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2028:
1973:
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36:
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the original Lorenz model (), Shen and his co-authors proposed a revised view that “weather possesses both chaos and order with distinct predictability”. The revised view, which is a build-up of the conventional view, is used to suggest that “the chaotic and regular features found in theoretical Lorenz models could better represent features of the Earth's atmosphere”.
1968:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-xy_{1}-\beta z,\\{\frac {\mathrm {d} y_{1}}{\mathrm {d} t}}&=xz-2xz_{1}-d_{0}y_{1},\\{\frac {\mathrm {d} z_{1}}{\mathrm {d} t}}&=2xy_{1}-4\beta z_{1}.\end{aligned}}}
402:
4974:: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts from
753:
6028:
4990:
The scientific community accepts that the chaotic features found in low-dimensional Lorenz models could represent features of the Earth's atmosphere (), yielding the statement of “weather is chaotic.” By comparison, based on the concept of attractor coexistence within the generalized Lorenz model and
5983:
and then apply the process recursively. Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal', leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.
215:
1593:
Over the past several years, a series of papers regarding high-dimensional Lorenz models have yielded a generalized Lorenz model, which can be simplified into the classical Lorenz model for three state variables or the following five-dimensional Lorenz model for five state variables:
6062:
587:
475:. The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.
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1036:
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The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model. Proving that this is indeed the case is the fourteenth problem on the list of
163:" in popular media may stem from the real-world implications of the Lorenz attractor, namely that tiny changes in initial conditions evolve to completely different trajectories. This underscores that chaotic systems can be completely
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1602:
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97:
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The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time:
397:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z.\end{aligned}}}
1584:
value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.
7226:
1407:
Evolution of three initially nearby trajectories of the Lorenz system. In this animation the equation is numerically integrated using a Runge-Kutta routine — made using starting from three initial conditions
220:
6066:
1090:
886:, the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set – the Lorenz attractor – a
1382:— show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10 in the
5064:
748:{\displaystyle \left({\sqrt {\beta (\rho -1)}},{\sqrt {\beta (\rho -1)}},\rho -1\right)\quad {\text{and}}\quad \left(-{\sqrt {\beta (\rho -1)}},-{\sqrt {\beta (\rho -1)}},\rho -1\right).}
1183:
1176:
1097:
1339:
1332:
1325:
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932:
2017:
1386:-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.
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6045:
5682:
5607:
5461:
5247:
4955:
As shown in Lorenz's original paper, the Lorenz system is a reduced version of a larger system studied earlier by Barry
Saltzman. The Lorenz equations are derived from the
5110:
4959:
to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above. This fluid circulation is known as
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and yet still be inherently impractical or even impossible to predict over longer periods of time. For example, even the small flap of a butterfly's wings could set the
6081:
5995:
1453:
5981:
5866:
5816:
5657:
5481:
5416:
5300:
5179:
6920:
Shen, Bo-Wen; Pielke, R. A.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R.; Reyes, T. A. L. (2021), Skiadas, Christos H.; Dimotikalis, Yiannis (eds.),
5846:
1539:
5766:
5739:
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5555:
5528:
5356:
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6162:
5139:
6065:
5276:
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then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin, which is a global
1568:
In Figure 4 of his paper, Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the
7378:
Leonov, G.A.; Kuznetsov, N.V.; Korzhemanova, N.A.; Kusakin, D.V. (2016). "Lyapunov dimension formula for the global attractor of the Lorenz system".
761:
54:
6816:
Shen, Bo-Wen; Pielke, Roger; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Kesarkar, Amit; Zeng, Xiping; Atlas, Robert (2022-11-12).
7135:
Gorman, M.; Widmann, P.J.; Robbins, K.A. (1986). "Nonlinear dynamics of a convection loop: A quantitative comparison of experiment with theory".
205:
who helped in the initial, numerical computations leading up to the findings of the Lorenz model, developed a simplified mathematical model for
8477:
8307:
926:. The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:
188:
7881:
5184:
Then the proof is split in three main points that are proved and imply the existence of a strange attractor. The three points are:
6011:
8187:
7947:
4963:. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.
1576:, which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the
7302:
7268:
7074:
6941:
202:
8140:
6970:
Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Baik, Jong-Jin; Faghih-Naini, Sara; Cui, Jialin; Atlas, Robert (2021-01-01).
8087:
4956:
7726:
7592:
Shen, B.-W. (2015-12-21). "Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term".
6150:
8420:
7908:
483:
5920:. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split
8655:
7463:
Mishra, Aashwin; Sanghi, Sanjeev (2006). "A study of the asymmetric Malkus waterwheel: The biased Lorenz equations".
7341:
Kolář, Miroslav; Gumbs, Godfrey (1992). "Theory for the experimental observation of chaos in a rotating waterwheel".
72:
8462:
8150:
4960:
8337:
486:. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.
17:
5018:
5012:
171:
on a vastly different trajectory, in which for example a hurricane occurs where it otherwise would have not (see
7584:
7559:
7872:
7529:
Poland, Douglas (1993). "Cooperative catalysis and chemical chaos: a chemical model for the Lorenz equations".
6797:
144:
8155:
8145:
7842:
7454:
7429:
1133:
1031:{\displaystyle 3-{\frac {2(\sigma +\beta +1)}{\sigma +1+{\sqrt {\left(\sigma -1\right)^{2}+4\sigma \rho }}}}}
5871:
8502:
8410:
8275:
895:
7837:
1542:
96:
1980:
8415:
7940:
7790:"On a high-precision method for studying attractors of dynamical systems and systems of explosive type"
8482:
7832:
5191:
8530:
7642:
209:. The model is a system of three ordinary differential equations now known as the Lorenz equations:
8234:
5217:
8179:
7622:
7500:
Pchelintsev, A.N. (2014). "Numerical and
Physical Modeling of the Dynamics of the Lorenz System".
5089:
5066:
that is cut transversely by the flow trajectories. From this, one can define the first-return map
1271:, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.
8395:
8160:
8067:
7260:
7090:
6928:, Springer Proceedings in Complexity, Cham: Springer International Publishing, pp. 805–825,
6052:
8135:
5305:
1484:
1458:
8435:
8047:
7637:
206:
5662:
5587:
5441:
1432:
8754:
8585:
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8117:
8052:
8027:
7933:
7903:
5953:
5851:
5801:
5642:
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5401:
5285:
5164:
1541:, beyond which the trajectories become uncorrelated. The full-sized graphic can be accessed
1316:
1304:
1292:
915:
567:
7088:(1993). "Circuit implementation of synchronized chaos with applications to communications".
6758:
6694:"Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term"
5923:
5821:
5771:
5687:
5612:
1518:
8595:
8347:
8247:
8092:
7571:
7538:
7472:
7441:
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7397:
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7321:
7204:
7173:
7144:
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6983:
6878:
6829:
6754:
6705:
6606:
6546:
6190:
6179:
5744:
5717:
5560:
5533:
5506:
5334:
2019:
has been applied to be consistent with the choice of the other parameters. See details in.
1253:
1219:
1167:
448:
194:
148:
90:
8425:
7878:
7164:
Grassberger, P.; Procaccia, I. (1983). "Measuring the strangeness of strange attractors".
5115:
755:
These correspond to steady convection. This pair of equilibrium points is stable only if
8:
8555:
8512:
8497:
8342:
8295:
8280:
8265:
8165:
8072:
8057:
8042:
6921:
5008:
3975:# Step through "time", calculating the partial derivatives at the current point
1043:
899:
168:
8097:
7919:
7904:
3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions
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7542:
7476:
7445:
7401:
7354:
7325:
7208:
7177:
7148:
7103:
6987:
6922:"Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models"
6882:
6833:
6709:
6610:
6550:
5258:
4999:
Smale's 14th problem says, 'Do the properties of the Lorenz attractor exhibit that of a
1572:
direction. This procedure later became known as a Lorenz map (not to be confused with a
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8430:
8317:
8312:
8204:
8082:
7984:
7801:
7764:
7699:
7668:
7655:
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7413:
7387:
7009:
6972:"Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model"
6947:
6902:
6790:
Application of Chaos to
Meteorology and Climate. In The Nature of Chaos; Mullin, T., Ed
6630:
6572:
6496:
6195:
5486:
5421:
5381:
5361:
5144:
5069:
5000:
903:
887:
50:
6742:
6397:"The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension"
1401:
843:. At the critical value, both equilibrium points lose stability through a subcritical
8605:
8570:
8560:
8457:
8077:
7999:
7868:
7854:
7851:
7703:
7667:
Tzenov, Stephan (2014). "Strange
Attractors Characterizing the Osmotic Instability".
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472:
464:
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6906:
6867:"Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model"
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6713:
6693:
6614:
6554:
6480:
6408:
844:
460:
160:
7659:
6051:
A solution in the Lorenz attractor rendered as a metal wire to show direction and
5255:
Vectors inside this invariant cone field are uniformly expanded by the derivative
2570:# build an animated gif by pushing new points to the plot, saving every 10th frame
1573:
471:. The Lorenz equations are also the governing equations in Fourier space for the
155:
solutions for certain parameter values and initial conditions. In particular, the
8645:
8540:
8467:
8300:
8112:
8102:
7885:
7753:"On differences and similarities in the analysis of Lorenz, Chen, and Lu systems"
6933:
6451:
6185:
4971:
4967:
468:
437:
8635:
8580:
7409:
7227:
IEEE Transactions on
Circuits and Systems I: Fundamental Theory and Applications
4199:
8728:
8695:
8690:
8685:
8487:
8377:
8372:
8270:
8219:
8009:
7789:
7294:
7193:
Haken, H. (1975). "Analogy between higher instabilities in fluids and lasers".
6413:
6396:
5557:
and using Euler integration method, one can estimate where the flow will bring
5004:
1047:
433:
7891:
7779:
7752:
7513:
7111:
6559:
6534:
1264:, the system is stable and evolves to one of two fixed point attractors. When
8748:
8723:
8680:
8670:
8665:
8565:
8545:
8327:
8224:
8032:
7282:
7278:
7247:
7119:
7005:
6996:
6971:
6898:
6851:
6774:
6743:"Geophysical flows as dynamical systems: the influence of Hide's experiments"
6727:
6626:
6594:
6568:
6492:
6075:
An animation showing the divergence of nearby solutions to the Lorenz system.
1089:
818:{\displaystyle \rho <\sigma {\frac {\sigma +\beta +3}{\sigma -\beta -1}},}
7362:
6718:
6618:
4985:
2027:
8675:
8640:
8550:
8507:
8362:
8357:
7956:
7492:
7127:
6453:
Attractor
Dimension Estimates for Dynamical Systems: Theory and Computation
6395:
Kuznetsov, N.V.; Mokaev, T.N.; Kuznetsova, O.A.; Kudryashova, E.V. (2020).
6039:
An animation showing trajectories of multiple solutions in a Lorenz system.
479:
452:
198:
172:
152:
86:
8322:
7913:
7651:
7370:
7257:
Chaos and
Nonlinear Dynamics: An Introduction for Scientists and Engineers
6842:
6817:
5007:
in 2002. To prove this result, Tucker used rigorous numerics methods like
8660:
8650:
8535:
8285:
8107:
8014:
7816:
7291:
Differential
Equations, Dynamical Systems, & An Introduction to Chaos
7046:
5798:. The goal is to do this method recursively until the flow comes back to
1561:
189:
Chaos theory § Lorenz's pioneering contributions to chaotic modeling
176:
164:
6473:
Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
8718:
8615:
8062:
7695:
6890:
6484:
1071:
7484:
7239:
542:. The system exhibits chaotic behavior for these (and nearby) values.
8630:
8590:
8332:
7994:
7979:
7859:
6866:
6001:
A solution in the Lorenz attractor plotted at high resolution in the
1564:. Points above the red line correspond to the system switching lobes.
553:
7673:
6394:
1182:
1175:
1096:
8367:
7806:
7682:
Viana, Marcelo (2000). "What's new on Lorenz strange attractors?".
7392:
6087:
A visualization of the Lorenz attractor near an intermittent cycle.
5331:. Tucker covers the location of these two arcs by small rectangles
1577:
1421:
1338:
1331:
1324:
456:
201:
who was responsible for the numerical simulations and figures, and
7894:. The implementation of Lorenz attractor in an electronic circuit.
7769:
7377:
7063:
Order within Chaos: Towards a
Deterministic Approach to Turbulence
6438:
3759:
Values of the Lorenz attractor's partial derivatives at *xyz*.
8037:
7989:
7614:
The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors
3161:% ''a'' is array containing x, y, and z variables
1057:, the system displays knotted periodic orbits. For example, with
891:
7560:"Finite Amplitude Free Convection as an Initial Value Problem—I"
7224:
Hemati, N. (1994). "Strange attractors in brushless DC motors".
7042:
Heitmann, S., Breakspear, M (2017-2022) Brain Dynamics Toolbox.
6168:
Animation of the Lorenz attractor in the Brain Dynamics Toolbox.
27:
System of ordinary differential equations with chaotic solutions
8610:
7925:
7892:
Synchronized Chaos and Private Communications, with Kevin Cuomo
1080:
Example solutions of the Lorenz system for different values of
7716:"The statistical prediction of solutions of dynamic equations"
7312:
Knobloch, Edgar (1981). "Chaos in the segmented disc dynamo".
159:
is a set of chaotic solutions of the Lorenz system. The term "
6926:
13th Chaotic Modeling and Simulation International Conference
6792:. Oxford, UK: Oxford Science Publications. pp. 220–260.
1556:
444:
175:). The shape of the Lorenz attractor itself, when plotted in
7897:
7380:
Communications in Nonlinear Science and Numerical Simulation
6535:"Aggregated Negative Feedback in a Generalized Lorenz Model"
7585:
10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2
5282:
To prove the first point, we notice that the cross section
7849:
7750:
6741:
Ghil, Michael; Read, Peter; Smith, Leonard (2010-07-23).
6018:
4986:
Model for the nature of chaos and order in the atmosphere
4966:
The partial differential equations modeling the system's
7916:(implemented in Ada with GTK+. Sources & executable)
7465:
Chaos: An Interdisciplinary Journal of Nonlinear Science
7455:
10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
7043:
3158:% ''f'' is set of differential equations
443:
The Lorenz equations can arise in simplified models for
6595:"Nonlinear Feedback in a Five-Dimensional Lorenz Model"
4982:, Appendix D; or Shen (2016), Supplementary Materials.
440:, and certain physical dimensions of the layer itself.
7061:
Bergé, Pierre; Pomeau, Yves; Vidal, Christian (1984).
6818:"The Dual Nature of Chaos and Order in the Atmosphere"
5684:
using Taylor expansion, this gives us a new rectangle
1983:
5956:
5926:
5874:
5854:
5824:
5804:
5774:
5747:
5720:
5690:
5665:
5645:
5615:
5590:
5563:
5536:
5509:
5489:
5469:
5444:
5424:
5404:
5384:
5364:
5337:
5308:
5288:
5261:
5252:
The return map admits a forward invariant cone field.
5220:
5194:
5167:
5147:
5118:
5092:
5072:
5021:
4994:
1998:
1600:
1521:
1487:
1461:
1435:
935:
764:
590:
420:
to the vertical temperature variation. The constants
218:
7163:
7134:
6466:
6427:
6390:
6388:
6278:
3155:% Solve over time interval with initial conditions
1046:. This problem was the first one to be resolved, by
5378:. Now, the goal is to prove that for all points in
4950:
45:
may be too technical for most readers to understand
7723:Symposium on Numerical Weather Prediction in Tokyo
7277:
6969:
6919:
6467:Guckenheimer, John; Williams, R. F. (1979-12-01).
6379:
6367:
6355:
6017:A solution in the Lorenz attractor rendered as an
5975:
5942:
5912:
5860:
5840:
5810:
5790:
5760:
5733:
5706:
5676:
5651:
5631:
5601:
5576:
5549:
5522:
5495:
5475:
5455:
5430:
5410:
5390:
5370:
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5323:
5294:
5270:
5241:
5206:
5173:
5153:
5133:
5104:
5078:
5058:
2011:
1967:
1533:
1507:
1473:
1447:
1030:
817:
747:
478:From a technical standpoint, the Lorenz system is
396:
7909:Lorenz Attractor implemented in analog electronic
6815:
6449:
6385:
6156:Animation of a Lorenz System with rho-dependence.
147:first studied by mathematician and meteorologist
8746:
7623:"A Rigorous ODE Solver and Smale's 14th Problem"
7060:
4979:
6976:Bulletin of the American Meteorological Society
6865:Yorke, James A.; Yorke, Ellen D. (1979-09-01).
6443:
100:A sample solution in the Lorenz attractor when
7083:
6539:International Journal of Bifurcation and Chaos
6512:
6510:
6300:
6241:
6239:
6237:
5214:invariant under the first-return map, meaning
5003:?'. The problem was answered affirmatively by
1588:
1280:Sensitive dependence on the initial condition
7941:
7027:
7025:
7023:
6740:
6450:Kuznetsov, Nikolay; Reitmann, Volker (2021).
5639:. Then, one can estimate where the points in
3741:Point of interest in three-dimensional space.
1515:. Siginificant divergence is seen at around
416:to the horizontal temperature variation, and
7462:
6344:
5059:{\displaystyle \Sigma \subset \{x_{3}=r-1\}}
5053:
5028:
4970:and temperature are subjected to a spectral
1560:A recreation of Lorenz's results created on
1400:
179:, may also be seen to resemble a butterfly.
7787:
7499:
6507:
6469:"Structural stability of Lorenz attractors"
6234:
3978:# and using them to estimate the next point
904:Lyapunov dimension (Kaplan-Yorke dimension)
412:is proportional to the rate of convection,
7948:
7934:
7340:
7020:
6864:
6333:
2453:# initialize a 3D plot with 1 empty series
1580:. Lorenz also found that when the maximum
898:with respect to all three equilibria. Its
432:are system parameters proportional to the
7815:
7805:
7778:
7768:
7751:G.A. Leonov & N.V. Kuznetsov (2015).
7672:
7641:
7583:
7453:
7391:
6995:
6841:
6717:
6558:
6412:
6099:Two streamlines in a Lorenz system, from
5398:, the flow will bring back the points in
3747:Parameters defining the Lorenz attractor.
3295:
1551:
584:two additional critical points appear at
494:One normally assumes that the parameters
73:Learn how and when to remove this message
57:, without removing the technical details.
7630:Foundations of Computational Mathematics
7557:
7311:
6658:
6267:
6214:
6212:
6210:
5015:. First, Tucker defined a cross section
4198:
3164:% ''t'' is time variable
2026:
1555:
1345:
1259:
95:
7611:
7254:
6245:
4975:
14:
8747:
7920:Interactive web based Lorenz Attractor
7914:Lorenz Attractor interactive animation
7898:Lorenz attractor interactive animation
7710:
7666:
7620:
7528:
7424:
7223:
6680:
6669:
6647:
6516:
6322:
6311:
6289:
6229:
6218:
5913:{\displaystyle P(R_{i})\subset Rf_{i}}
5358:, the union of these rectangles gives
3942:# Need one more for the initial values
3248:% Runge-Kutta 4th/5th order ODE solver
7929:
7900:(you need the Adobe Shockwave plugin)
7850:
7681:
7192:
7031:
6811:
6809:
6588:
6586:
6528:
6526:
6524:
6256:
6207:
506:are positive. Lorenz used the values
55:make it understandable to non-experts
6787:
6691:
6592:
6532:
6279:Gorman, Widmann & Robbins (1986)
3641:
3148:
2923:
29:
8088:Measure-preserving dynamical system
7970:
7757:Applied Mathematics and Computation
7564:Journal of the Atmospheric Sciences
7502:Numerical Analysis and Applications
7434:Journal of the Atmospheric Sciences
6599:Journal of the Atmospheric Sciences
2665:
2035:
2012:{\textstyle d_{0}={\dfrac {19}{3}}}
1395:Divergence of nearby trajectories.
482:, aperiodic, three-dimensional and
24:
7744:
6806:
6583:
6521:
6428:Grassberger & Procaccia (1983)
6380:Hirsch, Smale & Devaney (2003)
6368:Hirsch, Smale & Devaney (2003)
6356:Hirsch, Smale & Devaney (2003)
5855:
5805:
5741:. Thus we know that all points in
5667:
5646:
5592:
5470:
5446:
5405:
5315:
5289:
5201:
5168:
5099:
5022:
4995:Resolution of Smale's 14th problem
4787:# Graphics production with Plotly:
1906:
1889:
1813:
1796:
1737:
1727:
1675:
1665:
1619:
1609:
355:
345:
293:
283:
237:
227:
25:
8766:
8656:Oleksandr Mykolayovych Sharkovsky
7825:
7594:Nonlinear Processes in Geophysics
6698:Nonlinear Processes in Geophysics
5503:small, then by taking the center
4957:Oberbeck–Boussinesq approximation
3738:xyz : array-like, shape (3,)
828:which can hold only for positive
8186:
8178:
7955:
7430:"Deterministic nonperiodic flow"
6767:10.1111/j.1468-4004.2010.51428.x
6161:
6149:
6092:
6080:
6060:
6044:
6026:
6010:
5994:
5207:{\displaystyle N\subset \Sigma }
4980:Bergé, Pomeau & Vidal (1984)
4951:Model for atmospheric convection
3756:xyz_dot : array, shape (3,)
1337:
1330:
1323:
1181:
1174:
1095:
1088:
34:
7293:(Second ed.). Boston, MA:
7036:
6963:
6913:
6858:
6781:
6734:
6685:
6674:
6663:
6652:
6641:
6460:
6432:
6421:
6373:
6361:
6349:
6338:
6327:
6316:
4945:
4207:
902:is estimated from above by the
666:
660:
145:ordinary differential equations
8421:Rabinovich–Fabrikant equations
7873:Wolfram Demonstrations Project
7684:The Mathematical Intelligencer
7047:doi.org/10.5281/zenodo.5625923
6871:Journal of Statistical Physics
6305:
6294:
6283:
6272:
6261:
6250:
6223:
5891:
5878:
5318:
5312:
5230:
5224:
5128:
5122:
2022:
1707:
1695:
1651:
1639:
1276:
1076:
966:
948:
720:
708:
692:
680:
638:
626:
613:
601:
325:
313:
269:
257:
13:
1:
7054:
5438:. To do that, we take a plan
5302:is cut by two arcs formed by
5242:{\displaystyle P(N)\subset N}
2048:# define the Lorenz attractor
7551:10.1016/0167-2789(93)90006-M
7334:10.1016/0375-9601(81)90274-7
7217:10.1016/0375-9601(75)90353-9
7186:10.1016/0167-2789(83)90298-1
7157:10.1016/0167-2789(86)90022-9
6934:10.1007/978-3-030-70795-8_57
6301:Cuomo & Oppenheim (1993)
5105:{\displaystyle x\in \Sigma }
4179:"Lorenz Attractor"
2549:"Lorenz Attractor"
1336:
1329:
1322:
1308:
1296:
1284:
1223:
1189:
1180:
1173:
1137:
1103:
1094:
1087:
7:
8156:Poincaré recurrence theorem
7838:Encyclopedia of Mathematics
7410:10.1016/j.cnsns.2016.04.032
7255:Hilborn, Robert C. (2000).
6593:Shen, Bo-Wen (2014-04-28).
6533:Shen, Bo-Wen (2019-03-01).
6173:
5609:which gives us a new point
4658:# to assign color to points
1589:A Generalized Lorenz System
1346:These figures — made using
489:
182:
151:. It is notable for having
10:
8771:
8151:Poincaré–Bendixson theorem
7788:Pchelintsev, A.N. (2022).
7604::10.5194/npg-22-749-2015.
6747:Astronomy & Geophysics
6692:Shen, B.-W. (2015-12-21).
6414:10.1007/s11071-020-05856-4
6345:Mishra & Sanghi (2006)
5987:
5818:and we obtain a rectangle
5324:{\displaystyle P(\Sigma )}
4961:Rayleigh–Bénard convection
2912:`Lorenz Chaotic Attractor`
1977:A choice of the parameter
1508:{\displaystyle \beta =8/3}
1474:{\displaystyle \sigma =10}
186:
84:
8704:
8521:
8503:Swinging Atwood's machine
8448:
8386:
8256:
8243:
8195:
8176:
8146:Krylov–Bogolyubov theorem
8126:
8023:
7963:
7780:10.1016/j.amc.2014.12.132
7514:10.1134/S1995423914020098
7112:10.1103/PhysRevLett.71.65
6560:10.1142/S0218127419500378
6182:on the Lyapunov dimension
1279:
1079:
8411:Lotka–Volterra equations
8235:Synchronization of chaos
8038:axiom A dynamical system
7621:Tucker, Warwick (2002).
7558:Saltzman, Barry (1962).
6997:10.1175/BAMS-D-19-0165.1
6334:Kolář & Gumbs (1992)
6201:
5950:into smaller rectangles
5677:{\displaystyle \Sigma '}
5602:{\displaystyle \Sigma '}
5456:{\displaystyle \Sigma '}
5141:where the trajectory of
5086:, which assigns to each
4211:
3645:
3594:NonlinearStateSpaceModel
3585:
3302:
3152:
2927:
2669:
2039:
1448:{\displaystyle \rho =28}
85:Not to be confused with
8396:Double scroll attractor
8161:Stable manifold theorem
8068:False nearest neighbors
7612:Sparrow, Colin (1982).
7363:10.1103/PhysRevA.45.626
7261:Oxford University Press
7091:Physical Review Letters
6759:2010A&G....51d..28G
6719:10.5194/npg-22-749-2015
6619:10.1175/jas-d-13-0223.1
5976:{\displaystyle R_{i,j}}
5868:such that we know that
5861:{\displaystyle \Sigma }
5811:{\displaystyle \Sigma }
5652:{\displaystyle \Sigma }
5476:{\displaystyle \Sigma }
5411:{\displaystyle \Sigma }
5295:{\displaystyle \Sigma }
5174:{\displaystyle \Sigma }
8436:Van der Pol oscillator
8416:Mackey–Glass equations
8048:Box-counting dimension
6545:(3): 1950037–1950091.
5977:
5944:
5943:{\displaystyle R_{i}'}
5914:
5862:
5842:
5841:{\displaystyle Rf_{i}}
5812:
5792:
5791:{\displaystyle R_{i}'}
5762:
5735:
5708:
5707:{\displaystyle R_{i}'}
5678:
5653:
5633:
5632:{\displaystyle c_{i}'}
5603:
5578:
5551:
5524:
5497:
5477:
5457:
5432:
5412:
5392:
5372:
5352:
5325:
5296:
5272:
5243:
5208:
5188:There exists a region
5175:
5155:
5135:
5106:
5080:
5060:
4972:Galerkin approximation
4204:
3296:Mathematica simulation
2032:
2013:
1969:
1565:
1552:Connection to tent map
1535:
1534:{\displaystyle t=24.0}
1509:
1475:
1449:
1420:(red). Produced with
1405:
1032:
896:self-excited attractor
819:
749:
398:
207:atmospheric convection
136:
8586:Svetlana Jitomirskaya
8493:Multiscroll attractor
8338:Interval exchange map
8291:Dyadic transformation
8276:Complex quadratic map
8118:Topological conjugacy
8053:Correlation dimension
8028:Anosov diffeomorphism
7652:10.1007/s002080010018
7426:Lorenz, Edward Norton
7067:John Wiley & Sons
6843:10.3390/atmos13111892
5978:
5945:
5915:
5863:
5843:
5813:
5793:
5763:
5761:{\displaystyle R_{i}}
5736:
5734:{\displaystyle c_{i}}
5709:
5679:
5654:
5634:
5604:
5579:
5577:{\displaystyle c_{i}}
5552:
5550:{\displaystyle R_{i}}
5525:
5523:{\displaystyle c_{i}}
5498:
5478:
5458:
5433:
5413:
5393:
5373:
5353:
5351:{\displaystyle R_{i}}
5326:
5297:
5273:
5244:
5209:
5176:
5156:
5136:
5107:
5081:
5061:
4202:
2030:
2014:
1970:
1559:
1536:
1510:
1476:
1450:
1404:
1033:
916:correlation dimension
820:
750:
568:pitchfork bifurcation
399:
99:
8596:Edward Norton Lorenz
7817:10.3390/math10081207
6439:Leonov et al. (2016)
6191:List of chaotic maps
5954:
5924:
5872:
5852:
5822:
5802:
5772:
5745:
5718:
5688:
5663:
5643:
5613:
5588:
5561:
5534:
5507:
5487:
5467:
5442:
5422:
5402:
5382:
5362:
5335:
5306:
5286:
5259:
5218:
5192:
5165:
5145:
5134:{\displaystyle P(x)}
5116:
5090:
5070:
5019:
3972:# Set initial values
3744:s, r, b : float
1981:
1598:
1519:
1485:
1459:
1433:
1429:The parameters are:
1260:For small values of
1053:For other values of
933:
762:
588:
216:
91:Lorentz distribution
8556:Mitchell Feigenbaum
8498:Population dynamics
8483:Hénon–Heiles system
8343:Irrational rotation
8296:Dynamical billiards
8281:Coupled map lattice
8141:Liouville's theorem
8073:Hausdorff dimension
8058:Conservative system
8043:Bifurcation diagram
7576:1962JAtS...19..329S
7543:1993PhyD...65...86P
7477:2006Chaos..16a3114M
7446:1963JAtS...20..130L
7402:2016CNSNS..41...84L
7355:1992PhRvA..45..626K
7326:1981PhLA...82..439K
7259:(second ed.).
7209:1975PhLA...53...77H
7178:1983PhyD....9..189G
7149:1986PhyD...19..255G
7104:1993PhRvL..71...65C
6988:2021BAMS..102E.148S
6883:1979JSP....21..263Y
6834:2022Atmos..13.1892S
6710:2015NPGeo..22..749S
6611:2014JAtS...71.1701S
6551:2019IJBC...2950037S
5939:
5787:
5703:
5628:
5009:interval arithmetic
4859:'scatter3d'
918:is estimated to be
900:Hausdorff dimension
197:, with the help of
8734:Santa Fe Institute
8601:Aleksandr Lyapunov
8431:Three-body problem
8318:Gingerbreadman map
8205:Bifurcation theory
8083:Lyapunov stability
7884:2009-06-07 at the
7855:"Lorenz attractor"
7852:Weisstein, Eric W.
7833:"Lorenz attractor"
7696:10.1007/BF03025276
7086:Oppenheim, Alan V.
6891:10.1007/BF01011469
6485:10.1007/BF02684769
6401:Nonlinear Dynamics
5973:
5940:
5927:
5910:
5858:
5838:
5808:
5788:
5775:
5768:will be mapped in
5758:
5731:
5704:
5691:
5674:
5659:will be mapped in
5649:
5629:
5616:
5599:
5574:
5547:
5520:
5493:
5473:
5453:
5428:
5408:
5388:
5368:
5348:
5321:
5292:
5278:of the return map.
5271:{\displaystyle DP}
5268:
5239:
5204:
5171:
5151:
5131:
5102:
5076:
5056:
4205:
4161:"Z Axis"
4143:"Y Axis"
4125:"X Axis"
3762:"""
3729:"""
2033:
2009:
2007:
1965:
1963:
1566:
1531:
1505:
1471:
1445:
1406:
1028:
815:
745:
465:chemical reactions
394:
392:
169:earth's atmosphere
137:
8742:
8741:
8606:Benoît Mandelbrot
8571:Martin Gutzwiller
8561:Peter Grassberger
8444:
8443:
8426:Rössler attractor
8174:
8173:
8078:Invariant measure
8000:Lyapunov exponent
7888:on planetmath.org
7712:Lorenz, Edward N.
7485:10.1063/1.2154792
7343:Physical Review A
7314:Physics Letters A
7304:978-0-12-349703-1
7279:Hirsch, Morris W.
7270:978-0-19-850723-9
7240:10.1109/81.260218
7196:Physics Letters A
7084:Cuomo, Kevin M.;
7076:978-0-471-84967-4
6943:978-3-030-70794-1
6788:Read, P. (1993).
6456:. Cham: Springer.
6180:Eden's conjecture
6070:
6034:
5496:{\displaystyle h}
5431:{\displaystyle N}
5391:{\displaystyle N}
5371:{\displaystyle N}
5161:first intersects
5154:{\displaystyle x}
5079:{\displaystyle P}
5001:strange attractor
4595:# call ode solver
3651:matplotlib.pyplot
3642:Python simulation
3149:MATLAB simulation
3134:proportional_axes
2924:Maxima simulation
2006:
1914:
1821:
1745:
1683:
1627:
1549:
1548:
1390:
1389:
1275:
1274:
1026:
1023:
888:strange attractor
810:
723:
695:
664:
641:
616:
473:Malkus waterwheel
461:electric circuits
363:
301:
245:
203:Margaret Hamilton
83:
82:
75:
16:(Redirected from
8762:
8714:Butterfly effect
8626:Itamar Procaccia
8576:Brosl Hasslacher
8473:Elastic pendulum
8401:Duffing equation
8348:Kaplan–Yorke map
8266:Arnold's cat map
8254:
8253:
8230:Stability theory
8215:Dynamical system
8210:Control of chaos
8190:
8182:
8166:Takens's theorem
8098:Poincaré section
7968:
7967:
7950:
7943:
7936:
7927:
7926:
7922:made with Iodide
7869:Lorenz attractor
7865:
7864:
7846:
7821:
7819:
7809:
7784:
7782:
7772:
7740:
7738:
7737:
7731:
7725:. Archived from
7720:
7707:
7678:
7676:
7663:
7645:
7627:
7617:
7589:
7587:
7554:
7525:
7496:
7459:
7457:
7421:
7395:
7374:
7337:
7308:
7274:
7251:
7220:
7189:
7172:(1–2): 189–208.
7160:
7131:
7080:
7049:
7040:
7034:
7029:
7018:
7017:
6999:
6982:(1): E148–E158.
6967:
6961:
6960:
6959:
6958:
6917:
6911:
6910:
6862:
6856:
6855:
6845:
6813:
6804:
6803:
6785:
6779:
6778:
6753:(4): 4.28–4.35.
6738:
6732:
6731:
6721:
6689:
6683:
6678:
6672:
6667:
6661:
6656:
6650:
6645:
6639:
6638:
6605:(5): 1701–1723.
6590:
6581:
6580:
6562:
6530:
6519:
6514:
6505:
6504:
6464:
6458:
6457:
6447:
6441:
6436:
6430:
6425:
6419:
6418:
6416:
6392:
6383:
6377:
6371:
6365:
6359:
6353:
6347:
6342:
6336:
6331:
6325:
6320:
6314:
6309:
6303:
6298:
6292:
6287:
6281:
6276:
6270:
6265:
6259:
6254:
6248:
6243:
6232:
6227:
6221:
6216:
6165:
6153:
6143:
6142:
6140:
6139:
6136:
6133:
6122:
6120:
6112:
6105:
6096:
6084:
6072:
6071:
6048:
6036:
6035:
6014:
6004:
5998:
5982:
5980:
5979:
5974:
5972:
5971:
5949:
5947:
5946:
5941:
5935:
5919:
5917:
5916:
5911:
5909:
5908:
5890:
5889:
5867:
5865:
5864:
5859:
5847:
5845:
5844:
5839:
5837:
5836:
5817:
5815:
5814:
5809:
5797:
5795:
5794:
5789:
5783:
5767:
5765:
5764:
5759:
5757:
5756:
5740:
5738:
5737:
5732:
5730:
5729:
5713:
5711:
5710:
5705:
5699:
5683:
5681:
5680:
5675:
5673:
5658:
5656:
5655:
5650:
5638:
5636:
5635:
5630:
5624:
5608:
5606:
5605:
5600:
5598:
5583:
5581:
5580:
5575:
5573:
5572:
5556:
5554:
5553:
5548:
5546:
5545:
5529:
5527:
5526:
5521:
5519:
5518:
5502:
5500:
5499:
5494:
5482:
5480:
5479:
5474:
5462:
5460:
5459:
5454:
5452:
5437:
5435:
5434:
5429:
5417:
5415:
5414:
5409:
5397:
5395:
5394:
5389:
5377:
5375:
5374:
5369:
5357:
5355:
5354:
5349:
5347:
5346:
5330:
5328:
5327:
5322:
5301:
5299:
5298:
5293:
5277:
5275:
5274:
5269:
5248:
5246:
5245:
5240:
5213:
5211:
5210:
5205:
5180:
5178:
5177:
5172:
5160:
5158:
5157:
5152:
5140:
5138:
5137:
5132:
5111:
5109:
5108:
5103:
5085:
5083:
5082:
5077:
5065:
5063:
5062:
5057:
5040:
5039:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4295:# initial values
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3612:ParametricPlot3D
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3542:ParametricPlot3D
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2666:Maple simulation
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2036:Julia simulation
2031:Julia Simulation
2018:
2016:
2015:
2010:
2008:
1999:
1993:
1992:
1974:
1972:
1971:
1966:
1964:
1957:
1956:
1938:
1937:
1915:
1913:
1909:
1903:
1902:
1901:
1892:
1886:
1877:
1876:
1867:
1866:
1854:
1853:
1822:
1820:
1816:
1810:
1809:
1808:
1799:
1793:
1775:
1774:
1746:
1744:
1740:
1734:
1730:
1724:
1684:
1682:
1678:
1672:
1668:
1662:
1628:
1626:
1622:
1616:
1612:
1606:
1583:
1571:
1540:
1538:
1537:
1532:
1514:
1512:
1511:
1506:
1501:
1480:
1478:
1477:
1472:
1454:
1452:
1451:
1446:
1419:
1415:
1411:
1392:
1391:
1385:
1381:
1380:
1378:
1377:
1374:
1371:
1359:
1352:
1341:
1334:
1327:
1315:
1303:
1291:
1277:
1270:
1263:
1252:
1251:
1249:
1248:
1245:
1242:
1218:
1217:
1215:
1214:
1211:
1208:
1185:
1178:
1166:
1165:
1163:
1162:
1159:
1156:
1132:
1131:
1129:
1128:
1125:
1122:
1099:
1092:
1083:
1077:
1070:
1063:
1056:
1044:Smale's problems
1037:
1035:
1034:
1029:
1027:
1025:
1024:
1010:
1009:
1004:
1000:
984:
969:
943:
925:
923:
913:
911:
885:
884:
882:
881:
878:
875:
863:
856:
845:Hopf bifurcation
842:
831:
824:
822:
821:
816:
811:
809:
792:
775:
754:
752:
751:
746:
741:
737:
724:
704:
696:
676:
665:
662:
659:
655:
642:
622:
617:
597:
583:
576:
562:
551:
541:
534:
533:
531:
530:
527:
524:
512:
505:
501:
497:
431:
427:
423:
419:
415:
411:
403:
401:
400:
395:
393:
364:
362:
358:
352:
348:
342:
302:
300:
296:
290:
286:
280:
246:
244:
240:
234:
230:
224:
161:butterfly effect
157:Lorenz attractor
135:
134:
132:
131:
128:
125:
113:
106:
78:
71:
67:
64:
58:
38:
37:
30:
21:
18:Lorenz attractor
8770:
8769:
8765:
8764:
8763:
8761:
8760:
8759:
8745:
8744:
8743:
8738:
8706:
8700:
8646:Caroline Series
8541:Mary Cartwright
8523:
8517:
8468:Double pendulum
8450:
8440:
8389:
8382:
8308:Exponential map
8259:
8245:
8239:
8197:
8191:
8184:
8170:
8136:Ergodic theorem
8129:
8122:
8113:Stable manifold
8103:Recurrence plot
8019:
7973:
7959:
7954:
7886:Wayback Machine
7879:Lorenz equation
7871:by Rob Morris,
7831:
7828:
7747:
7745:Further reading
7735:
7733:
7729:
7718:
7643:10.1.1.545.3996
7625:
7305:
7287:Devaney, Robert
7271:
7077:
7057:
7052:
7041:
7037:
7030:
7021:
6968:
6964:
6956:
6954:
6944:
6918:
6914:
6863:
6859:
6814:
6807:
6800:
6786:
6782:
6739:
6735:
6690:
6686:
6679:
6675:
6668:
6664:
6659:Saltzman (1962)
6657:
6653:
6646:
6642:
6591:
6584:
6531:
6522:
6515:
6508:
6465:
6461:
6448:
6444:
6437:
6433:
6426:
6422:
6393:
6386:
6378:
6374:
6366:
6362:
6354:
6350:
6343:
6339:
6332:
6328:
6321:
6317:
6310:
6306:
6299:
6295:
6288:
6284:
6277:
6273:
6268:Knobloch (1981)
6266:
6262:
6255:
6251:
6244:
6235:
6228:
6224:
6217:
6208:
6204:
6196:Takens' theorem
6186:Lorenz 96 model
6176:
6169:
6166:
6157:
6154:
6145:
6137:
6134:
6131:
6130:
6128:
6123:
6115:
6113:
6107:
6100:
6097:
6088:
6085:
6076:
6073:
6061:
6056:
6049:
6040:
6037:
6027:
6022:
6015:
6006:
6002:
5999:
5990:
5961:
5957:
5955:
5952:
5951:
5931:
5925:
5922:
5921:
5904:
5900:
5885:
5881:
5873:
5870:
5869:
5853:
5850:
5849:
5832:
5828:
5823:
5820:
5819:
5803:
5800:
5799:
5779:
5773:
5770:
5769:
5752:
5748:
5746:
5743:
5742:
5725:
5721:
5719:
5716:
5715:
5695:
5689:
5686:
5685:
5666:
5664:
5661:
5660:
5644:
5641:
5640:
5620:
5614:
5611:
5610:
5591:
5589:
5586:
5585:
5568:
5564:
5562:
5559:
5558:
5541:
5537:
5535:
5532:
5531:
5514:
5510:
5508:
5505:
5504:
5488:
5485:
5484:
5468:
5465:
5464:
5445:
5443:
5440:
5439:
5423:
5420:
5419:
5403:
5400:
5399:
5383:
5380:
5379:
5363:
5360:
5359:
5342:
5338:
5336:
5333:
5332:
5307:
5304:
5303:
5287:
5284:
5283:
5260:
5257:
5256:
5219:
5216:
5215:
5193:
5190:
5189:
5166:
5163:
5162:
5146:
5143:
5142:
5117:
5114:
5113:
5091:
5088:
5087:
5071:
5068:
5067:
5035:
5031:
5020:
5017:
5016:
4997:
4988:
4978:, Appendix C;
4968:stream function
4953:
4948:
4943:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4871:'lines'
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4693:
4690:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4197:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3639:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3581:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3298:
3293:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3146:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2921:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2663:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2025:
1997:
1988:
1984:
1982:
1979:
1978:
1962:
1961:
1952:
1948:
1933:
1929:
1916:
1905:
1904:
1897:
1893:
1888:
1887:
1885:
1882:
1881:
1872:
1868:
1862:
1858:
1849:
1845:
1823:
1812:
1811:
1804:
1800:
1795:
1794:
1792:
1789:
1788:
1770:
1766:
1747:
1736:
1735:
1726:
1725:
1723:
1720:
1719:
1685:
1674:
1673:
1664:
1663:
1661:
1658:
1657:
1629:
1618:
1617:
1608:
1607:
1605:
1601:
1599:
1596:
1595:
1591:
1581:
1569:
1554:
1520:
1517:
1516:
1497:
1486:
1483:
1482:
1460:
1457:
1456:
1434:
1431:
1430:
1417:
1413:
1409:
1383:
1375:
1372:
1369:
1368:
1366:
1361:
1354:
1347:
1310:
1298:
1286:
1265:
1261:
1246:
1243:
1240:
1239:
1237:
1224:
1212:
1209:
1206:
1205:
1203:
1190:
1160:
1157:
1154:
1153:
1151:
1138:
1126:
1123:
1120:
1119:
1117:
1104:
1081:
1065:
1058:
1054:
1005:
990:
986:
985:
983:
970:
944:
942:
934:
931:
930:
921:
919:
909:
907:
879:
876:
873:
872:
870:
865:
862: = 10
858:
855: = 28
851:
833:
829:
793:
776:
774:
763:
760:
759:
703:
675:
671:
667:
661:
621:
596:
595:
591:
589:
586:
585:
578:
571:
557:
546:
536:
528:
525:
522:
521:
519:
514:
507:
503:
499:
495:
492:
469:forward osmosis
438:Rayleigh number
429:
425:
421:
417:
413:
409:
391:
390:
365:
354:
353:
344:
343:
341:
338:
337:
303:
292:
291:
282:
281:
279:
276:
275:
247:
236:
235:
226:
225:
223:
219:
217:
214:
213:
191:
185:
143:is a system of
129:
126:
123:
122:
120:
115:
112: = 10
108:
105: = 28
101:
94:
79:
68:
62:
59:
51:help improve it
48:
39:
35:
28:
23:
22:
15:
12:
11:
5:
8768:
8758:
8757:
8740:
8739:
8737:
8736:
8731:
8729:Predictability
8726:
8721:
8716:
8710:
8708:
8702:
8701:
8699:
8698:
8696:Lai-Sang Young
8693:
8691:James A. Yorke
8688:
8686:Amie Wilkinson
8683:
8678:
8673:
8668:
8663:
8658:
8653:
8648:
8643:
8638:
8633:
8628:
8623:
8621:Henri Poincaré
8618:
8613:
8608:
8603:
8598:
8593:
8588:
8583:
8578:
8573:
8568:
8563:
8558:
8553:
8548:
8543:
8538:
8533:
8527:
8525:
8519:
8518:
8516:
8515:
8510:
8505:
8500:
8495:
8490:
8488:Kicked rotator
8485:
8480:
8475:
8470:
8465:
8460:
8458:Chua's circuit
8454:
8452:
8446:
8445:
8442:
8441:
8439:
8438:
8433:
8428:
8423:
8418:
8413:
8408:
8403:
8398:
8392:
8390:
8387:
8384:
8383:
8381:
8380:
8378:Zaslavskii map
8375:
8373:Tinkerbell map
8370:
8365:
8360:
8355:
8350:
8345:
8340:
8335:
8330:
8325:
8320:
8315:
8310:
8305:
8304:
8303:
8293:
8288:
8283:
8278:
8273:
8268:
8262:
8260:
8257:
8251:
8241:
8240:
8238:
8237:
8232:
8227:
8222:
8220:Ergodic theory
8217:
8212:
8207:
8201:
8199:
8193:
8192:
8177:
8175:
8172:
8171:
8169:
8168:
8163:
8158:
8153:
8148:
8143:
8138:
8132:
8130:
8127:
8124:
8123:
8121:
8120:
8115:
8110:
8105:
8100:
8095:
8090:
8085:
8080:
8075:
8070:
8065:
8060:
8055:
8050:
8045:
8040:
8035:
8030:
8024:
8021:
8020:
8018:
8017:
8012:
8010:Periodic point
8007:
8002:
7997:
7992:
7987:
7982:
7976:
7974:
7971:
7965:
7961:
7960:
7953:
7952:
7945:
7938:
7930:
7924:
7923:
7917:
7911:
7906:
7901:
7895:
7889:
7876:
7866:
7847:
7827:
7826:External links
7824:
7823:
7822:
7785:
7746:
7743:
7742:
7741:
7708:
7679:
7664:
7618:
7609:
7600:(6): 749–764.
7590:
7570:(4): 329–341.
7555:
7526:
7508:(2): 159–167.
7497:
7460:
7440:(2): 130–141.
7422:
7375:
7349:(2): 626–637.
7338:
7320:(9): 439–440.
7309:
7303:
7295:Academic Press
7283:Smale, Stephen
7275:
7269:
7252:
7221:
7190:
7161:
7143:(2): 255–267.
7132:
7081:
7075:
7056:
7053:
7051:
7050:
7035:
7019:
6962:
6942:
6912:
6877:(3): 263–277.
6857:
6805:
6798:
6780:
6733:
6704:(6): 749–764.
6684:
6673:
6662:
6651:
6640:
6582:
6520:
6506:
6459:
6442:
6431:
6420:
6407:(2): 713–732.
6384:
6372:
6360:
6348:
6337:
6326:
6315:
6304:
6293:
6282:
6271:
6260:
6249:
6246:Sparrow (1982)
6233:
6222:
6205:
6203:
6200:
6199:
6198:
6193:
6188:
6183:
6175:
6172:
6171:
6170:
6167:
6160:
6158:
6155:
6148:
6146:
6098:
6091:
6089:
6086:
6079:
6077:
6074:
6059:
6057:
6050:
6043:
6041:
6038:
6025:
6023:
6016:
6009:
6007:
6000:
5993:
5989:
5986:
5970:
5967:
5964:
5960:
5938:
5934:
5930:
5907:
5903:
5899:
5896:
5893:
5888:
5884:
5880:
5877:
5857:
5835:
5831:
5827:
5807:
5786:
5782:
5778:
5755:
5751:
5728:
5724:
5702:
5698:
5694:
5672:
5669:
5648:
5627:
5623:
5619:
5597:
5594:
5571:
5567:
5544:
5540:
5517:
5513:
5492:
5483:at a distance
5472:
5451:
5448:
5427:
5407:
5387:
5367:
5345:
5341:
5320:
5317:
5314:
5311:
5291:
5280:
5279:
5267:
5264:
5253:
5250:
5238:
5235:
5232:
5229:
5226:
5223:
5203:
5200:
5197:
5170:
5150:
5130:
5127:
5124:
5121:
5101:
5098:
5095:
5075:
5055:
5052:
5049:
5046:
5043:
5038:
5034:
5030:
5027:
5024:
5005:Warwick Tucker
4996:
4993:
4987:
4984:
4976:Hilborn (2000)
4952:
4949:
4947:
4944:
4212:
4209:
4206:
3646:
3643:
3640:
3586:
3583:Less verbose:
3303:
3300:Standard way:
3297:
3294:
3153:
3150:
3147:
2928:
2925:
2922:
2670:
2667:
2664:
2040:
2037:
2034:
2024:
2021:
2005:
2002:
1996:
1991:
1987:
1960:
1955:
1951:
1947:
1944:
1941:
1936:
1932:
1928:
1925:
1922:
1919:
1917:
1912:
1908:
1900:
1896:
1891:
1884:
1883:
1880:
1875:
1871:
1865:
1861:
1857:
1852:
1848:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1824:
1819:
1815:
1807:
1803:
1798:
1791:
1790:
1787:
1784:
1781:
1778:
1773:
1769:
1765:
1762:
1759:
1756:
1753:
1750:
1748:
1743:
1739:
1733:
1729:
1722:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1686:
1681:
1677:
1671:
1667:
1660:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1630:
1625:
1621:
1615:
1611:
1604:
1603:
1590:
1587:
1553:
1550:
1547:
1546:
1530:
1527:
1524:
1504:
1500:
1496:
1493:
1490:
1470:
1467:
1464:
1444:
1441:
1438:
1426:
1425:
1397:
1396:
1388:
1387:
1343:
1342:
1335:
1328:
1320:
1319:
1307:
1295:
1282:
1281:
1273:
1272:
1257:
1256:
1222:
1187:
1186:
1179:
1171:
1170:
1136:
1101:
1100:
1093:
1085:
1084:
1048:Warwick Tucker
1039:
1038:
1022:
1019:
1016:
1013:
1008:
1003:
999:
996:
993:
989:
982:
979:
976:
973:
968:
965:
962:
959:
956:
953:
950:
947:
941:
938:
826:
825:
814:
808:
805:
802:
799:
796:
791:
788:
785:
782:
779:
773:
770:
767:
744:
740:
736:
733:
730:
727:
722:
719:
716:
713:
710:
707:
702:
699:
694:
691:
688:
685:
682:
679:
674:
670:
658:
654:
651:
648:
645:
640:
637:
634:
631:
628:
625:
620:
615:
612:
609:
606:
603:
600:
594:
491:
488:
434:Prandtl number
405:
404:
389:
386:
383:
380:
377:
374:
371:
368:
366:
361:
357:
351:
347:
340:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
304:
299:
295:
289:
285:
278:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
248:
243:
239:
233:
229:
222:
221:
184:
181:
81:
80:
42:
40:
33:
26:
9:
6:
4:
3:
2:
8767:
8756:
8753:
8752:
8750:
8735:
8732:
8730:
8727:
8725:
8724:Edge of chaos
8722:
8720:
8717:
8715:
8712:
8711:
8709:
8703:
8697:
8694:
8692:
8689:
8687:
8684:
8682:
8681:Marcelo Viana
8679:
8677:
8674:
8672:
8671:Audrey Terras
8669:
8667:
8666:Floris Takens
8664:
8662:
8659:
8657:
8654:
8652:
8649:
8647:
8644:
8642:
8639:
8637:
8634:
8632:
8629:
8627:
8624:
8622:
8619:
8617:
8614:
8612:
8609:
8607:
8604:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8577:
8574:
8572:
8569:
8567:
8566:Celso Grebogi
8564:
8562:
8559:
8557:
8554:
8552:
8549:
8547:
8546:Chen Guanrong
8544:
8542:
8539:
8537:
8534:
8532:
8531:Michael Berry
8529:
8528:
8526:
8520:
8514:
8511:
8509:
8506:
8504:
8501:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8479:
8476:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8455:
8453:
8447:
8437:
8434:
8432:
8429:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8407:
8406:Lorenz system
8404:
8402:
8399:
8397:
8394:
8393:
8391:
8385:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8353:Langton's ant
8351:
8349:
8346:
8344:
8341:
8339:
8336:
8334:
8331:
8329:
8328:Horseshoe map
8326:
8324:
8321:
8319:
8316:
8314:
8311:
8309:
8306:
8302:
8299:
8298:
8297:
8294:
8292:
8289:
8287:
8284:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8255:
8252:
8249:
8242:
8236:
8233:
8231:
8228:
8226:
8225:Quantum chaos
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8206:
8203:
8202:
8200:
8194:
8189:
8185:
8181:
8167:
8164:
8162:
8159:
8157:
8154:
8152:
8149:
8147:
8144:
8142:
8139:
8137:
8134:
8133:
8131:
8125:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8099:
8096:
8094:
8091:
8089:
8086:
8084:
8081:
8079:
8076:
8074:
8071:
8069:
8066:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8033:Arnold tongue
8031:
8029:
8026:
8025:
8022:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7981:
7978:
7977:
7975:
7969:
7966:
7962:
7958:
7951:
7946:
7944:
7939:
7937:
7932:
7931:
7928:
7921:
7918:
7915:
7912:
7910:
7907:
7905:
7902:
7899:
7896:
7893:
7890:
7887:
7883:
7880:
7877:
7874:
7870:
7867:
7862:
7861:
7856:
7853:
7848:
7844:
7840:
7839:
7834:
7830:
7829:
7818:
7813:
7808:
7803:
7799:
7795:
7791:
7786:
7781:
7776:
7771:
7766:
7762:
7758:
7754:
7749:
7748:
7732:on 2019-05-23
7728:
7724:
7717:
7713:
7709:
7705:
7701:
7697:
7693:
7689:
7685:
7680:
7675:
7670:
7665:
7661:
7657:
7653:
7649:
7644:
7639:
7636:(1): 53–117.
7635:
7631:
7624:
7619:
7615:
7610:
7607:
7603:
7599:
7595:
7591:
7586:
7581:
7577:
7573:
7569:
7565:
7561:
7556:
7552:
7548:
7544:
7540:
7536:
7532:
7527:
7523:
7519:
7515:
7511:
7507:
7503:
7498:
7494:
7490:
7486:
7482:
7478:
7474:
7471:(1): 013114.
7470:
7466:
7461:
7456:
7451:
7447:
7443:
7439:
7435:
7431:
7427:
7423:
7419:
7415:
7411:
7407:
7403:
7399:
7394:
7389:
7385:
7381:
7376:
7372:
7368:
7364:
7360:
7356:
7352:
7348:
7344:
7339:
7335:
7331:
7327:
7323:
7319:
7315:
7310:
7306:
7300:
7296:
7292:
7288:
7284:
7280:
7276:
7272:
7266:
7262:
7258:
7253:
7249:
7245:
7241:
7237:
7233:
7229:
7228:
7222:
7218:
7214:
7210:
7206:
7202:
7198:
7197:
7191:
7187:
7183:
7179:
7175:
7171:
7167:
7162:
7158:
7154:
7150:
7146:
7142:
7138:
7133:
7129:
7125:
7121:
7117:
7113:
7109:
7105:
7101:
7097:
7093:
7092:
7087:
7082:
7078:
7072:
7068:
7064:
7059:
7058:
7048:
7045:
7044:bdtoolbox.org
7039:
7033:
7028:
7026:
7024:
7015:
7011:
7007:
7003:
6998:
6993:
6989:
6985:
6981:
6977:
6973:
6966:
6953:
6949:
6945:
6939:
6935:
6931:
6927:
6923:
6916:
6908:
6904:
6900:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6861:
6853:
6849:
6844:
6839:
6835:
6831:
6827:
6823:
6819:
6812:
6810:
6801:
6795:
6791:
6784:
6776:
6772:
6768:
6764:
6760:
6756:
6752:
6748:
6744:
6737:
6729:
6725:
6720:
6715:
6711:
6707:
6703:
6699:
6695:
6688:
6682:
6681:Lorenz (1963)
6677:
6671:
6670:Lorenz (1963)
6666:
6660:
6655:
6649:
6648:Lorenz (1963)
6644:
6636:
6632:
6628:
6624:
6620:
6616:
6612:
6608:
6604:
6600:
6596:
6589:
6587:
6578:
6574:
6570:
6566:
6561:
6556:
6552:
6548:
6544:
6540:
6536:
6529:
6527:
6525:
6518:
6517:Tucker (2002)
6513:
6511:
6502:
6498:
6494:
6490:
6486:
6482:
6478:
6474:
6470:
6463:
6455:
6454:
6446:
6440:
6435:
6429:
6424:
6415:
6410:
6406:
6402:
6398:
6391:
6389:
6382:, pp. 307–308
6381:
6376:
6370:, pp. 306+307
6369:
6364:
6358:, pp. 303–305
6357:
6352:
6346:
6341:
6335:
6330:
6324:
6323:Tzenov (2014)
6319:
6313:
6312:Poland (1993)
6308:
6302:
6297:
6291:
6290:Hemati (1994)
6286:
6280:
6275:
6269:
6264:
6258:
6253:
6247:
6242:
6240:
6238:
6231:
6230:Lorenz (1960)
6226:
6220:
6219:Lorenz (1963)
6215:
6213:
6211:
6206:
6197:
6194:
6192:
6189:
6187:
6184:
6181:
6178:
6177:
6164:
6159:
6152:
6147:
6126:
6118:
6110:
6103:
6095:
6090:
6083:
6078:
6058:
6054:
6047:
6042:
6024:
6020:
6013:
6008:
5997:
5992:
5991:
5985:
5968:
5965:
5962:
5958:
5936:
5932:
5928:
5905:
5901:
5897:
5894:
5886:
5882:
5875:
5833:
5829:
5825:
5784:
5780:
5776:
5753:
5749:
5726:
5722:
5700:
5696:
5692:
5670:
5625:
5621:
5617:
5595:
5569:
5565:
5542:
5538:
5515:
5511:
5490:
5449:
5425:
5385:
5365:
5343:
5339:
5309:
5265:
5262:
5254:
5251:
5236:
5233:
5227:
5221:
5198:
5195:
5187:
5186:
5185:
5182:
5148:
5125:
5119:
5096:
5093:
5073:
5050:
5047:
5044:
5041:
5036:
5032:
5025:
5014:
5010:
5006:
5002:
4992:
4983:
4981:
4977:
4973:
4969:
4964:
4962:
4958:
4733:as.data.frame
4201:
3606:StateResponse
3584:
3301:
3086:points_joined
2029:
2020:
2003:
2000:
1994:
1989:
1985:
1975:
1958:
1953:
1949:
1945:
1942:
1939:
1934:
1930:
1926:
1923:
1920:
1918:
1910:
1898:
1894:
1878:
1873:
1869:
1863:
1859:
1855:
1850:
1846:
1842:
1839:
1836:
1833:
1830:
1827:
1825:
1817:
1805:
1801:
1785:
1782:
1779:
1776:
1771:
1767:
1763:
1760:
1757:
1754:
1751:
1749:
1741:
1731:
1716:
1713:
1710:
1704:
1701:
1698:
1692:
1689:
1687:
1679:
1669:
1654:
1648:
1645:
1642:
1636:
1633:
1631:
1623:
1613:
1586:
1579:
1575:
1574:Poincaré plot
1563:
1558:
1544:
1528:
1525:
1522:
1502:
1498:
1494:
1491:
1488:
1468:
1465:
1462:
1442:
1439:
1436:
1428:
1427:
1423:
1403:
1399:
1398:
1394:
1393:
1364:
1357:
1350:
1344:
1340:
1333:
1326:
1321:
1318:
1313:
1306:
1301:
1294:
1289:
1283:
1278:
1268:
1258:
1255:
1235:
1231:
1227:
1221:
1201:
1197:
1193:
1188:
1184:
1177:
1172:
1169:
1149:
1145:
1141:
1135:
1115:
1111:
1107:
1102:
1098:
1091:
1086:
1078:
1075:
1073:
1068:
1064:it becomes a
1061:
1051:
1049:
1045:
1020:
1017:
1014:
1011:
1006:
1001:
997:
994:
991:
987:
980:
977:
974:
971:
963:
960:
957:
954:
951:
945:
939:
936:
929:
928:
927:
917:
905:
901:
897:
893:
889:
869: =
868:
861:
854:
848:
846:
840:
836:
812:
806:
803:
800:
797:
794:
789:
786:
783:
780:
777:
771:
768:
765:
758:
757:
756:
742:
738:
734:
731:
728:
725:
717:
714:
711:
705:
700:
697:
689:
686:
683:
677:
672:
668:
656:
652:
649:
646:
643:
635:
632:
629:
623:
618:
610:
607:
604:
598:
592:
581:
574:
569:
564:
560:
555:
549:
543:
539:
517:
510:
487:
485:
484:deterministic
481:
476:
474:
470:
466:
462:
458:
454:
453:thermosyphons
450:
446:
441:
439:
435:
387:
384:
381:
378:
375:
372:
369:
367:
359:
349:
334:
331:
328:
322:
319:
316:
310:
307:
305:
297:
287:
272:
266:
263:
260:
254:
251:
249:
241:
231:
212:
211:
210:
208:
204:
200:
196:
195:Edward Lorenz
190:
180:
178:
174:
173:Saddle points
170:
166:
165:deterministic
162:
158:
154:
150:
149:Edward Lorenz
146:
142:
141:Lorenz system
119: =
118:
111:
104:
98:
92:
88:
77:
74:
66:
63:December 2023
56:
52:
46:
43:This article
41:
32:
31:
19:
8755:Chaotic maps
8676:Mary Tsingou
8641:David Ruelle
8636:Otto Rössler
8581:Michel Hénon
8551:Leon O. Chua
8508:Tilt-A-Whirl
8478:FPUT problem
8405:
8363:Standard map
8358:Logistic map
8183:
7957:Chaos theory
7858:
7836:
7797:
7793:
7760:
7756:
7734:. Retrieved
7727:the original
7722:
7687:
7683:
7633:
7629:
7613:
7597:
7593:
7567:
7563:
7537:(1): 86–99.
7534:
7530:
7505:
7501:
7468:
7464:
7437:
7433:
7383:
7379:
7346:
7342:
7317:
7313:
7290:
7256:
7234:(1): 40–45.
7231:
7225:
7203:(1): 77–78.
7200:
7194:
7169:
7165:
7140:
7136:
7098:(1): 65–68.
7095:
7089:
7065:. New York:
7062:
7038:
7032:Viana (2000)
6979:
6975:
6965:
6955:, retrieved
6925:
6915:
6874:
6870:
6860:
6828:(11): 1892.
6825:
6821:
6789:
6783:
6750:
6746:
6736:
6701:
6697:
6687:
6676:
6665:
6654:
6643:
6602:
6598:
6542:
6538:
6479:(1): 59–72.
6476:
6472:
6462:
6452:
6445:
6434:
6423:
6404:
6400:
6375:
6363:
6351:
6340:
6329:
6318:
6307:
6296:
6285:
6274:
6263:
6257:Haken (1975)
6252:
6225:
6124:
6116:
6108:
6101:
5714:centered on
5281:
5183:
5013:normal forms
4998:
4989:
4965:
4954:
4946:Applications
4238:# parameters
4208:R simulation
4203:R Simulation
4068:'3d'
3582:
3536:NDSolveValue
3299:
1976:
1592:
1567:
1362:
1355:
1348:
1311:
1299:
1287:
1266:
1233:
1229:
1225:
1199:
1195:
1191:
1147:
1143:
1139:
1113:
1109:
1105:
1066:
1059:
1052:
1040:
866:
859:
852:
849:
838:
834:
827:
579:
572:
565:
558:
547:
544:
537:
515:
508:
493:
477:
455:, brushless
442:
406:
199:Ellen Fetter
192:
156:
140:
138:
116:
109:
102:
87:Lorenz curve
69:
60:
44:
8661:Nina Snaith
8651:Yakov Sinai
8536:Rufus Bowen
8286:Duffing map
8271:Baker's map
8196:Theoretical
8108:SRB measure
8015:Phase space
7985:Bifurcation
7800:(8): 1207.
7794:Mathematics
7763:: 334–343.
7690:(3): 6–19.
7674:1406.0979v1
7616:. Springer.
4056:add_subplot
2897:orientation
2023:Simulations
1562:Mathematica
1416:(blue) and
177:phase space
8719:Complexity
8616:Edward Ott
8463:Convection
8388:Continuous
8063:Ergodicity
7807:2206.08195
7736:2020-09-16
7608:1607-7946.
7393:1508.07498
7386:: 84–103.
7055:References
6957:2022-12-22
6822:Atmosphere
6799:0198539541
6055:structure.
5112:the point
4928:reverscale
4155:set_zlabel
4137:set_ylabel
4119:set_xlabel
4062:projection
3735:----------
3732:Parameters
3098:point_type
2852:linecolour
1269:> 24.74
1072:torus knot
914:, and the
577:, and for
570:occurs at
187:See also:
8631:Mary Rees
8591:Bryna Kra
8524:theorists
8333:Ikeda map
8323:Hénon map
8313:Gauss map
7995:Limit set
7980:Attractor
7860:MathWorld
7843:EMS Press
7770:1409.8649
7704:121427433
7638:CiteSeerX
7531:Physica D
7522:123023929
7418:119614076
7248:1057-7122
7166:Physica D
7137:Physica D
7120:0031-9007
7014:208369617
7006:0003-0007
6952:245197840
6899:1572-9613
6852:2073-4433
6775:1366-8781
6728:1607-7946
6635:123683839
6627:0022-4928
6577:132494234
6569:0218-1274
6493:0073-8301
5895:⊂
5856:Σ
5806:Σ
5668:Σ
5647:Σ
5593:Σ
5471:Σ
5447:Σ
5406:Σ
5316:Σ
5290:Σ
5234:⊂
5202:Σ
5199:⊂
5169:Σ
5100:Σ
5097:∈
5048:−
5026:⊂
5023:Σ
4667:function
4352:function
4173:set_title
3996:num_steps
3924:num_steps
3897:num_steps
2885:thickness
2642:attractor
2630:attractor
2618:attractor
2600:attractor
2441:attractor
1946:β
1940:−
1856:−
1837:−
1780:β
1777:−
1761:−
1711:−
1702:−
1699:ρ
1646:−
1637:σ
1489:β
1463:σ
1437:ρ
1418:(1.1,0,0)
1414:(1.0,0,0)
1412:(green),
1410:(0.9,0,0)
1317:(Enlarge)
1305:(Enlarge)
1293:(Enlarge)
1254:(Enlarge)
1220:(Enlarge)
1168:(Enlarge)
1134:(Enlarge)
1050:in 2002.
1021:ρ
1018:σ
995:−
992:σ
972:σ
958:β
952:σ
940:−
804:−
801:β
798:−
795:σ
784:β
778:σ
772:σ
766:ρ
732:−
729:ρ
715:−
712:ρ
706:β
701:−
687:−
684:ρ
678:β
673:−
650:−
647:ρ
633:−
630:ρ
624:β
608:−
605:ρ
599:β
554:attractor
480:nonlinear
457:DC motors
382:β
379:−
329:−
320:−
317:ρ
264:−
255:σ
193:In 1963,
8749:Category
8707:articles
8449:Physical
8368:Tent map
8258:Discrete
8198:branches
8128:Theorems
7964:Concepts
7882:Archived
7714:(1960).
7493:16599745
7428:(1963).
7289:(2003).
7128:10054374
6907:12172750
6501:55218285
6174:See also
5937:′
5785:′
5701:′
5671:′
5626:′
5596:′
5450:′
3056:makelist
3026:makelist
2996:makelist
2780:stepsize
2696:DEplot3d
2177:function
1578:tent map
1422:WxMaxima
894:, and a
490:Analysis
183:Overview
8705:Related
8513:Weather
8451:systems
8244:Chaotic
7990:Fractal
7845:, 2001
7572:Bibcode
7539:Bibcode
7473:Bibcode
7442:Bibcode
7398:Bibcode
7371:9907027
7351:Bibcode
7322:Bibcode
7205:Bibcode
7174:Bibcode
7145:Bibcode
7100:Bibcode
6984:Bibcode
6879:Bibcode
6830:Bibcode
6755:Bibcode
6706:Bibcode
6607:Bibcode
6547:Bibcode
6141:
6129:
5988:Gallery
4877:opacity
4790:plot_ly
4763:rainbow
4700:ceiling
4385:as.list
4226:library
4220:deSolve
4214:library
3753:-------
3750:Returns
2687:DEtools
2165:Float64
2150:Float64
2135:Float64
2114:Float64
2099:Float64
2084:Float64
2069:Float64
2054:mutable
1379:
1367:
1250:
1238:
1216:
1204:
1164:
1152:
1130:
1118:
1062:= 99.96
892:fractal
883:
871:
556:, when
532:
520:
449:dynamos
153:chaotic
133:
121:
49:Please
8611:Hee Oh
8246:maps (
8093:Mixing
7702:
7660:353254
7658:
7640:
7520:
7491:
7416:
7369:
7301:
7267:
7246:
7126:
7118:
7073:
7012:
7004:
6950:
6940:
6905:
6897:
6850:
6796:
6773:
6726:
6633:
6625:
6575:
6567:
6499:
6491:
6005:plane.
5463:below
4718:repArr
4712:length
4694:repArr
4673:repArr
4640:Lorenz
4616:varini
4505:return
4346:Lorenz
4298:varini
4232:plotly
4047:figure
4032:# Plot
4014:lorenz
3873:return
3675:lorenz
3660:import
3648:import
3588:lorenz
3110:points
3080:draw3d
2978:length
2555:marker
2462:plot3d
2447:Lorenz
2192:Lorenz
2060:Lorenz
2057:struct
2051:@kwdef
1232:= 10,
1228:= 28,
1198:= 10,
1194:= 15,
1146:= 10,
1142:= 13,
1112:= 10,
1108:= 14,
864:, and
582:> 1
561:< 1
550:< 1
502:, and
445:lasers
428:, and
114:, and
8522:Chaos
8301:outer
8005:Orbit
7802:arXiv
7765:arXiv
7730:(PDF)
7719:(PDF)
7700:S2CID
7669:arXiv
7656:S2CID
7626:(PDF)
7518:S2CID
7414:S2CID
7388:arXiv
7010:S2CID
6948:S2CID
6903:S2CID
6631:S2CID
6573:S2CID
6497:S2CID
6202:Notes
5418:, in
4934:FALSE
4922:color
4913:color
4901:width
4757:gfill
4754:<-
4751:color
4730:<-
4664:<-
4661:gfill
4646:parms
4628:times
4622:times
4601:<-
4550:<-
4547:times
4475:<-
4436:<-
4412:sigma
4403:<-
4349:<-
4301:<-
4253:sigma
4244:<-
3990:range
3918:empty
3903:10000
3882:array
3846:z_dot
3813:y_dot
3786:x_dot
3723:2.667
3663:numpy
3497:->
3485:->
3473:->
3392:'
3356:'
3329:'
3251:plot3
3236:ode45
3167:sigma
2906:title
2657:every
2606:push!
2594:step!
2543:title
2180:step!
2045:Plots
2042:using
1309:Time
1297:Time
1285:Time
1069:(3,2)
850:When
837:>
8248:list
7972:Core
7606:ISSN
7489:PMID
7367:PMID
7299:ISBN
7265:ISBN
7244:ISSN
7124:PMID
7116:ISSN
7071:ISBN
7002:ISSN
6938:ISBN
6895:ISSN
6848:ISSN
6794:ISBN
6771:ISSN
6724:ISSN
6623:ISSN
6565:ISSN
6489:ISSN
6119:= 10
6111:= 28
5011:and
4895:list
4889:line
4865:mode
4853:type
4802:dout
4796:data
4781:dout
4775:nrow
4745:dout
4727:dout
4706:long
4679:long
4634:func
4589:0.01
4559:from
4511:list
4496:beta
4391:vars
4379:with
4364:vars
4277:beta
4247:list
4191:show
4089:xyzs
4080:plot
4020:xyzs
4008:xyzs
4002:xyzs
3966:1.05
3945:xyzs
3906:xyzs
3894:0.01
3600:soln
3575:tend
3461:pars
3416:init
3305:tend
3245:,,);
3179:beta
3092:true
2786:0.01
2681:with
2591:1500
2573:@gif
2522:zlim
2498:ylim
2474:xlim
2075:0.02
1543:here
1529:24.0
1481:and
1360:and
1358:= 10
1351:= 28
924:0.01
920:2.05
912:0.01
908:2.06
890:, a
769:<
540:= 28
535:and
511:= 10
467:and
139:The
7812:doi
7775:doi
7761:256
7692:doi
7648:doi
7602:doi
7580:doi
7547:doi
7510:doi
7481:doi
7450:doi
7406:doi
7359:doi
7330:doi
7236:doi
7213:doi
7182:doi
7153:doi
7108:doi
6992:doi
6980:102
6930:doi
6887:doi
6838:doi
6763:doi
6714:doi
6615:doi
6555:doi
6481:doi
6409:doi
6405:102
6106:to
6104:= 0
6019:SVG
5848:in
5584:in
5530:of
4739:out
4721:)))
4688:rep
4652:prm
4604:ode
4598:out
4577:100
4553:seq
4538:)))
4490:prm
4454:rho
4448:prm
4406:prm
4370:prm
4265:rho
4241:prm
4185:plt
4107:0.6
4041:plt
3981:for
3783:xyz
3681:xyz
3672:def
3657:plt
3284:(:,
3272:(:,
3260:(:,
3197:rho
3143:)$
3140:xyz
3077:)$
3074:len
3062:sol
3047:)$
3044:len
3032:sol
3017:)$
3014:len
3002:sol
2987:)$
2984:sol
2972:len
2969:)$
2945:sol
2858:sin
2768:100
2702:deq
2672:deq
2654:end
2612:plt
2576:for
2456:plt
2438:end
2174:end
1314:= 3
1302:= 2
1290:= 1
906:as
841:+ 1
832:if
663:and
575:= 1
545:If
89:or
53:to
8751::
7857:.
7841:,
7835:,
7810:.
7798:10
7796:.
7792:.
7773:.
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7688:22
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7596:.
7578:.
7568:19
7566:.
7562:.
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7535:65
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7504:.
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7412:.
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7328:.
7318:82
7316:.
7297:.
7285:;
7281:;
7263:.
7242:.
7232:41
7230:.
7211:.
7201:53
7199:.
7180:.
7168:.
7151:.
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7139:.
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7114:.
7106:.
7096:71
7094:.
7069:.
7022:^
7008:.
7000:.
6990:.
6978:.
6974:.
6946:,
6936:,
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6901:.
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6875:21
6873:.
6869:.
6846:.
6836:.
6826:13
6824:.
6820:.
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6769:.
6761:.
6751:51
6749:.
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6541:.
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6509:^
6495:.
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6475:.
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6144:).
6127:=
6053:3D
6003:xz
5181:.
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4772:),
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4748:$
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4571:to
4541:})
4535:dZ
4529:dY
4523:dX
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4394:),
4271:28
4259:10
4194:()
4167:ax
4149:ax
4131:ax
4113:ax
4101:lw
4074:ax
4050:()
4035:ax
4029:dt
3999:):
3987:in
3960:1.
3954:0.
3939:))
3921:((
3912:np
3888:dt
3885:()
3876:np
3726:):
3711:28
3699:10
3669:np
3666:as
3654:as
3636:}]
3633:50
3578:}]
3557:},
3554:zs
3548:ys
3527:zs
3521:ys
3515:xs
3509:};
3488:28
3476:10
3458:};
3455:10
3452:==
3443:10
3440:==
3431:10
3428:==
3413:};
3395:==
3359:==
3350:),
3332:==
3317:eq
3311:50
3290:))
3278:),
3266:),
3215:@(
3203:28
3173:10
3131:),
3101:=-
2957:eq
2951:rk
2942:$
2936:eq
2933:$
2870:Pi
2846:50
2843:..
2828:25
2825:..
2822:25
2807:20
2804:..
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2765:..
2750:)}
2675::=
2660:10
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2537:60
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2435:dz
2429:dt
2420:+=
2357:dz
2354:dy
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2270:dy
2267:dx
2261:dt
2252:+=
2240:);
2198:dx
2189:::
2162:::
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2132:::
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2105:28
2096:::
2090:10
2081:::
2066:::
2063:dt
2001:19
1545:.
1469:10
1455:,
1443:28
1424:.
1365:=
1353:,
1236:=
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857:,
847:.
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563:.
518:=
513:,
498:,
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6138:3
6135:/
6132:8
6125:β
6121:,
6117:σ
6114:(
6109:ρ
6102:ρ
6021:.
5969:j
5966:,
5963:i
5959:R
5933:i
5929:R
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4931:=
4925:,
4919:~
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4883:1
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4856:=
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4832:Y
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4565:0
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