Butterfly effect - Knowledge (XXG)

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step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics.

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afterward differed by one and then several units in the last place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution.

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new type of butterfly effect is introduced, highlighting the potential impact of small-scale processes on finite predictability within the Lorenz 1969 model. Additionally, the identification of ill-conditioned aspects of the Lorenz 1969 model points to a practical form of finite predictability. These two distinct mechanisms suggesting finite predictability in the Lorenz 1969 model are collectively referred to as the third kind of butterfly effect. The authors in have considered Palmer et al.'s suggestions and have aimed to present their perspective without raising specific contentions.

1326:. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. David Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect". Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given 4369: 2959: 4361: 38: 1211:

to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong."

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respectively. As a result, when two kayaks move along strong currents, their paths display SDIC. On the other hand, when two kayaks move into a stagnant area, they become trapped, showing no typical SDIC (although a chaotic transient may occur). Such features of SDIC or no SDIC suggest two types of solutions and illustrate the nature of multistability.

1241:, it is acknowledged that the second kind of butterfly effect (BE2) has never been rigorously verified using a realistic weather model. While the studies suggest that BE2 is unlikely in the real atmosphere, its invalidity in this context does not negate the applicability of BE1 in other areas, such as pandemics or historical events. 62: 1330:, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to initial conditions are known as 1164:

never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2 shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared

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While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied to work he published in 1969 which took the idea a

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By revealing coexisting chaotic and non-chaotic attractors within Lorenz models, Shen and his colleagues proposed a revised view that "weather possesses chaos and order", in contrast to the conventional view of "weather is chaotic". As a result, sensitive dependence on initial conditions (SDIC) does

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A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore's Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa

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The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions, and the ability of a tiny perturbation to create an organized circulation at large distances, are not exactly the same. In Palmer et al., a

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The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive

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At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about

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The first kind of butterfly effect (BE1), known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its

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the tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing represents a small change in the initial condition of the system, which

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According to Lighthill (1986), the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review, it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement:

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published in 1993, Lorenz defined butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." This feature is the same as sensitive dependence of

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The third kind of butterfly effect with finite predictability, as discussed in, was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in.

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or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon

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In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather

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system) contains more than one bounded attractor that depends only on initial conditions. The multistability was illustrated using kayaking in Figure on the right side (i.e., Figure 1 of ) where the appearance of strong currents and a stagnant area suggests instability and local stability,

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A comparison of the two kinds of butterfly effects and the third kind of butterfly effect has been documented. In recent studies, it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is

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solutions on initial conditions (SDIC) in . In the same book, Lorenz applied the activity of skiing and developed an idealized skiing model for revealing the sensitivity of time-varying paths to initial positions. A predictability horizon is determined before the onset of SDIC.

1315:. Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by 226:

More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published:

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By taking into consideration time-varying multistability that is associated with the modulation of large-scale processes (e.g., seasonal forcing) and aggregated feedback of small-scale processes (e.g., convection), the above revised view is refined as follows:

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displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz defined sensitive dependence as follows:

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These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, and the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ by only 10 in the

65: 69: 68: 64: 63: 1261:(B). In the 1960s, the two-week predictability limit was originally estimated based on a doubling time of five days in real-world models. Since then, this finding has been documented in Charney et al. (1966) and has become a consensus. 116:

flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 1972. He discovered the effect when he observed runs of his

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or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can

1258:(A). The Lorenz 1963 model qualitatively revealed the essence of a finite predictability within a chaotic system such as the atmosphere. However, it did not determine a precise limit for the predictability of the atmosphere. 478:, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making 731: 1282:

motion provides an analogy. For large angles of swing the motion of the pendulum is often chaotic. By comparison, for small angles of swing, motions are non-chaotic. Multistability is defined when a system (e.g., the

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under Charney's leadership. The hypothesis supports the investigation into extended-range predictions using both partial differential equation (PDE)-based physics methods and Artificial Intelligence (AI) techniques.

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as a title. Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.

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For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article and by system ill-conditioning in another more recent study.

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The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of

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The concept of the butterfly effect has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences.

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not always appear. Namely, SDIC appears when two orbits (i.e., solutions) become the chaotic attractor; it does not appear when two orbits move toward the same point attractor. The above animation for

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Other authors suggest that the butterfly effect can be observed in quantum systems. Zbyszek P. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different

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noted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping."

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Poulin, David; Blume-Kohout, Robin; Laflamme, Raymond & Ollivier, Harold (2004). "Exponential Speedup with a Single Bit of Quantum Information: Measuring the Average Fidelity Decay".

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are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations. Recent studies using generalized

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and John B. Delos and co-workers. The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.

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of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.

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would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.

443: > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at 303: 256:

an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak

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wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.

78:. In each recording, the pendulum starts with almost the same initial condition. Over time, the differences in the dynamics grow from almost unnoticeable to drastic. 1860: 1190: 1162: 1088: 562: 535: 4658: 66: 4488: 3375:

Shen, Bo-Wen; Pielke Sr., Roger Pielke; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Kesarkar, Amit; Zeng, Xiping; Atlas, Robert (2022-11-12).

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The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances.

231:"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world." 180:

says "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".

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Karkuszewski, Zbyszek P.; Jarzynski, Christopher; Zurek, Wojciech H. (2002). "Quantum Chaotic Environments, the Butterfly Effect, and Decoherence".

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argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.

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The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the

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that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos.

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onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing

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and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the

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Hilborn, Robert C. (2004). "Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect in nonlinear dynamics".

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The idea that small causes may have large effects in weather was earlier acknowledged by the French mathematician and physicist

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The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in

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Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.

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Medium-range weather prediction: The European approach; The story of the European Centre for Medium-Range Weather Forecasts

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Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Baik, Jong-Jin; Faghih-Naini, Sara; Cui, Jialin; Atlas, Robert (2021-01-01).

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for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a

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Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Atlas, Robert (2022-07-04).

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Shen, Bo-Wen; Pielke, R. A. Sr.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R.; Reyes, T. A. L. (2021).

3079:"Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz's Predictability Studies in the 1960s" 4596: 4663: 5405: 5076: 4121: 1472: 3686:

Yan, Bin; Sinitsyn, Nikolai A. (2020). "Recovery of Damaged Information and the Out-of-Time-Ordered Correlators".

3630: 302:. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the 5451: 5290: 4940: 4711: 1757: 1437: 3285:, Springer Series in Synergetics, vol. 21, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 86–97, 4985: 4415: 4360: 398: 2796: 2747: 112:(the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant 1957: 1709:

The Restless Universe Applications of Gravitational N-Body Dynamics to Planetary Stellar and Galactic Systems

1620: 1595: 5375: 5239: 4576: 4341: 4248: 3749: 2456: 1502: 748:. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems. 582: 2482: 5400: 5066: 5000: 4616: 4228: 2547: 177: 17: 190:

by Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology.

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wings on tornado formation, as indicated in two 2024 articles. In more recent discussions published by

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with initial condition data that were rounded in a seemingly inconsequential manner. He noted that the

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Lorenz, Edward N. (June 1969). "The predictability of a flow which possesses many scales of motion".

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In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called

105: 95: 435:. Initially, the two trajectories seem coincident, as indicated by the small difference between the 5295: 5260: 4736: 4693: 4678: 4523: 4476: 4461: 4446: 4346: 4278: 4253: 4238: 4223: 1462: 740:. The definition does not require that all points from a neighborhood separate from the base point 315: 172: 141: 1265:

Recently, a short video has been created to present Lorenz's perspective on predictability limit.

108:. He noted that the butterfly effect is derived from the metaphorical example of the details of a 5426: 5380: 5360: 5143: 4914: 4781: 4611: 4498: 4493: 4385: 4263: 4165: 3853: 3819: 3792: 3758: 3729: 3695: 3668: 3574: 3210: 3191:. Springer Proceedings in Complexity. Cham: Springer International Publishing. pp. 805–825. 3184: 3165: 2906: 2518: 2436: 2291: 2257: 2183: 1843: 1817: 1492: 1457: 1397: 1350: 1327: 216: 187: 145: 5355: 3278: 1988: 1876: 298:

Following proposals from colleagues, in later speeches and papers, Lorenz used the more poetic

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characterized by brief but significant exponential growth resulting from a small disturbance.

1061: 937: 745: 129: 91: 50: 42: 1120: 1100: 1043: 943: 5395: 5305: 5192: 5157: 5045: 5037: 4806: 4756: 4653: 4581: 4533: 4410: 4395: 4390: 4185: 4023: 3939: 3837: 3776: 3713: 3652: 3611: 3566: 3474: 3396: 3340: 3286: 3249: 3192: 3147: 3098: 3032: 2983: 2937: 2890: 2851: 2816: 2767: 2715: 2676: 2632: 2593: 2556: 2502: 2379: 2221: 2173: 2074: 2033: 1980: 1916: 1835: 1654: 1615: 1532:"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" 1452: 1447: 1427: 1382: 490: 453: 194: 98: 46: 1739: 1168: 1140: 1066: 540: 513: 5250: 5187: 4826: 4721: 4648: 4481: 4293: 4283: 2963: 2958: 1694: 1487: 1387: 1362: 1284: 1279: 352: 75: 5330: 5310: 5167: 4816: 4761: 4019: 3935: 3833: 3772: 3709: 3648: 3607: 3562: 3470: 3392: 3336: 3245: 3185:"Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models" 3143: 3094: 3028: 2962: This article incorporates text from this source, which is available under the 2886: 2812: 2763: 2711: 2628: 2498: 2375: 2360:"One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models" 2217: 2169: 2070: 2029: 1831: 1650: 1611: 314:

The phrase refers to the idea that a butterfly's wings might create tiny changes in the

5340: 5325: 5255: 5177: 5162: 5129: 4909: 4876: 4871: 4866: 4668: 4558: 4553: 4451: 4400: 4316: 4190: 4047:. A short documentary that explains the "butterfly effect" in context of Lorenz's work. 3592:

Gutzwiller, Martin C. (1971). "Periodic Orbits and Classical Quantization Conditions".

3128:"Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model" 2410:

Nonlinear ordinary differential equations: an introduction for scientists and engineers

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The term is closely associated with the work of the mathematician and meteorologist

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The butterfly effect has appeared across mediums such as literature (for instance,

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Experimental demonstration of the butterfly effect with six recordings of the same

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Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences

2586:"Revisiting Lorenz's and Lilly's Empirical Formulas for Predictability Estimates" 1707: 5390: 5385: 5285: 5172: 5109: 5050: 4956: 4841: 4831: 4716: 4466: 4288: 4195: 3290: 2942: 2925: 2855: 2843: 2664: 2597: 2561: 2542: 2079: 2054: 1417: 280: 257: 205: 3421: 3037: 3012: 5280: 5081: 4899: 4796: 4243: 3435: 2283: 2249: 2178: 2153: 1659: 1095: 415: 333: 211:

The idea that the death of one butterfly could eventually have a far-reaching

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MIT Department of Earth, Atmospheric, and Planetary Sciences Youtube channel

1402: 1368: 299: 113: 54: 3953: 3849: 3788: 3725: 2894: 2665:"The Butterfly Effect: Can a butterfly in Brazil cause a tornado in Texas?" 2038: 2013: 5208: 3664: 3401: 3376: 3254: 3229: 3103: 3078: 3052:

Shen, Bo-Wen; Pielke, Sr., Roger; Zeng, Xubin; Zeng, Xiping (2023-09-13).

2637: 2612: 2408: 2384: 2359: 2055:"Role of the metric in forecast error growth: How chaotic is the weather?" 308:

Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?

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One meteorologist remarked that if the theory were correct, one flap of a

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noted general divergence of trajectories in spaces of negative curvature.

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Chaos theory § Lorenz's pioneering contributions to chaotic modeling

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on subsequent historical events made its earliest known appearance in "

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also contributed to this theory. Lorenz's work placed the concept of

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Shinbrot, Troy, Celso A Grebogi, Jack Wisdom, James A Yorke (1992).

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Orrell, David; Smith, Leonard; Barkmeijer, Jan; Palmer, Tim (2001).

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displays sensitive dependence to initial conditions if for any x in

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New England Complex Systems Institute - Concepts: Butterfly Effect

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Motter, Adilson E.; Campbell, David K. (2013). "Chaos at fifty".

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that, starting from any of various arbitrarily close alternative

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The Feasibility of a Global Observation and Analysis Experiment

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Chaos theory § A popular but inaccurate analogy for chaos

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Text was copied from this source, which is available under a

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Stochastic Phenomena and Chaotic Behaviour in Complex Systems

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13th Chaotic Modeling and Simulation International Conference

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discussed the possible general significance of this in 1908.

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10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2

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coordinate of the blue and yellow trajectories, but for

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cascades to large-scale alterations of events (compare:

2543:"Three Kinds of Butterfly Effects within Lorenz Models" 2483:"Nonlinear Feedback in a Five-Dimensional Lorenz Model" 1311:, including atoms in strong fields and the anisotropic 1273:

Revised perspectives on chaotic and non-chaotic systems

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Creative Commons Attribution 4.0 International License

3377:"The Dual Nature of Chaos and Order in the Atmosphere" 2842:

Shen, Bo-Wen; Pielke, Roger; Xubin Zeng (2024-09-02).

974: 3918:"Quantum amplifier: Measurement with entangled spins" 2250:"The Butterfly Effect – What Does It Really Signify?" 1171: 1143: 1123: 1103: 1069: 1046: 966: 946: 874: 764: 635: 585: 543: 516: 384: 1753:"The Physics of Ray Bradbury's "A Sound of Thunder"" 1683: 1681: 5138: 5095: 5059: 5036: 4999: 4963: 4885: 4702: 4629: 4567: 4437: 4424: 4376: 4307: 4151: 4144: 4084:. Advanced graduate textbook on chaos (no fractals) 3279:"Chaos in Classical Mechanics: The Double Pendulum" 304:

American Association for the Advancement of Science

57:

will become arbitrarily spread out from each other.

2120: 1706:Steves, Bonnie; Maciejewski, AJ (September 2001). 1372:), AI-driven expansive language models, and more. 1184: 1156: 1129: 1109: 1082: 1052: 1032: 952: 925: 850: 725: 618: 556: 529: 101:can result in large differences in a later state. 2098:. New Haven: Yale University Press. p. 208. 926:{\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )} 2611:Saiki, Yoshitaka; Yorke, James A. (2023-05-02). 2096:Truth or Beauty: Science and the Quest for Order 1909:Transactions of the New York Academy of Sciences 332:). Had the butterfly not flapped its wings, the 4045:Weather and Chaos: The Work of Edward N. Lorenz 3132:Bulletin of the American Meteorological Society 288: 270:, U. Washington Press, Seattle (1993), page 134 253: 229: 4053:. An introductory primer on chaos and fractals 2696:"The real butterfly effect and maggoty apples" 2254:Oxford U. Dept. of Mathematics Youtube Channel 2158:International Journal of Bifurcation and Chaos 1688:Some Historical Notes: History of Chaos Theory 5224: 4941: 4122: 1950:"The Butterfly Effects: Variations on a Meme" 1895: 1893: 1564:"When Lorenz Discovered the Butterfly Effect" 388:The butterfly effect in the Lorenz attractor 223:. "A Sound of Thunder" features time travel. 132:. The American mathematician and philosopher 27:Idea that small causes can have large effects 8: 3916:Lee, Jae-Seung & Khitrin, A. K. (2004). 3529:Notices of the American Mathematical Society 3357:: CS1 maint: multiple names: authors list ( 2441:: CS1 maint: multiple names: authors list ( 1875:. Cs.ualberta.ca. 1960-11-22. Archived from 3054:"Lorenz's View on the Predictability Limit" 2844:"Summary of Two Kinds of Butterfly Effects" 568:and any δ > 0, there are y in 5231: 5217: 5209: 4948: 4934: 4926: 4434: 4148: 4129: 4115: 4107: 1215:Differentiating types of butterfly effects 94:in which a small change in one state of a 18:Sensitive dependence on initial conditions 3987:Introduction to Chaotic Dynamical Systems 3943: 3869: 3867: 3823: 3762: 3699: 3400: 3253: 3151: 3102: 3036: 2941: 2771: 2719: 2636: 2560: 2383: 2177: 2078: 2037: 1902:"The Predictability of Hydrodynamic Flow" 1821: 1658: 1619: 1176: 1170: 1148: 1142: 1122: 1102: 1074: 1068: 1045: 1017: 1013: 1008: 989: 973: 965: 945: 908: 892: 879: 873: 833: 810: 791: 769: 763: 704: 695: 690: 668: 646: 634: 584: 548: 542: 521: 515: 3494:Chaos in Classical and Quantum Mechanics 1249:Finite predictability in chaotic systems 318:that may ultimately alter the path of a 1514: 861:which, unlike most chaotic maps, has a 3350: 3277:Richter, P. H.; Scholz, H.-J. (1984), 2741: 2739: 2434: 619:{\displaystyle 0<d(x,y)<\delta } 279:(the calculations were performed on a 3370: 3368: 2658: 2656: 2536: 2534: 2532: 2353: 2351: 2349: 2347: 2311: 2309: 2277: 2275: 2243: 2241: 2239: 2237: 2235: 2199: 2197: 7: 3902:Quantum Theory: Concepts and Methods 2014:"Model error in weather forecasting" 1930:from the original on 10 October 2014 1740:Computing Machinery and Intelligence 1589: 1587: 1585: 1583: 1581: 1526: 1524: 1522: 1520: 1518: 1117:are irrational, and, for irrational 4269:Measure-preserving dynamical system 3629:Gao, J. & Delos, J. B. (1992). 2487:Journal of the Atmospheric Sciences 1600:Journal of the Atmospheric Sciences 1344:Butterfly effect in popular culture 1232:Recent debates on butterfly effects 306:in 1972, Philip Merilees concocted 4981:Novikov self-consistency principle 2226:10.1111/j.2153-3490.1969.tb00444.x 1921:10.1111/j.2164-0947.1963.tb01464.x 691: 465:Theory and mathematical definition 25: 4837:Oleksandr Mykolayovych Sharkovsky 3978:, New York: Viking, 1987. 368 pp. 3228:Anthes, Richard A. (2022-08-14). 2018:Nonlinear Processes in Geophysics 1354:), films and television (such as 32:Butterfly effect (disambiguation) 4991:Quantum mechanics of time travel 4971:Chronology protection conjecture 4367: 4359: 3876:"A Rough Guide to Quantum Chaos" 3538:from the original on 2009-10-02. 3415: 3230:"Predictability and Predictions" 2957: 2459:. RealClimate. 4 November 2005. 2282:Emanuel, Kerry (26 March 2018). 1596:"Deterministic Nonperiodic Flow" 1594:Lorenz, Edward N. (March 1963). 456:shows the continuous evolution. 421: 414: 286:computer). Elsewhere he stated: 3595:Journal of Mathematical Physics 3520:Rudnick, Ze'ev (January 2008). 2869:Lighthill, James (1986-09-08). 2463:from the original on 2014-07-02 2294:from the original on 2021-10-31 2260:from the original on 2021-10-31 1765:from the original on 2015-09-24 1667:from the original on 2016-01-02 1544:from the original on 2022-10-09 1192:folded within the range . 822: 744:, but it requires one positive 234:-- The Princess of Pure Reason 90:is the sensitive dependence on 5442:Metaphors referring to insects 4602:Rabinovich–Fabrikant equations 3718:10.1103/PhysRevLett.125.040605 3492:Gutzwiller, Martin C. (1990). 1987:. New York: Springer. p. 1861:Google Scholar citation record 1027: 1001: 920: 901: 816: 797: 720: 708: 683: 680: 674: 658: 652: 639: 607: 595: 277:Deterministic Nonperiodic Flow 1: 5366:Rebound effect (conservation) 5087:Parallel universes in fiction 4057:Dizikes, Peter (2008-06-08). 3905:. Dordrecht: Kluwer Academic. 3842:10.1103/PhysRevLett.92.177906 3781:10.1103/PhysRevLett.89.170405 3496:. New York: Springer-Verlag. 2681:10.1080/00431672.2024.2329521 1645:. Vol. 4. p. 1720. 5346:Parable of the broken window 5097:Philosophy of space and time 3434:Shen, Bo-Wen (21 Feb 2023). 3321:"Chaos in a double pendulum" 3197:10.1007/978-3-030-70795-8_57 2407:W., Jordan, Dominic (2011). 1478:Representativeness heuristic 736:for some positive parameter 4337:Poincaré recurrence theorem 4007:American Journal of Physics 3975:Chaos: Making a New Science 3923:Journal of Chemical Physics 3522:"What is... Quantum Chaos?" 3325:American Journal of Physics 3291:10.1007/978-3-642-69591-9_9 2943:10.3390/encyclopedia3030063 2856:10.13140/RG.2.2.32401.24163 2598:10.13140/RG.2.2.32941.15849 2562:10.3390/encyclopedia2030084 2481:Shen, Bo-Wen (2014-05-01). 2248:Tim, Palmer (19 May 2017). 2080:10.3402/tellusa.v54i4.12159 1785:Chaos: Making a New Science 1060:, after a finite number of 5468: 5406:Tyranny of small decisions 5077:Many-worlds interpretation 4964:General terms and concepts 4332:Poincaré–Bendixson theorem 3571:10.1088/0031-8949/40/3/013 3038:10.1175/1520-0477-50.3.136 2795:Palmer, Tim (2024-09-01). 2694:Palmer, Tim (2024-05-01). 2316:Lorenz, Edward N. (1993). 1900:Lorenz, Edward N. (1963). 1473:Potentiality and actuality 1341: 468: 163: 29: 5291:Excess burden of taxation 5246: 4684:Swinging Atwood's machine 4357: 4327:Krylov–Bogolyubov theorem 4204: 2179:10.1142/S0218127419500378 2127:. Wolfram Media. p. 2119:Wolfram, Stephen (2002). 1758:The Philadelphia Inquirer 1751:Flam, Faye (2012-06-15). 1660:10.4249/scholarpedia.1720 1438:Great Stirrup Controversy 451: 429: 413: 392: 387: 219:", a 1952 short story by 4986:Self-fulfilling prophecy 4592:Lotka–Volterra equations 4416:Synchronization of chaos 4219:axiom A dynamical system 3657:10.1103/PhysRevA.46.1455 3153:10.1175/BAMS-D-19-0165.1 1360:), video games (such as 5376:Self-defeating prophecy 5240:Unintended consequences 4577:Double scroll attractor 4342:Stable manifold theorem 4249:False nearest neighbors 4065:. Boston, Massachusetts 4051:The Chaos Hypertextbook 3812:Physical Review Letters 3750:Physical Review Letters 3688:Physical Review Letters 3017:Bull. Amer. Meteor. Soc 2507:10.1175/JAS-D-13-0223.1 1503:Unintended consequences 1130:{\displaystyle \theta } 1110:{\displaystyle \theta } 1053:{\displaystyle \theta } 953:{\displaystyle \theta } 5401:Tragedy of the commons 5150:closed timelike curves 5001:Time travel in fiction 4617:Van der Pol oscillator 4597:Mackey–Glass equations 4229:Box-counting dimension 2895:10.1098/rspa.1986.0082 2413:. Oxford Univ. Press. 2164:(3): 1950037–1950091. 2094:Orrell, David (2012). 2053:Orrell, David (2002). 2039:10.5194/npg-8-357-2001 1979:Woods, Austin (2005). 1954:AP42 ...and everything 1787:. Viking. p. 16. 1783:Gleick, James (1987). 1366:), webcomics (such as 1186: 1158: 1131: 1111: 1084: 1054: 1034: 954: 927: 852: 727: 620: 558: 531: 296: 273: 245: 178:Johann Gottlieb Fichte 79: 58: 53:on the attractor, the 5351:Paradox of enrichment 5198:Traversable wormholes 4976:Closed timelike curve 4767:Svetlana Jitomirskaya 4674:Multiscroll attractor 4519:Interval exchange map 4472:Dyadic transformation 4457:Complex quadratic map 4299:Topological conjugacy 4234:Correlation dimension 4209:Anosov diffeomorphism 3402:10.3390/atmos13111892 3255:10.3390/atmos13081292 3104:10.3390/atmos15070837 2638:10.3390/atmos14050821 2385:10.3390/atmos13050753 2322:. London: UCL Press. 2152:Shen, Bo-Wen (2019). 2123:A New Kind of Science 1498:Tropical cyclogenesis 1187: 1185:{\displaystyle x_{n}} 1159: 1157:{\displaystyle x_{n}} 1132: 1112: 1085: 1083:{\displaystyle x_{n}} 1055: 1035: 955: 928: 853: 728: 621: 559: 557:{\displaystyle f^{t}} 532: 530:{\displaystyle f^{t}} 370:In the book entitled 241:The Phantom Tollbooth 73: 40: 5321:Inverse consequences 5009:Timelines in fiction 4777:Edward Norton Lorenz 2821:10.1063/pt.oktn.zdwa 2773:10.1063/pt.ifge.djjy 2721:10.1063/pt.eike.hsbz 2319:The essence of chaos 1443:Innovation butterfly 1299:In quantum mechanics 1169: 1165:sine function keeps 1141: 1121: 1101: 1067: 1044: 964: 944: 872: 863:closed-form solution 762: 633: 583: 541: 514: 452:An animation of the 372:The Essence of Chaos 355:also notes that the 342:ensemble forecasting 268:The Essence of Chaos 106:Edward Norton Lorenz 30:For other uses, see 5296:Four Pests campaign 5046:Grandfather paradox 4737:Mitchell Feigenbaum 4679:Population dynamics 4664:Hénon–Heiles system 4524:Irrational rotation 4477:Dynamical billiards 4462:Coupled map lattice 4322:Liouville's theorem 4254:Hausdorff dimension 4239:Conservative system 4224:Bifurcation diagram 4020:2004AmJPh..72..425H 3936:2004JChPh.121.3949L 3834:2004PhRvL..92q7906P 3773:2002PhRvL..89q0405K 3710:2020PhRvL.125d0605Y 3649:1992PhRvA..46.1455G 3608:1971JMP....12..343G 3563:1989PhyS...40..335B 3471:1993PhT....46g..38H 3393:2022Atmos..13.1892S 3337:1992AmJPh..60..491S 3246:2022Atmos..13.1292A 3144:2021BAMS..102E.148S 3095:2024Atmos..15..837S 3029:1969BAMS...50..136. 3011:GARP (1969-03-01). 2887:1986RSPSA.407...35L 2813:2024PhT....77R..10P 2797:"Butterfly effects" 2764:2024PhT....77Q..10P 2748:"Butterfly effects" 2712:2024PhT....77e..30P 2629:2023Atmos..14..821S 2499:2014JAtS...71.1701S 2457:"Chaos and Climate" 2376:2022Atmos..13..753S 2218:1969Tell...21..289L 2170:2019IJBC...2950037S 2071:2002TellA..54..350O 2030:2001NPGeo...8..357O 1960:on 11 November 2011 1832:2013PhT....66e..27M 1651:2009SchpJ...4.1720R 1612:1963JAtS...20..130L 1463:Point of divergence 1196:In physical systems 1026: 393:time 0 ≤ 173:The Vocation of Man 150:deterministic chaos 5447:Physical phenomena 5381:Self-refuting idea 5361:Perverse incentive 5144:general relativity 5067:Alternative future 5060:Parallel timelines 5038:Temporal paradoxes 5014:in science fiction 4915:Santa Fe Institute 4782:Aleksandr Lyapunov 4612:Three-body problem 4499:Gingerbreadman map 4386:Bifurcation theory 4264:Lyapunov stability 4093:"Butterfly Effect" 4090:Weisstein, Eric W. 3989:. Westview Press. 3983:Devaney, Robert L. 3899:Peres, A. (1995). 1712:. USA: CRC Press. 1693:2006-07-19 at the 1637:"Butterfly effect" 1493:Traffic congestion 1458:Numerical analysis 1398:Catastrophe theory 1351:A Sound of Thunder 1338:In popular culture 1328:Hamiltonian system 1182: 1154: 1127: 1107: 1080: 1050: 1030: 1004: 983: 950: 923: 848: 723: 616: 554: 527: 217:A Sound of Thunder 188:three-body problem 146:nonlinear dynamics 142:Earth's atmosphere 92:initial conditions 80: 59: 51:initial conditions 41:A plot of Lorenz' 5414: 5413: 5371:Risk compensation 5206: 5205: 5158:Alcubierre metric 5072:Alternate history 4923: 4922: 4787:Benoît Mandelbrot 4752:Martin Gutzwiller 4742:Peter Grassberger 4625: 4624: 4607:Rössler attractor 4355: 4354: 4259:Invariant measure 4181:Lyapunov exponent 4028:10.1119/1.1636492 3945:10.1063/1.1788661 3636:Physical Review A 3616:10.1063/1.1665596 3300:978-3-642-69593-3 3206:978-3-030-70795-8 3013:"A Guide to GARP" 2997:978-0-309-35922-1 2420:978-0-19-920825-8 1840:10.1063/PT.3.1977 1468:Positive feedback 1393:Cascading failure 1317:Martin Gutzwiller 1092:periodic sequence 982: 938:initial condition 746:Lyapunov exponent 460: 459: 447: = 30. 71: 43:strange attractor 16:(Redirected from 5459: 5452:Stability theory 5396:Streisand effect 5306:Hawthorne effect 5266:Butterfly effect 5261:Braess's paradox 5233: 5226: 5219: 5210: 5193:van Stockum dust 5105:Butterfly effect 4950: 4943: 4936: 4927: 4895:Butterfly effect 4807:Itamar Procaccia 4757:Brosl Hasslacher 4654:Elastic pendulum 4582:Duffing equation 4529:Kaplan–Yorke map 4447:Arnold's cat map 4435: 4411:Stability theory 4396:Dynamical system 4391:Control of chaos 4371: 4363: 4347:Takens's theorem 4279:Poincaré section 4149: 4131: 4124: 4117: 4108: 4103: 4102: 4073: 4071: 4070: 4063:The Boston Globe 4031: 4000: 3958: 3957: 3947: 3913: 3907: 3906: 3896: 3890: 3889: 3887: 3881:. 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Archived from 1946: 1940: 1939: 1937: 1935: 1929: 1906: 1897: 1888: 1887: 1885: 1884: 1869: 1863: 1858: 1852: 1851: 1825: 1805: 1799: 1798: 1780: 1774: 1773: 1771: 1770: 1748: 1742: 1737: 1731: 1730: 1728: 1726: 1703: 1697: 1685: 1676: 1675: 1673: 1672: 1662: 1632: 1626: 1625: 1623: 1591: 1576: 1575: 1573: 1571: 1560: 1554: 1553: 1551: 1549: 1543: 1536: 1528: 1448:Kessler syndrome 1428:Dynamical system 1383:Avalanche effect 1191: 1189: 1188: 1183: 1181: 1180: 1163: 1161: 1160: 1155: 1153: 1152: 1136: 1134: 1133: 1128: 1116: 1114: 1113: 1108: 1089: 1087: 1086: 1081: 1079: 1078: 1059: 1057: 1056: 1051: 1039: 1037: 1036: 1031: 1025: 1021: 1012: 997: 996: 984: 975: 959: 957: 956: 951: 932: 930: 929: 924: 913: 912: 897: 896: 884: 883: 857: 855: 854: 849: 838: 837: 815: 814: 796: 795: 780: 779: 732: 730: 729: 724: 703: 702: 694: 673: 672: 651: 650: 625: 623: 622: 617: 578: 572:, with distance 563: 561: 560: 555: 553: 552: 536: 534: 533: 528: 526: 525: 491:dynamical system 454:Lorenz attractor 425: 418: 397: ≤ 30 385: 357:Lorenz equations 271: 243: 195:Jacques Hadamard 99:nonlinear system 88:butterfly effect 72: 47:dynamical system 21: 5467: 5466: 5462: 5461: 5460: 5458: 5457: 5456: 5417: 5416: 5415: 5410: 5356:Parkinson's law 5251:Abilene paradox 5242: 5237: 5207: 5202: 5188:Tipler cylinder 5147: 5134: 5091: 5055: 5032: 4995: 4959: 4954: 4924: 4919: 4887: 4881: 4827:Caroline Series 4722:Mary Cartwright 4704: 4698: 4649:Double pendulum 4631: 4621: 4570: 4563: 4489:Exponential map 4440: 4426: 4420: 4378: 4372: 4365: 4351: 4317:Ergodic theorem 4310: 4303: 4294:Stable manifold 4284:Recurrence plot 4200: 4154: 4140: 4135: 4088: 4087: 4068: 4066: 4056: 4041: 4003: 3997: 3981: 3966: 3964:Further reading 3961: 3915: 3914: 3910: 3898: 3897: 3893: 3885: 3878: 3874:Poulin, David. 3873: 3872: 3865: 3809: 3808: 3804: 3746: 3745: 3741: 3685: 3684: 3680: 3628: 3627: 3623: 3591: 3590: 3586: 3551:Physica Scripta 3548: 3547: 3543: 3535: 3524: 3519: 3518: 3511: 3504: 3491: 3490: 3486: 3454: 3453: 3449: 3433: 3432: 3428: 3374: 3373: 3366: 3349: 3345:10.1119/1.16860 3318: 3317: 3313: 3305: 3303: 3301: 3276: 3275: 3271: 3227: 3226: 3222: 3207: 3182: 3181: 3177: 3125: 3124: 3120: 3076: 3075: 3071: 3062: 3060: 3051: 3050: 3046: 3010: 3009: 3005: 2998: 2976: 2975: 2971: 2923: 2922: 2918: 2881:(1832): 35–50. 2868: 2867: 2863: 2841: 2840: 2836: 2794: 2793: 2789: 2745: 2744: 2737: 2693: 2692: 2688: 2662: 2661: 2654: 2610: 2609: 2605: 2583: 2582: 2578: 2540: 2539: 2530: 2480: 2479: 2475: 2466: 2464: 2455: 2454: 2450: 2433: 2421: 2406: 2405: 2401: 2357: 2356: 2345: 2330: 2315: 2314: 2307: 2297: 2295: 2281: 2280: 2273: 2263: 2261: 2247: 2246: 2233: 2203: 2202: 2195: 2151: 2150: 2146: 2139: 2118: 2117: 2113: 2106: 2093: 2092: 2088: 2052: 2051: 2047: 2011: 2010: 2006: 1999: 1978: 1977: 1973: 1963: 1961: 1948: 1947: 1943: 1933: 1931: 1927: 1904: 1899: 1898: 1891: 1882: 1880: 1871: 1870: 1866: 1859: 1855: 1807: 1806: 1802: 1795: 1782: 1781: 1777: 1768: 1766: 1750: 1749: 1745: 1738: 1734: 1724: 1722: 1720: 1705: 1704: 1700: 1695:Wayback Machine 1686: 1679: 1670: 1668: 1634: 1633: 1629: 1593: 1592: 1579: 1569: 1567: 1562: 1561: 1557: 1547: 1545: 1541: 1534: 1530: 1529: 1516: 1512: 1507: 1488:Snowball effect 1388:Behavioral cusp 1378: 1363:Life Is Strange 1346: 1340: 1309:quantum physics 1301: 1285:double pendulum 1280:double pendulum 1275: 1251: 1234: 1217: 1208: 1203: 1198: 1172: 1167: 1166: 1144: 1139: 1138: 1119: 1118: 1099: 1098: 1070: 1065: 1064: 1042: 1041: 1040:. For rational 985: 962: 961: 942: 941: 904: 888: 875: 870: 869: 829: 806: 787: 765: 760: 759: 689: 664: 642: 631: 630: 581: 580: 573: 544: 539: 538: 517: 512: 511: 480:complex systems 473: 467: 381: 353:Stephen Wolfram 272: 265: 244: 239:Norton Juster, 238: 168: 162: 76:double pendulum 61: 55:iterated points 35: 28: 23: 22: 15: 12: 11: 5: 5465: 5463: 5455: 5454: 5449: 5444: 5439: 5434: 5429: 5419: 5418: 5412: 5411: 5409: 5408: 5403: 5398: 5393: 5388: 5383: 5378: 5373: 5368: 5363: 5358: 5353: 5348: 5343: 5341:Osborne effect 5338: 5333: 5328: 5326:Jevons paradox 5323: 5318: 5313: 5308: 5303: 5301:Goodhart's law 5298: 5293: 5288: 5283: 5278: 5273: 5271:Campbell's law 5268: 5263: 5258: 5256:Adverse effect 5253: 5247: 5244: 5243: 5238: 5236: 5235: 5228: 5221: 5213: 5204: 5203: 5201: 5200: 5195: 5190: 5185: 5180: 5178:Krasnikov tube 5175: 5170: 5165: 5163:BTZ black hole 5160: 5154: 5152: 5136: 5135: 5133: 5132: 5130:Predestination 5127: 5122: 5117: 5112: 5107: 5101: 5099: 5093: 5092: 5090: 5089: 5084: 5079: 5074: 5069: 5063: 5061: 5057: 5056: 5054: 5053: 5048: 5042: 5040: 5034: 5033: 5031: 5030: 5029: 5028: 5018: 5017: 5016: 5005: 5003: 4997: 4996: 4994: 4993: 4988: 4983: 4978: 4973: 4967: 4965: 4961: 4960: 4955: 4953: 4952: 4945: 4938: 4930: 4921: 4920: 4918: 4917: 4912: 4910:Predictability 4907: 4902: 4897: 4891: 4889: 4883: 4882: 4880: 4879: 4877:Lai-Sang Young 4874: 4872:James A. Yorke 4869: 4867:Amie Wilkinson 4864: 4859: 4854: 4849: 4844: 4839: 4834: 4829: 4824: 4819: 4814: 4809: 4804: 4802:Henri Poincaré 4799: 4794: 4789: 4784: 4779: 4774: 4769: 4764: 4759: 4754: 4749: 4744: 4739: 4734: 4729: 4724: 4719: 4714: 4708: 4706: 4700: 4699: 4697: 4696: 4691: 4686: 4681: 4676: 4671: 4669:Kicked rotator 4666: 4661: 4656: 4651: 4646: 4641: 4639:Chua's circuit 4635: 4633: 4627: 4626: 4623: 4622: 4620: 4619: 4614: 4609: 4604: 4599: 4594: 4589: 4584: 4579: 4573: 4571: 4568: 4565: 4564: 4562: 4561: 4559:Zaslavskii map 4556: 4554:Tinkerbell map 4551: 4546: 4541: 4536: 4531: 4526: 4521: 4516: 4511: 4506: 4501: 4496: 4491: 4486: 4485: 4484: 4474: 4469: 4464: 4459: 4454: 4449: 4443: 4441: 4438: 4432: 4422: 4421: 4419: 4418: 4413: 4408: 4403: 4401:Ergodic theory 4398: 4393: 4388: 4382: 4380: 4374: 4373: 4358: 4356: 4353: 4352: 4350: 4349: 4344: 4339: 4334: 4329: 4324: 4319: 4313: 4311: 4308: 4305: 4304: 4302: 4301: 4296: 4291: 4286: 4281: 4276: 4271: 4266: 4261: 4256: 4251: 4246: 4241: 4236: 4231: 4226: 4221: 4216: 4211: 4205: 4202: 4201: 4199: 4198: 4193: 4191:Periodic point 4188: 4183: 4178: 4173: 4168: 4163: 4157: 4155: 4152: 4146: 4142: 4141: 4136: 4134: 4133: 4126: 4119: 4111: 4105: 4104: 4085: 4079: 4074: 4054: 4048: 4040: 4039:External links 4037: 4036: 4035: 4032: 4014:(4): 425–427. 4001: 3995: 3979: 3965: 3962: 3960: 3959: 3930:(9): 3949–51. 3908: 3891: 3888:on 2010-11-04. 3863: 3818:(17): 177906. 3802: 3757:(17): 170405. 3739: 3678: 3621: 3584: 3557:(3): 335–336. 3541: 3509: 3502: 3484: 3447: 3426: 3364: 3331:(6): 491–499. 3311: 3299: 3269: 3220: 3205: 3175: 3118: 3069: 3044: 3023:(3): 136–141. 3003: 2996: 2988:10.17226/21272 2982:. 1966-01-01. 2969: 2936:(3): 887–899. 2916: 2861: 2834: 2787: 2735: 2686: 2652: 2603: 2576: 2528: 2473: 2448: 2419: 2399: 2343: 2328: 2305: 2271: 2231: 2212:(3): 289–297. 2193: 2144: 2138:978-1579550080 2137: 2111: 2105:978-0300186611 2104: 2086: 2065:(4): 350–362. 2045: 2024:(6): 357–371. 2004: 1998:978-0387269283 1997: 1971: 1941: 1915:(4): 409–432. 1889: 1864: 1853: 1800: 1793: 1775: 1743: 1732: 1718: 1698: 1677: 1627: 1606:(2): 130–141. 1577: 1555: 1513: 1511: 1508: 1506: 1505: 1500: 1495: 1490: 1485: 1480: 1475: 1470: 1465: 1460: 1455: 1450: 1445: 1440: 1435: 1430: 1425: 1420: 1415: 1410: 1408:Chain reaction 1405: 1400: 1395: 1390: 1385: 1379: 1377: 1374: 1342:Main article: 1339: 1336: 1313:Kepler problem 1300: 1297: 1274: 1271: 1263: 1262: 1259: 1250: 1247: 1233: 1230: 1216: 1213: 1207: 1204: 1202: 1199: 1197: 1194: 1179: 1175: 1151: 1147: 1126: 1106: 1077: 1073: 1049: 1029: 1024: 1020: 1016: 1011: 1007: 1003: 1000: 995: 992: 988: 981: 978: 972: 969: 949: 934: 933: 922: 919: 916: 911: 907: 903: 900: 895: 891: 887: 882: 878: 859: 858: 847: 844: 841: 836: 832: 828: 825: 821: 818: 813: 809: 805: 802: 799: 794: 790: 786: 783: 778: 775: 772: 768: 734: 733: 722: 719: 716: 713: 710: 707: 701: 698: 693: 688: 685: 682: 679: 676: 671: 667: 663: 660: 657: 654: 649: 645: 641: 638: 626:and such that 615: 612: 609: 606: 603: 600: 597: 594: 591: 588: 551: 547: 524: 520: 482:, such as the 466: 463: 462: 461: 458: 457: 449: 448: 427: 426: 419: 411: 410: 401: 390: 389: 380: 377: 266:E. N. Lorenz, 263: 251:Lorenz wrote: 236: 161: 158: 134:Norbert Wiener 130:Henri Poincaré 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 5464: 5453: 5450: 5448: 5445: 5443: 5440: 5438: 5435: 5433: 5430: 5428: 5425: 5424: 5422: 5407: 5404: 5402: 5399: 5397: 5394: 5392: 5389: 5387: 5384: 5382: 5379: 5377: 5374: 5372: 5369: 5367: 5364: 5362: 5359: 5357: 5354: 5352: 5349: 5347: 5344: 5342: 5339: 5337: 5334: 5332: 5329: 5327: 5324: 5322: 5319: 5317: 5314: 5312: 5309: 5307: 5304: 5302: 5299: 5297: 5294: 5292: 5289: 5287: 5284: 5282: 5279: 5277: 5274: 5272: 5269: 5267: 5264: 5262: 5259: 5257: 5254: 5252: 5249: 5248: 5245: 5241: 5234: 5229: 5227: 5222: 5220: 5215: 5214: 5211: 5199: 5196: 5194: 5191: 5189: 5186: 5184: 5181: 5179: 5176: 5174: 5171: 5169: 5166: 5164: 5161: 5159: 5156: 5155: 5153: 5151: 5145: 5141: 5137: 5131: 5128: 5126: 5123: 5121: 5118: 5116: 5113: 5111: 5108: 5106: 5103: 5102: 5100: 5098: 5094: 5088: 5085: 5083: 5080: 5078: 5075: 5073: 5070: 5068: 5065: 5064: 5062: 5058: 5052: 5049: 5047: 5044: 5043: 5041: 5039: 5035: 5027: 5024: 5023: 5022: 5019: 5015: 5012: 5011: 5010: 5007: 5006: 5004: 5002: 4998: 4992: 4989: 4987: 4984: 4982: 4979: 4977: 4974: 4972: 4969: 4968: 4966: 4962: 4958: 4951: 4946: 4944: 4939: 4937: 4932: 4931: 4928: 4916: 4913: 4911: 4908: 4906: 4905:Edge of chaos 4903: 4901: 4898: 4896: 4893: 4892: 4890: 4884: 4878: 4875: 4873: 4870: 4868: 4865: 4863: 4862:Marcelo Viana 4860: 4858: 4855: 4853: 4852:Audrey Terras 4850: 4848: 4847:Floris Takens 4845: 4843: 4840: 4838: 4835: 4833: 4830: 4828: 4825: 4823: 4820: 4818: 4815: 4813: 4810: 4808: 4805: 4803: 4800: 4798: 4795: 4793: 4790: 4788: 4785: 4783: 4780: 4778: 4775: 4773: 4770: 4768: 4765: 4763: 4760: 4758: 4755: 4753: 4750: 4748: 4747:Celso Grebogi 4745: 4743: 4740: 4738: 4735: 4733: 4730: 4728: 4727:Chen Guanrong 4725: 4723: 4720: 4718: 4715: 4713: 4712:Michael Berry 4710: 4709: 4707: 4701: 4695: 4692: 4690: 4687: 4685: 4682: 4680: 4677: 4675: 4672: 4670: 4667: 4665: 4662: 4660: 4657: 4655: 4652: 4650: 4647: 4645: 4642: 4640: 4637: 4636: 4634: 4628: 4618: 4615: 4613: 4610: 4608: 4605: 4603: 4600: 4598: 4595: 4593: 4590: 4588: 4587:Lorenz system 4585: 4583: 4580: 4578: 4575: 4574: 4572: 4566: 4560: 4557: 4555: 4552: 4550: 4547: 4545: 4542: 4540: 4537: 4535: 4534:Langton's ant 4532: 4530: 4527: 4525: 4522: 4520: 4517: 4515: 4512: 4510: 4509:Horseshoe map 4507: 4505: 4502: 4500: 4497: 4495: 4492: 4490: 4487: 4483: 4480: 4479: 4478: 4475: 4473: 4470: 4468: 4465: 4463: 4460: 4458: 4455: 4453: 4450: 4448: 4445: 4444: 4442: 4436: 4433: 4430: 4423: 4417: 4414: 4412: 4409: 4407: 4406:Quantum chaos 4404: 4402: 4399: 4397: 4394: 4392: 4389: 4387: 4384: 4383: 4381: 4375: 4370: 4366: 4362: 4348: 4345: 4343: 4340: 4338: 4335: 4333: 4330: 4328: 4325: 4323: 4320: 4318: 4315: 4314: 4312: 4306: 4300: 4297: 4295: 4292: 4290: 4287: 4285: 4282: 4280: 4277: 4275: 4272: 4270: 4267: 4265: 4262: 4260: 4257: 4255: 4252: 4250: 4247: 4245: 4242: 4240: 4237: 4235: 4232: 4230: 4227: 4225: 4222: 4220: 4217: 4215: 4214:Arnold tongue 4212: 4210: 4207: 4206: 4203: 4197: 4194: 4192: 4189: 4187: 4184: 4182: 4179: 4177: 4174: 4172: 4169: 4167: 4164: 4162: 4159: 4158: 4156: 4150: 4147: 4143: 4139: 4132: 4127: 4125: 4120: 4118: 4113: 4112: 4109: 4100: 4099: 4094: 4091: 4086: 4083: 4082:ChaosBook.org 4080: 4078: 4075: 4064: 4060: 4055: 4052: 4049: 4046: 4043: 4042: 4038: 4033: 4029: 4025: 4021: 4017: 4013: 4009: 4008: 4002: 3998: 3992: 3988: 3984: 3980: 3977: 3976: 3971: 3968: 3967: 3963: 3955: 3951: 3946: 3941: 3937: 3933: 3929: 3925: 3924: 3919: 3912: 3909: 3904: 3903: 3895: 3892: 3884: 3877: 3870: 3868: 3864: 3859: 3855: 3851: 3847: 3843: 3839: 3835: 3831: 3826: 3821: 3817: 3813: 3806: 3803: 3798: 3794: 3790: 3786: 3782: 3778: 3774: 3770: 3765: 3760: 3756: 3752: 3751: 3743: 3740: 3735: 3731: 3727: 3723: 3719: 3715: 3711: 3707: 3702: 3697: 3694:(4): 040605. 3693: 3689: 3682: 3679: 3674: 3670: 3666: 3662: 3658: 3654: 3650: 3646: 3642: 3638: 3637: 3632: 3625: 3622: 3617: 3613: 3609: 3605: 3601: 3597: 3596: 3588: 3585: 3580: 3576: 3572: 3568: 3564: 3560: 3556: 3552: 3545: 3542: 3534: 3530: 3523: 3516: 3514: 3510: 3505: 3503:0-387-97173-4 3499: 3495: 3488: 3485: 3480: 3476: 3472: 3468: 3464: 3460: 3459: 3458:Physics Today 3451: 3448: 3443: 3442: 3437: 3430: 3427: 3423: 3418: 3412: 3408: 3403: 3398: 3394: 3390: 3386: 3382: 3378: 3371: 3369: 3365: 3360: 3354: 3346: 3342: 3338: 3334: 3330: 3326: 3322: 3315: 3312: 3302: 3296: 3292: 3288: 3284: 3280: 3273: 3270: 3265: 3261: 3256: 3251: 3247: 3243: 3239: 3235: 3231: 3224: 3221: 3216: 3212: 3208: 3202: 3198: 3194: 3190: 3186: 3179: 3176: 3171: 3167: 3163: 3159: 3154: 3149: 3145: 3141: 3137: 3133: 3129: 3122: 3119: 3114: 3110: 3105: 3100: 3096: 3092: 3088: 3084: 3080: 3073: 3070: 3059: 3055: 3048: 3045: 3039: 3034: 3030: 3026: 3022: 3018: 3014: 3007: 3004: 2999: 2993: 2989: 2985: 2981: 2980: 2973: 2970: 2967: 2965: 2960: 2953: 2949: 2944: 2939: 2935: 2931: 2927: 2920: 2917: 2912: 2908: 2904: 2900: 2896: 2892: 2888: 2884: 2880: 2876: 2872: 2865: 2862: 2857: 2853: 2849: 2845: 2838: 2835: 2830: 2826: 2822: 2818: 2814: 2810: 2806: 2802: 2801:Physics Today 2798: 2791: 2788: 2783: 2779: 2774: 2769: 2765: 2761: 2757: 2753: 2752:Physics Today 2749: 2742: 2740: 2736: 2731: 2727: 2722: 2717: 2713: 2709: 2705: 2701: 2700:Physics Today 2697: 2690: 2687: 2682: 2678: 2674: 2670: 2666: 2659: 2657: 2653: 2648: 2644: 2639: 2634: 2630: 2626: 2622: 2618: 2614: 2607: 2604: 2599: 2595: 2591: 2587: 2580: 2577: 2572: 2568: 2563: 2558: 2554: 2550: 2549: 2544: 2537: 2535: 2533: 2529: 2524: 2520: 2516: 2512: 2508: 2504: 2500: 2496: 2492: 2488: 2484: 2477: 2474: 2462: 2458: 2452: 2449: 2444: 2438: 2430: 2426: 2422: 2416: 2412: 2411: 2403: 2400: 2395: 2391: 2386: 2381: 2377: 2373: 2369: 2365: 2361: 2354: 2352: 2350: 2348: 2344: 2339: 2335: 2331: 2329:0-203-21458-7 2325: 2321: 2320: 2312: 2310: 2306: 2293: 2289: 2285: 2278: 2276: 2272: 2259: 2255: 2251: 2244: 2242: 2240: 2238: 2236: 2232: 2227: 2223: 2219: 2215: 2211: 2207: 2200: 2198: 2194: 2189: 2185: 2180: 2175: 2171: 2167: 2163: 2159: 2155: 2148: 2145: 2140: 2134: 2130: 2125: 2124: 2115: 2112: 2107: 2101: 2097: 2090: 2087: 2081: 2076: 2072: 2068: 2064: 2060: 2056: 2049: 2046: 2040: 2035: 2031: 2027: 2023: 2019: 2015: 2008: 2005: 2000: 1994: 1990: 1985: 1984: 1975: 1972: 1959: 1955: 1951: 1945: 1942: 1926: 1922: 1918: 1914: 1910: 1903: 1896: 1894: 1890: 1879:on 2009-07-17 1878: 1874: 1868: 1865: 1862: 1857: 1854: 1849: 1845: 1841: 1837: 1833: 1829: 1824: 1819: 1815: 1811: 1810:Physics Today 1804: 1801: 1796: 1794:0-8133-4085-3 1790: 1786: 1779: 1776: 1764: 1760: 1759: 1754: 1747: 1744: 1741: 1736: 1733: 1721: 1715: 1711: 1710: 1702: 1699: 1696: 1692: 1689: 1684: 1682: 1678: 1666: 1661: 1656: 1652: 1648: 1644: 1643: 1638: 1631: 1628: 1622: 1617: 1613: 1609: 1605: 1601: 1597: 1590: 1588: 1586: 1584: 1582: 1578: 1566:. 22 May 2015 1565: 1559: 1556: 1540: 1533: 1527: 1525: 1523: 1521: 1519: 1515: 1509: 1504: 1501: 1499: 1496: 1494: 1491: 1489: 1486: 1484: 1483:Ripple effect 1481: 1479: 1476: 1474: 1471: 1469: 1466: 1464: 1461: 1459: 1456: 1454: 1453:Norton's dome 1451: 1449: 1446: 1444: 1441: 1439: 1436: 1434: 1431: 1429: 1426: 1424: 1423:Domino effect 1421: 1419: 1416: 1414: 1411: 1409: 1406: 1404: 1401: 1399: 1396: 1394: 1391: 1389: 1386: 1384: 1381: 1380: 1375: 1373: 1371: 1370: 1365: 1364: 1359: 1358: 1353: 1352: 1345: 1337: 1335: 1333: 1332:quantum chaos 1329: 1325: 1320: 1318: 1314: 1310: 1306: 1305:semiclassical 1298: 1296: 1293: 1289: 1286: 1281: 1272: 1270: 1266: 1260: 1257: 1256: 1255: 1248: 1246: 1242: 1240: 1239:Physics Today 1231: 1229: 1225: 1221: 1214: 1212: 1205: 1200: 1195: 1193: 1177: 1173: 1149: 1145: 1124: 1104: 1097: 1093: 1075: 1071: 1063: 1047: 1022: 1018: 1014: 1009: 1005: 998: 993: 990: 986: 979: 976: 970: 967: 947: 939: 917: 914: 909: 905: 898: 893: 889: 885: 880: 876: 868: 867: 866: 864: 845: 842: 839: 834: 830: 826: 823: 819: 811: 807: 803: 800: 792: 788: 784: 781: 776: 773: 770: 766: 758: 757: 756: 754: 749: 747: 743: 739: 717: 714: 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4857:Mary Tsingou 4822:David Ruelle 4817:Otto Rössler 4762:Michel Hénon 4732:Leon O. 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