:
232:
228:
223:
221:
217:
213:
209:
205:
201:
197:
193:
189:
188:geometrically
184:
182:
178:
174:
170:
166:
162:
160:
155:
152:-dimensional
151:
147:
146:algebraically
142:
139:
135:
131:
123:
119:
115:
110:
100:
97:
89:
79:
75:
69:
68:
62:
57:
48:
47:
44:
40:
33:
19:
5673:Mary Tsingou
5638:David Ruelle
5633:Otto Rössler
5578:Michel HĂ©non
5548:Leon O. Chua
5505:Tilt-A-Whirl
5475:FPUT problem
5360:Standard map
5355:Logistic map
5180:
4976:
4954:Chaos theory
4912:Chaos theory
4907:Kaleidoscope
4898:
4890:
4882:
4808:Gaston Julia
4788:Georg Cantor
4613:Escape-time
4545:Gosper curve
4493:LĂ©vy C curve
4478:Dragon curve
4357:Box-counting
4225:
4222:James Gleick
4209:
4172:
4168:
4139:
4135:
4116:. Retrieved
4114:(7): 764â765
4111:
4105:
4076:
4043:
4039:
3999:
3995:
3988:David Ruelle
3979:Scholarpedia
3977:
3952:
3944:
3934:
3929:
3896:
3892:
3886:
3851:
3847:
3837:
3804:
3800:
3790:
3776:cite journal
3735:
3731:
3725:
3712:
3705:
3672:
3668:
3662:
3629:
3625:
3616:
3607:
3601:
3592:
3586:
3573:. Retrieved
3567:
3554:
3541:
3502:Steady state
3476:
3469:
3460:
3456:
3451:
3431:
3408:
3309:
3119:
3049:in terms of
2972:
2848:
2817:
2156:logistic map
2153:
1982:logistic map
1951:
1935:bifurcations
1927:David Ruelle
1922:
1920:
1898:if it has a
1895:
1893:
1887:
1883:
1879:
1842:
1835:
1821:
1817:
1801:
1757:
1742:logistic map
1731:
1699:), and even
1670:
1651:
1627:
1596:intersection
1585:
1530:
1489:
1389:
1081:and denoted
1058:
1035:neighborhood
906:
854:
765:
761:
757:
753:
386:
370:
361:
345:
329:
320:
310:
304:
300:
294:
275:
259:differential
252:
242:
238:
224:
220:chaos theory
207:
185:
168:
158:
149:
143:
137:
130:mathematical
127:
92:
83:
64:
43:
5658:Nina Snaith
5648:Yakov Sinai
5533:Rufus Bowen
5283:Duffing map
5268:Baker's map
5193:Theoretical
5105:SRB measure
5012:Phase space
4982:Bifurcation
4903:(1987 book)
4895:(1986 book)
4887:(1982 book)
4873:Fractal art
4793:Bill Gosper
4757:LĂ©vy flight
4503:Peano curve
4498:Moore curve
4384:Topological
4369:Correlation
3974:"Attractor"
3970:John Milnor
3622:John Milnor
3564:"Attractor"
3074:eigenvalues
2820:phase space
2070:bifurcation
1947:Cantor dust
1798:Limit torus
1787:Van der Pol
1760:limit cycle
1754:Limit cycle
1748:Limit cycle
1720:considered
1689:deformation
1654:fixed point
1636:Fixed point
1592:limit cycle
1588:fixed point
1537:phase space
1394:containing
969:then so is
880:phase space
290:phase space
78:introducing
5752:Limit sets
5716:Complexity
5613:Edward Ott
5460:Convection
5385:Continuous
5060:Ergodicity
4711:Orbit trap
4706:Buddhabrot
4699:techniques
4687:Mandelbulb
4488:Koch curve
4421:Cantor set
4132:Edward Ott
4118:16 January
3533:References
3507:Wada basin
2730:values of
2341:values of
2295:values of
1874:A plot of
1776:escapement
1722:stationary
1697:plasticity
1581:Cantor set
1292:such that
1004:, for all
297:limit sets
227:trajectory
118:this video
86:March 2013
61:references
18:Attractors
5628:Mary Rees
5588:Bryna Kra
5521:theorists
5330:Ikeda map
5320:HĂ©non map
5310:Gauss map
4992:Limit set
4977:Attractor
4818:Paul LĂ©vy
4697:Rendering
4682:Mandelbox
4628:Julia set
4540:Hexaflake
4471:Minkowski
4391:Recursion
4374:Hausdorff
4185:CiteSeerX
4169:Physica D
4136:Physica D
3864:CiteSeerX
3848:Physica D
3697:137430238
3654:120688149
3569:MathWorld
3372:−
3356:−
3312:nonlinear
3032:−
2881:−
2708:×
2614:≈
2588:≈
2430:≈
2204:−
2109:≈
2083:≈
1921:The term
1709:spherical
1624:manifolds
1318:∈
1164:∞
1161:→
746:Julia set
427:⋅
305:limit set
271:iteration
267:integrate
138:attractor
132:field of
122:available
5746:Category
5704:articles
5446:Physical
5365:Tent map
5255:Discrete
5195:branches
5125:Theorems
4961:Concepts
4728:fractals
4615:fractals
4583:L-system
4525:T-square
4333:Fractals
4278:Archived
4068:55827557
4024:17074317
3921:16486826
3829:17074317
3760:17816542
3481:See also
3442:fractals
1764:isolated
1681:friction
1677:stiction
1662:pendulum
1612:surfaces
1590:and the
1557:surfaces
1392:open set
750:iterates
748:, which
659:, where
316:pendulum
243:repellor
239:repeller
231:periodic
200:manifold
179:and the
5702:Related
5510:Weather
5448:systems
5241:Chaotic
4987:Fractal
4677:Tricorn
4530:n-flake
4379:Packing
4362:Higuchi
4352:Assouad
4224:(1988)
4208:(1996)
4177:Bibcode
4165:M. Ghil
4144:Bibcode
4048:Bibcode
4004:Bibcode
3972:(ed.).
3901:Bibcode
3856:Bibcode
3844:Ghil M.
3809:Bibcode
3768:1586349
3740:Bibcode
3732:Science
3677:Bibcode
3634:Bibcode
1908:chaotic
1900:fractal
1896:strange
1693:elastic
1620:toroids
1616:spheres
1535:of the
1533:subsets
1492:measure
878:of the
235:chaotic
204:fractal
128:In the
74:improve
5608:Hee Oh
5243:maps (
5090:Mixing
4776:People
4726:Random
4633:Filled
4601:H tree
4520:String
4408:system
4231:
4215:
4187:
4083:
4066:
4022:
3919:
3866:
3827:
3766:
3758:
3695:
3652:
3575:30 May
3459:, the
2979:vector
2832:stable
1964:, and
1910:, but
1717:deform
1691:(both
1659:damped
1606:(e.g.
1577:robust
1549:points
1456:metric
909:under
857:subset
840:
313:damped
216:scalar
167:, the
154:vector
148:as an
63:, but
5519:Chaos
5298:outer
5002:Orbit
4852:Other
4102:(PDF)
4064:S2CID
4036:(PDF)
4020:S2CID
3825:S2CID
3764:S2CID
3717:(PDF)
3693:S2CID
3650:S2CID
2617:0.799
2591:0.513
2433:0.615
1813:torus
1732:In a
1608:lines
1602:) of
1600:union
1553:lines
1539:of a
929:: if
210:(see
196:curve
192:point
163:. In
136:, an
5245:list
4969:Core
4273:and
4229:ISBN
4213:ISBN
4120:2008
4081:ISBN
3917:PMID
3782:link
3756:PMID
3577:2021
3320:real
3224:<
3198:>
2955:<
2919:>
2603:and
2562:<
2556:<
2399:the
2384:<
2378:<
2352:>
2306:<
2135:>
2021:<
2015:<
1943:like
1929:and
1856:sine
1695:and
1598:and
1575:was
1344:>
1061:for
1015:>
387:Let
241:(or
198:, a
4195:doi
4173:240
4152:doi
4056:doi
4012:doi
3909:doi
3874:doi
3852:240
3817:doi
3748:doi
3736:238
3685:doi
3673:295
3642:doi
3076:of
2263:2.6
2158:,
2138:3.6
2112:3.5
2086:3.0
1715:or
1675:of
1252:of
1037:of
905:is
261:or
245:).
233:or
5748::
4862:"
4193:.
4183:.
4171:.
4150:.
4140:13
4138:.
4130:;
4112:53
4110:.
4104:.
4062:.
4054:.
4044:82
4042:.
4038:.
4018:.
4010:.
4000:20
3998:.
3990:;
3976:.
3915:.
3907:.
3897:96
3895:.
3872:.
3862:.
3850:.
3823:.
3815:.
3805:20
3803:.
3799:.
3778:}}
3774:{{
3762:.
3754:.
3746:.
3734:.
3691:.
3683:.
3671:.
3648:.
3640:.
3630:99
3628:.
3566:.
3444:.
3403:A
3384:12
3375:11
1968:.
1960:,
1956:,
1945:a
1758:A
1687:,
1683:,
1679:,
1652:A
1632:.
1622:,
1618:,
1614:,
1610:,
1583:.
1555:,
1551:,
543:,
284:,
253:A
225:A
222:.
183:.
5247:)
4946:e
4939:t
4932:v
4858:"
4325:e
4318:t
4311:v
4201:.
4197::
4179::
4158:.
4154::
4146::
4122:.
4089:.
4070:.
4058::
4050::
4026:.
4014::
4006::
3982:.
3923:.
3911::
3903::
3880:.
3876::
3858::
3831:.
3819::
3811::
3784:)
3770:.
3750::
3742::
3699:.
3687::
3679::
3656:.
3644::
3636::
3579:.
3409:x
3381:+
3378:x
3367:2
3363:x
3359:2
3351:3
3347:x
3343:=
3340:)
3337:x
3334:(
3331:f
3290:A
3270:X
3267:A
3264:=
3261:t
3258:d
3254:/
3250:X
3247:d
3227:0
3221:a
3201:0
3195:a
3175:x
3155:x
3152:a
3149:=
3146:t
3143:d
3139:/
3135:x
3132:d
3105:n
3084:A
3060:A
3035:1
3029:t
3025:X
3021:A
3018:=
3013:t
3009:X
2988:X
2958:1
2951:|
2947:a
2943:|
2922:1
2915:|
2911:a
2907:|
2884:1
2878:t
2874:x
2870:a
2867:=
2862:t
2858:x
2798:r
2778:x
2758:r
2738:x
2716:n
2712:2
2705:k
2681:x
2661:r
2637:r
2611:x
2585:x
2565:1
2559:x
2553:0
2533:r
2513:x
2493:r
2473:x
2453:r
2427:x
2407:x
2387:1
2381:x
2375:0
2355:1
2349:x
2329:x
2309:0
2303:x
2283:x
2260:=
2257:r
2237:r
2217:)
2212:n
2208:x
2201:1
2198:(
2193:n
2189:x
2185:r
2182:=
2177:1
2174:+
2171:n
2167:x
2132:r
2106:r
2080:r
2056:)
2053:x
2050:,
2047:r
2044:(
2024:1
2018:x
2012:0
1992:r
1888:ÎČ
1884:Ï
1880:Ï
1851:t
1849:N
1830:t
1828:N
1822:t
1818:N
1511:)
1508:A
1505:(
1502:B
1473:n
1468:R
1442:A
1422:A
1402:A
1374:A
1359:.
1347:T
1341:t
1321:N
1315:)
1312:b
1309:,
1306:t
1303:(
1300:f
1280:T
1260:A
1240:N
1213:b
1193:)
1190:A
1187:(
1184:B
1158:t
1138:A
1118:b
1098:)
1095:A
1092:(
1089:B
1069:A
1045:A
1030:.
1018:0
1012:t
992:)
989:a
986:,
983:t
980:(
977:f
957:A
937:a
917:f
893:A
866:A
837:.
834:)
831:v
828:,
825:v
822:t
819:+
816:x
813:(
810:=
807:)
804:)
801:v
798:,
795:x
792:(
789:,
786:t
783:(
780:f
766:c
762:z
758:z
756:(
754:f
725:)
722:v
719:,
716:x
713:(
710:=
707:a
687:v
667:x
647:)
644:v
641:,
638:x
635:(
613:2
608:R
586:t
566:)
563:a
560:,
557:t
554:(
551:f
531:t
511:a
508:=
505:)
502:a
499:,
496:0
493:(
490:f
470:n
450:a
430:)
424:,
421:t
418:(
415:f
395:t
365:0
362:x
356:0
354:x
349:1
346:x
340:0
338:x
333:1
330:x
324:0
321:x
169:n
159:n
150:n
124:.
99:)
93:(
88:)
84:(
70:.
41:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
