Chaos theory - Knowledge

3611:, to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome. Lorenz's discovery, which gave its name to 4346: 4249:

over an infinite time span. Considering the nature of Lorenz's chaotic solutions, the committee led by Charney et al. in 1966 extrapolated a doubling time of five days from a general circulation model, suggesting a predictability limit of two weeks. This connection between the five-day doubling time and the two-week predictability limit was also recorded in a 1969 report by the Global Atmospheric Research Program (GARP). To acknowledge the combined direct and indirect influences from the Mintz and Arakawa model and Lorenz's models, as well as the leadership of Charney et al., Shen et al. refer to the two-week predictability limit as the "Predictability Limit Hypothesis," drawing an analogy to Moore's Law.

565:, published in 1993, "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the 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." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions. A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories). 1649: 12022: 12305: 4383: 3428: 12994: 4240:. In fact, Orlando et al. by the means of the so-called recurrence quantification correlation index were able detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (regular) to turbulent (chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics. Finally, chaos theory could help in modeling how an economy operates as well as in embedding shocks due to external events such as COVID-19. 312: 3950:

review of Lorenz's model progression spanning from 1960 to 2008 revealed his adeptness at employing varied physical systems to illustrate chaotic phenomena. These systems encompassed Quasi-geostrophic systems, the Conservative Vorticity Equation, the Rayleigh-Bénard Convection Equations, and the Shallow Water Equations. Moreover, Lorenz can be credited with the early application of the logistic map to explore chaotic solutions, a milestone he achieved ahead of his colleagues (e.g. Lorenz 1964).

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graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model.

4397: 1102: 12297: 2074:, which have only one nonlinear term out of seven. Sprott found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are 1631: 38: 3981:

described true chaos but that it better illustrated the simpler phenomenon of instability and that the verse implicitly suggests that subsequent small events will not reverse the outcome. Based on the analysis, the verse only indicates divergence, not boundedness. Boundedness is important for the finite size of a butterfly pattern. In a recent study, the characteristic of the aforementioned verse was recently denoted as "finite-time sensitive dependence".

3544: 795: 4345: 4111:. The majority of these algorithms are based on uni-modal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys. From a wider perspective, without loss of generality, the similarities between the chaotic maps and the cryptographic systems is the main motivation for the design of chaos based cryptographic algorithms. One type of encryption, secret key or 1536: 4233:

natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.

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amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r.

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admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.

3043:. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behavior. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems. 1917: 569:

predictable only about a week ahead. This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach 100 °C (212 °F) or fall below −130 °C (−202 °F) on earth (during the current

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configurations but different initial conditions. The findings of attractor coexistence, obtained from classical and generalized Lorenz models, suggested a revised view that "the entirety of weather possesses a dual nature of chaos and order with distinct predictability", in contrast to the conventional view of "weather is chaotic".

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initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

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The red cars and blue cars take turns to move; the red ones only move upwards, and the blue ones move rightwards. Every time, all the cars of the same colour try to move one step if there is no car in front of it. Here, the model has self-organized in a somewhat geometric pattern where there are some

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is helped by constraint. There is always potential difficulty in distinguishing real chaos from chaos that is only in the model. Hence both constraint in the model and or duplicate time series data for comparison will be helpful in constraining the model to something close to the reality, for example

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In 1972, Lorenz coined the term "butterfly effect" as a metaphor to discuss whether a small perturbation could eventually create a tornado with a three-dimensional, organized, and coherent structure. While connected to the original butterfly effect based on sensitive dependence on initial conditions,

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small measuring device. Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An

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is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest

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of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This

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data), such as rainfall and streamflow. These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into

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benefitted greatly from chaos theory. Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled,

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of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda

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A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally

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with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot

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Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other

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Coexisting chaotic and non-chaotic attractors within the generalized Lorenz model. There are 128 orbits in different colors, beginning with different initial conditions for dimensionless time between 0.625 and 5 and a heating parameter r = 680. Chaotic orbits recurrently return close to the saddle

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In their 1995 paper, Metcalf and Allen maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual

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Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were

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Due to the sensitive dependence of solutions on initial conditions (SDIC), also known as the butterfly effect, chaotic systems like the Lorenz 1963 model imply a finite predictability horizon. This means that while accurate predictions are possible over a finite time period, they are not feasible

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The above set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model. Since 1963, higher-dimensional Lorenz models have been developed in numerous studies for examining the impact of an increased degree of nonlinearity, as well as its collective effect

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It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task. Economic and financial systems are fundamentally different from those in the classical

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Based on the above, many people mistakenly believe that the impact of a tiny initial perturbation monotonically increases with time and that any tiny perturbation can eventually produce a large impact on numerical integrations. However, in 2008, Lorenz stated that he did not feel that this verse

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weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a

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Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: how much

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Throughout his career, Professor Edward Lorenz authored a total of 61 research papers, out of which 58 were solely authored by him. Commencing with the 1960 conference in Japan, Lorenz embarked on a journey of developing diverse models aimed at uncovering the SDIC and chaotic features. A recent

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In contrast to single type chaotic solutions, recent studies using Lorenz models have emphasized the importance of considering various types of solutions. For example, coexisting chaotic and non-chaotic may appear within the same model (e.g., the double pendulum system) using the same modeling

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Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an

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In AI-driven large language models, responses can exhibit sensitivities to factors like alterations in formatting and variations in prompts. These sensitivities are akin to butterfly effects. Although classifying AI-powered large language models as classical deterministic chaotic systems poses

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A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the

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Traffic forecasting may benefit from applications of chaos theory. Better predictions of when a congestion will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term

3872:, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public. Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of 1736: 4325:

By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions. Modern organizations are increasingly seen as open

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Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been

558:. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different. 538:

means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.

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Li, Mengshan; Xingyuan Huanga; Hesheng Liua; Bingxiang Liub; Yan Wub; Aihua Xiongc; Tianwen Dong (25 October 2013). "Prediction of gas solubility in polymers by back propagation artificial neural network based on self-adaptive particle swarm optimization algorithm and chaos theory".

3638:, a set of points with infinite roughness and detail Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). In 1967, he published " 3954:

its metaphorical variant carries distinct nuances. To commemorate this milestone, a reprint book containing invited papers that deepen our understanding of both butterfly effects was officially published to celebrate the 50th anniversary of the metaphorical butterfly effect.

3859:. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws. 4310:'s theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member. 835:

occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation

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Chaos theory is a method of qualitative and quantitative analysis to investigate the behavior of dynamic systems that cannot be explained and predicted by single data relationships, but must be explained and predicted by whole, continuous data relationships.

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Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

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for instance, whereas the latter concerns minor differentiations in models, as opposed to the former focusing upon slight changes in states. Furthermore, time also holds different roles within the definitions of chaos as well as structural theory.

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was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.

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In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by

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Some say the chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.

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behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including

3053:, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another example of a jerk equation with nonlinearity in the magnitude of 2288: 2089:

can still exhibit some chaotic properties. Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional. A theory of linear chaos is being developed in a branch of mathematical analysis known as

294:. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years. In chaotic systems, the uncertainty in a forecast increases 3504:. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing. 709: 530:

respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial

4123:, when paired with chaos theory, offers a way to encrypt images and other information. Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient. 3079: 10010:

Dingqi, Li; Yuanping Chenga; Lei Wanga; Haifeng Wanga; Liang Wanga; Hongxing Zhou (May 2011). "Prediction method for risks of coal and gas outbursts based on spatial chaos theory using gas desorption index of drill cuttings".

13188: 3639: 1912:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma y-\sigma x,\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=\rho x-xz-y,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z.\end{aligned}}} 830:

passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that

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in open mirror traps. This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos.

3700:'s article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections. Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the 3032: 2917: 1635:

point at the origin. Nonchaotic orbits eventually approach one of two stable critical points, as shown with large blue dots. Chaotic and nonchaotic orbits occupy different regions of attraction within the phase space.

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one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

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Kyrtsou, C.; M. Terraza (2003). "Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series".

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Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a

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An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if

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The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, involving many different disciplines such as

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Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as

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Moreover, from the theoretical physics standpoint, dynamical chaos itself, in its most general manifestation, is a spontaneous order. The essence here is that most orders in nature arise from the

1741: 9379: 1519:(or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by 3855:). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of 4164:. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory. Another biological application is found in 82: 4306:

Researchers have continued to apply chaos theory to psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in

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is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.

1312: 1142:. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space. 1517: 1476: 1435: 3507:

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that

178:, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are 1614:, which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a 159:

can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in

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A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.

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One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the

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are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an

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challenges, chaos-inspired approaches and techniques (such as ensemble modeling) may be employed to extract reliable information from these expansive language models (see also "

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Adachihara, H; McLaughlin, D W; Moloney, J V; Newell, A C (1988). "Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis".

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Although chaos theory was born from observing weather patterns, it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are

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measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions. More specifically, given two starting

4282:, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets. Four of the five 2522: 3344: 10527: 3696:, and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and 1203: 752: 4204:

himself did some of these foundational crop co-evolution studies, and this in turn helped shape the entire field. Even for a steady environment, merely combining one

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uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the

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Salinas, Abel; Morstatter, Fred (2024-01-01). "The Butterfly Effect of Altering Prompts: How Small Changes and Jailbreaks Affect Large Language Model Performance".

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In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In the discrete-time case, this is true for all

8503: 4278:(PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck. In 2070:. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the 10200:

Thompson, Jamie; Johnstone, James; Banks, Curt (2018). "An examination of initiation rituals in a UK sporting institution and the impact on group development".

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attempted). Glass and Mandell and Selz have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior.

3360:, one basis for chaotic true random number generators. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system. 3071: 2715: 2417: 2004: 1980: 1960: 1940: 925: 732: 11360: 8063:"The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models" 4096: 828: 6710: 3223:{\displaystyle {\frac {\mathrm {d} ^{3}x}{\mathrm {d} t^{3}}}+A{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\frac {\mathrm {d} x}{\mathrm {d} t}}-|x|+1=0.} 2106:

The straightforward generalization of coupled discrete maps is based upon convolution integral which mediates interaction between spatially distributed maps:

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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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displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.

626: 416:, the second property implies the other two. An alternative and a generally weaker definition of chaos uses only the first two properties in the above list. 7707: 11274: 3469:". Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive 4259: 10260:

Wang, Jin; Qixin Shi (February 2013). "Short-term traffic speed forecasting hybrid model based on Chaos–Wavelet Analysis-Support Vector Machine theory".

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50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models

3642:", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an 551: 409:. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition. 3350:

0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative.

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eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored

11096: 4168:. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of 7697:

Coullet, Pierre, and Charles Tresser. "Iterations d'endomorphismes et groupe de renormalisation." Le Journal de Physique Colloques 39.C5 (1978): C5-25

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states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three

131:. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of 3465:

published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "

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Dal Forno, Arianna; Merlone, Ugo (2013). "Chaotic Dynamics in Organization Theory". In Bischi, Gian Italo; Chiarella, Carl; Shusko, Irina (eds.).

3890:(1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. 2947: 10046:

Glass, L (1997). "Dynamical disease: The impact of nonlinear dynamics and chaos on cardiology and medicine". In Grebogi, C; Yorke, J. A. (eds.).

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The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated

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of various symmetries. This large family of phenomena includes elasticity, superconductivity, ferromagnetism, and many others. According to the

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Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (March 2017). "Effect of size on the chaotic behavior of nano resonators".

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Redington, D. J.; Reidbord, S. P. (1992). "Chaotic dynamics in autonomic nervous system activity of a patient during a psychotherapy session".

9766:"Business cycle modeling between financial crises and black swans: Ornstein–Uhlenbeck stochastic process vs Kaldor deterministic chaotic model" 3453:", and is seen as being one of the earliest to discuss chaos theory, with work in the 1860s and 1870s. An early proponent of chaos theory was 9237:

Goswami, Ambarish; Thuilot, Benoit; Espiau, Bernard (1998). "A Study of the Passive Gait of a Compass-Like Biped Robot: Symmetry and Chaos".

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Behnia, S.; Akhshani, A.; Mahmodi, H.; Akhavan, A. (2008-01-01). "A novel algorithm for image encryption based on mixture of chaotic maps".

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Akhavan, A.; Samsudin, A.; Akhshani, A. (2011-10-01). "A symmetric image encryption scheme based on combination of nonlinear chaotic maps".

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Okulov, A Yu; Oraevskiĭ, A N (1988). "Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps". In N.G. Basov (ed.).

12883: 12594: 12424: 4540: 3400: 11978:– article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos. 8404:

Kyrtsou C., Labys W.; Labys (2007). "Detecting positive feedback in multivariate time series: the case of metal prices and US inflation".

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wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the

12039:, Bo-Wen Shen, Roger A. Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, and Robert Atlas, MDPI. Encyclopedia. 11478:

Cunningham, Lawrence A. (1994). "From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis".

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Liu, Yuansheng; Tang, Jie; Xie, Tao (2014-08-01). "Cryptanalyzing a RGB image encryption algorithm based on DNA encoding and chaos map".

758:

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example,

6926:

vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)

9543: 9428: 3403:, chaos, or more precisely, its stochastic generalization, is also part of this family. The corresponding symmetry being broken is the 374: 186:

elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as

7837:

Laurson, Lasse; Alava, Mikko J.; Zapperi, Stefano (15 September 2005). "Letter: Power spectra of self-organized critical sand piles".

2530: 9284: 3511:, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the 10154:

Metcalf, B. R.; Allen, J. D. (1995). "In search of chaos in schedule-induced polydipsia". In Abraham, F. D.; Gilgen, A. R. (eds.).

4065:. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing. 9419:

Perry, Joe; Smith, Robert; Woiwod, Ian; Morse, David (2000). Perry, Joe N; Smith, Robert H; Woiwod, Ian P; Morse, David R (eds.).

9075:

Xu, Ming (2017-06-01). "Cryptanalysis of an Image Encryption Algorithm Based on DNA Sequence Operation and Hyper-chaotic System".

7100: 4585: 3886: 3461:, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898, 11914: 10662: 9180: 4330:

with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance,

13248: 12304: 12064: 10386: 8862:

Dilão, R.; Domingos, T. (2001). "Periodic and Quasi-Periodic Behavior in Resource Dependent Age Structured Population Models".

6965:

Kolmogorov, A. N. (1991). "The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers".

5710: 4156:, but recently scientists have been able to implement chaotic models in certain populations. For example, a study on models of 9040:

Akhavan, A.; Samsudin, A.; Akhshani, A. (2017-10-01). "Cryptanalysis of an image encryption algorithm based on DNA encoding".

11676: 11451: 11430: 11406: 11372: 11344: 11325: 11306: 11264: 11245: 11213: 11192: 11171: 11147: 11118: 11086: 11065: 11044: 11023: 10999: 10978: 10959: 10933: 10901: 10861: 10839: 10816: 10795: 10776: 10244: 9837: 9559: 9444: 8846: 8211: 7459: 7088: 6073: 5528: 5433: 5408: 5383: 5136: 4937: 3601: 182:, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no 9659: 8507: 8439:

Kyrtsou, C.; Vorlow, C. (2005). "Complex dynamics in macroeconomics: A novel approach". In Diebolt, C.; Kyrtsou, C. (eds.).

7112:

Cartwright, Mary L.; Littlewood, John E. (1945). "On non-linear differential equations of the second order, I: The equation

12257: 11970: 8232:"The butterfly effect. In Premio Felice Pietro Chisesi E Caterina Tomassoni Award Lecture; University of Rome: Rome, Italy" 3962:

The sensitive dependence on initial conditions (i.e., butterfly effect) has been illustrated using the following folklore:

3798:, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display 3795: 3241:=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes: 8454:

Hernández-Acosta, M. A.; Trejo-Valdez, M.; Castro-Chacón, J. H.; Miguel, C. R. Torres-San; Martínez-Gutiérrez, H. (2018).

5888:

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

1149:(or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or 13169: 12204: 11975: 7063:

Kolmogorov, A. N. (1979). "Preservation of conditionally periodic movements with small change in the Hamilton function".

3735: 8837:

Progonati, E (2018). "Brexit in the Light of Chaos Theory and Some Assumptions About the Future of the European Union".

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

4237: 179: 10173:(June 2008). "Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers". 9874:"Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz's Predictability Studies in the 1960s" 7782:

Bak, Per; Tang, Chao; Wiesenfeld, Kurt (27 July 1987). "Self-organized criticality: An explanation of the 1/f noise".

6052:

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

202:

Chaos: When the present determines the future but the approximate present does not approximately determine the future.

12772: 11782: 11748: 11580: 10880: 10362: 8599: 8020: 7535: 6861: 5212: 5058: 4709: 12579: 12267: 6854:

The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory

6179:

Heidel, J.; Fu, Z. (1999). "Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case".

3720: 2296: 1723: 1705: 13207: 12876: 12454: 4160:

showed there was chaotic behavior in the population growth. Chaos can also be found in ecological systems, such as

3701: 3566: 3408: 3404: 1269: 11100: 6600:

K. E. Chlouverakis and J. C. Sprott, Chaos Solitons & Fractals 28, 739–746 (2005), Chaotic Hyperjerk Systems,

174:

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in

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Thompson, John; Burdon, Jeremy (1992). "Gene-for-gene coevolution between plants and parasites". Review Article.

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question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.

3685: 3615:, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions. 1481: 1440: 1399: 3673: 1701: 12272: 11946: 10296:"Dr. Gregory B. Pasternack – Watershed Hydrology, Geomorphology, and Ecohydraulics :: Chaos in Hydrology" 9585:"Applying the theory of chaos and a complex model of health to establish relations among financial indicators" 9311:

Lai, Dejian (1996). "Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic".

1377:

means that every point in the space is approached arbitrarily closely by periodic orbits. The one-dimensional

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Dal Forno, Arianna; Merlone, Ugo (2013). "Nonlinear dynamics in work groups with Bion's basic assumptions".

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Bozóki, Zsolt (February 1997). "Chaos theory and power spectrum analysis in computerized cardiotocography".

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Mandell, A. J.; Selz, K. A. (1997). "Is the EEG a strange attractor?". In Grebogi, C; Yorke, J. A. (eds.).

4501: 4471: 3681: 1035: 17: 11991:. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience. 9710: 9005:

Babaei, Majid (2013). "A novel text and image encryption method based on chaos theory and DNA computing".

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Wotherspoon, T.; Hubler, A. (2009). "Adaptation to the Edge of Chaos in the Self-Adjusting Logistic Map".

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Kyrtsou C.; Labys W. (2006). "Evidence for chaotic dependence between US inflation and commodity prices".

7904: 6663: 12869: 11874: 11791: 11515: 10319:

Pasternack, Gregory B. (1999-11-01). "Does the river run wild? Assessing chaos in hydrological systems".

8806:"Challenges of Instability: The Concepts of Synergetics in Studying the Historical Development of Russia" 7449: 7008:

Kolmogorov, A. N. (1941). "On degeneration of isotropic turbulence in an incompressible viscous liquid".

6821:"Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt" 4275: 3868: 3807:, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the 1681: 930: 11858: 7645:

Feigenbaum, Mitchell (July 1978). "Quantitative universality for a class of nonlinear transformations".

7424: 12532: 12057: 6945: 6890: 3931: 3784: 1983: 12599: 8338: 6736: 6401: 6271:

Bonet, J.; Martínez-Giménez, F.; Peris, A. (2001). "A Banach space which admits no chaotic operator".

4148:

For over a hundred years, biologists have been keeping track of populations of different species with

3592:

was an early pioneer of the theory. His interest in chaos came about accidentally through his work on

2663: 989: 12973: 12647: 10650: 10170: 6402:"Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations" 4654: 3808: 1570:, since then a large set of initial conditions leads to orbits that converge to this chaotic region. 10416: 9200: 7667: 7297: 6636: 6362: 5740: 4083:. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of 3254: 782:

sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for

89:

at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different

12351: 11233: 10295: 9340:

Sivakumar, B (31 January 2000). "Chaos theory in hydrology: important issues and interpretations".

9251: 8349: 6762:"Chaos theory and applications: a retrospective on lessons learned and missed or new opportunities" 5642: 4630: 3739: 3439: 3385: 1709: 1520: 12983: 12296: 5627: 1211: 12512: 12277: 12184: 10496: 9616: 9484: 8642:

Chaos and order in the capital markets : a new view of cycles, prices, and market volatility

8456:"Chaotic signatures of photoconductive Cu 2 ZnSnS 4 nanostructures explored by Lorenz attractors" 8238: 7711: 6060:. Springer Proceedings in Complexity. Cham: Springer International Publishing. pp. 805–825. 6053: 4714: 4327: 4116: 4084: 3779: 3693: 3497: 3466: 2485: 770: 155:, an underlying principle of chaos, describes how a small change in one state of a deterministic 12252: 12021: 7830: 6639:, A. S. Andreatos*, and A. P. Leros, Journal of Engineering Science and Technology Review, 2013. 4778: 3758:

in the 1980s through the application of chaos theory, for example, in the study of pathological

2718: 12552: 12164: 10411: 9421:

Chaos in Real Data : The Analysis of Non-Linear Dynamics from Short Ecological Time Series

9246: 9195: 7662: 7167: 6136: 5735: 5520: 4929: 4635: 4431: 4402: 4291: 4221: 3844: 3728: 3481: 3372:, four conditions suffice to produce synchronization in a chaotic system. Examples include the 3312: 2086: 2081:

While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean

1648: 1347: 413: 132: 9282:

Eduardo, Liz; Ruiz-Herrera, Alfonso (2012). "Chaos in discrete structured population models".

8822: 8805: 7527: 7213: 1176: 737: 13238: 12702: 12609: 12407: 12234: 12169: 12144: 12050: 10602: 7475:

Berger J.M.; Mandelbrot B. (1963). "A new model for error clustering in telephone circuits".

7411: 4961: 4824: 4525: 4088: 3724: 3485: 3347: 2283:{\displaystyle \psi _{n+1}({\vec {r}},t)=\int K({\vec {r}}-{\vec {r}}^{,},t)fd{\vec {r}}^{,}} 2009: 1727: 1351: 759: 481: 235: 11490: 11278: 8970:

Wang, Xingyuan; Zhao, Jianfeng (2012). "An improved key agreement protocol based on chaos".

5619: 5512: 5204:

Everyday Chaos – Technology, Complexity and How We're Thriving in a New World of Possibility

2049: 1243: 501: 12712: 12464: 12364: 12209: 11869: 11630: 11620: 11460: 11386: 11286: 10730: 10677: 10626: 10510: 10484: 10464: 10403: 10328: 10269: 10020: 9944: 9777: 9722: 9488: 9351: 9342: 9145: 9084: 9049: 8979: 8944: 8770: 8719: 8674: 8467: 8413: 8301: 8162: 8115: 8074: 7969: 7926: 7791: 7740: 7654: 7564: 7329: 7068: 7029: 6974: 6954: 6904:

Hadamard, Jacques (1898). "Les surfaces à courbures opposées et leurs lignes géodesiques".

6675: 6567: 6514: 6456: 6413: 6363:"The onset of chaos in nonlinear dynamical systems determined with a new fractal technique" 6319: 6307: 6188: 6145: 6134:

Fu, Z.; Heidel, J. (1997). "Non-chaotic behaviour in three-dimensional quadratic systems".

6108: 6010: 5961: 5901: 5839: 5787: 5727: 5582: 5261: 5099: 4486: 4169: 4133: 4104: 3843:(where SOC has been invoked, for example, as the dynamical mechanism behind the theory of " 3840: 3493: 3416: 2029: 1603: 762: 461: 277: 117: 12542: 11465:

Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness

6941:"Local structure of turbulence in an incompressible fluid for very large Reynolds numbers" 5012:

Werndl, Charlotte (2009). "What are the New Implications of Chaos for Unpredictability?".

4491: 3656: 2634: 573:), but we cannot predict exactly which day will have the hottest temperature of the year. 556:

Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?

97:. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions. 8: 13217: 13177: 12961: 12672: 12629: 12614: 12459: 12412: 12397: 12382: 12282: 12189: 12174: 12159: 11385:. World Scientific Series on Nonlinear Science Series A. Vol. 39. World Scientific. 10827: 10586: 10501: 5620: 5252:

Wisdom, Jack; Sussman, Gerald Jay (1992-07-03). "Chaotic Evolution of the Solar System".

4729: 4461: 4446: 4421: 4295: 4279: 4217: 4184: 4153: 4058: 3708: 3697: 3446: 3389: 3373: 2091: 2082: 2075: 1715:

are never chaotic; for a dynamical system to display chaotic behavior, it must be either

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Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político

11390: 11290: 10734: 10681: 10630: 10514: 10468: 10407: 10332: 10273: 10024: 9948: 9781: 9726: 9492: 9355: 9149: 9088: 9053: 8983: 8948: 8774: 8723: 8678: 8471: 8417: 8305: 8166: 8119: 8078: 7973: 7930: 7795: 7744: 7658: 7568: 7553:"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" 7333: 7072: 7033: 6978: 6958: 6679: 6571: 6518: 6460: 6417: 6323: 6192: 6149: 6112: 6054:"Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models" 6014: 5965: 5905: 5843: 5791: 5731: 5586: 5571:"One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models" 5265: 5103: 3519:" was considered by chaos theorists as a full component of the studied systems. In 1959 3368:

Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the

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Kolmogorov, A. N. (1991). "Dissipation of Energy in the Locally Isotropic Turbulence".

6990: 6801: 6583: 6557: 6530: 6343: 6288: 6235: 6204: 6161: 6079: 6034: 5927: 5890:"Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model" 5865: 5753: 5493: 5476:

Vellekoop, Michel; Berglund, Raoul (April 1994). "On Intervals, Transitivity = Chaos".

5303: 5039: 5021: 4922: 4689: 4620: 4605: 4555: 4530: 4426: 4388: 4197: 4193: 4176: 4074: 4002: 3927: 3716: 3623: 3528: 3524: 3516: 3458: 3377: 3056: 2700: 2362: 1989: 1965: 1945: 1925: 1315: 1146: 1139: 910: 717: 704:{\displaystyle |\delta \mathbf {Z} (t)|\approx e^{\lambda t}|\delta \mathbf {Z} _{0}|,} 434:

Lorenz equations used to generate plots for the y variable. The initial conditions for

295: 211: 175: 128: 13189:

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

10522: 10340: 9392: 9363: 8523:"Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns" 8037: 7960:

Hubler, A.; Foster, G.; Phelps, K. (2007). "Managing chaos: Thinking out of the box".

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How long is the coast of Britain? Statistical self-similarity and fractional dimension

801: 369:, says that to classify a dynamical system as chaotic, it must have these properties: 13182: 13082: 13077: 13018: 13008: 12946: 12722: 12687: 12677: 12574: 12194: 12116: 11927: 11778: 11744: 11713: 11672: 11644: 11576: 11447: 11426: 11402: 11368: 11340: 11321: 11302: 11260: 11241: 11227: 11209: 11188: 11167: 11161: 11143: 11128: 11114: 11082: 11061: 11040: 11019: 10995: 10974: 10955: 10943: 10929: 10897: 10876: 10857: 10849: 10835: 10812: 10791: 10772: 10693: 10642: 10598: 10594: 10555: 10240: 10221: 10128: 10124: 10093: 10085: 9895: 9833: 9805: 9793: 9738: 9655: 9643: 9555: 9512: 9440: 9396: 9326: 9213: 9112: 9100: 8879: 8842: 8786: 8735: 8645: 8620: 8595: 8576: 8485: 8319: 8290:"Can the Flap of a Butterfly's Wings Shift a Tornado into Texas—Without Chaos?" 8207: 8135: 8016: 7883: 7807: 7756: 7732: 7684: 7625: 7619: 7580: 7531: 7520: 7455: 7378: 7357: 7345: 7259: 7084: 7049: 6994: 6867: 6857: 6856:. Popp, Bruce D. (Translator). Cham, Switzerland: Springer International Publishing. 6805: 6793: 6691: 6587: 6534: 6487: 6429: 6382: 6347: 6335: 6292: 6253: 6208: 6200: 6165: 6157: 6099: 6083: 6069: 6026: 5979: 5931: 5919: 5869: 5857: 5803: 5600: 5524: 5513: 5429: 5404: 5379: 5295: 5287: 5208: 5174: 5132: 4933: 4892: 4674: 4436: 4416: 4382: 4356: 4335: 4200:

incorporating additional variables to reflect additional facets of real populations.

3938: 3852: 3743: 3668: 3647:

object whose irregularity is constant over different scales ("self-similarity") is a

3619: 3582: 3559: 3470: 3357: 2936: 1619: 1595: 907:. To expand the state-space of the logistic map into two dimensions, a second state, 577: 402: 366: 316: 262: 258: 148: 109: 105: 90: 12737: 10750: 10437: 9983: 9750: 9454: 9165: 9157: 9061: 9026: 8921: 8747: 7946: 7768: 7592: 7271: 7197: 7180: 6038: 5999:"Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model" 5761: 5307: 3454: 3427: 13033: 12830: 12742: 12692: 12589: 12517: 12469: 12346: 12331: 12326: 12121: 11997:, BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson ( 11853: 11770: 11736: 11664: 11568: 11394: 11294: 11106: 11009: 10738: 10685: 10634: 10582: 10543: 10518: 10472: 10421: 10336: 10277: 10209: 10182: 10120: 10028: 9979: 9952: 9885: 9825: 9785: 9730: 9689: 9635: 9596: 9569: 9547: 9524: 9496: 9475: 9432: 9388: 9359: 9322: 9293: 9268: 9256: 9205: 9153: 9092: 9057: 9014: 8987: 8952: 8917: 8871: 8839:

Chaos, complexity and leadership 2018 explorations of chaotic and complexity theory

8817: 8778: 8727: 8682: 8564: 8534: 8475: 8421: 8386: 8309: 8202: 8197: 8170: 8123: 8082: 8008: 8000: 7977: 7934: 7819: 7799: 7748: 7672: 7572: 7484: 7337: 7317: 7251: 7192: 7157: 7076: 7037: 6982: 6832: 6783: 6773: 6683: 6627:, J. C. Sprott, IEEE Transactions on Circuits and Systems—II: Express Briefs, 2010. 6624: 6612: 6601: 6575: 6522: 6464: 6421: 6374: 6327: 6280: 6245: 6196: 6153: 6116: 6061: 6018: 5969: 5909: 5847: 5795: 5745: 5590: 5485: 5462: 5458: 5277: 5269: 5175:"Covid-19 Pandemic and Chaos Theory: Applications based on a Bibliometric Analysis" 5129:

Complex nonlinearity: chaos, phase transitions, topology change, and path integrals

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Field of mathematics and science based on non-linear systems and initial conditions

10638: 10365:(1964). "Co-existence of cycles of a continuous mapping of the line into itself". 10213: 10140: 9765: 8891: 8012: 5799: 5043: 4749: 3704:

in chaos, permitting the application of chaos theory to many different phenomena.

3523:

proposed a criterion for the emergence of classical chaos in Hamiltonian systems (

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Banks; Brooks; Cairns; Davis; Stacey (1992). "On Devaney's definition of chaos".

5202: 4625: 4520: 4466: 4451: 4287: 3774: 1688:

increases, eventually producing chaos. Darker points are visited more frequently.

1587: 1374: 1338:. This implies that it is impossible to decompose the system into two open sets. 786:

initial conditions, the variable evolves chaotically with non-periodic behavior.

215: 140: 86: 12752: 12697: 11982: 11540:

Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature

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Hubler, A.; Phelps, K. (2007). "Guiding a self-adjusting system through chaos".

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Fractal market analysis : applying chaos theory to investment and economics

8425: 6687: 6579: 6425: 5273: 4887: 4870: 4699: 4684: 219: 13144: 13139: 13072: 12845: 12812: 12807: 12802: 12604: 12494: 12489: 12387: 12336: 12126: 12030: 11695: 11640: 11610: 10891: 10492: 10382: 10032: 9260: 8731: 8480: 8455: 8453: 8175: 8150: 7803: 7161: 6778: 6761: 6249: 5706: 4907: 4724: 4694: 4610: 4580: 4550: 4283: 4100: 4092: 3907: 3881: 3848: 3689: 3520: 3477: 3369: 2932: 391: 136: 120: 12034: 12004: 11966: 11668: 11110: 10547: 10281: 9694: 9677: 9639: 9551: 9436: 9181:"Quantitative description of robot–environment interaction using chaos theory" 9096: 9018: 8568: 8127: 6378: 5852: 5827: 3050: 2071: 13232: 13134: 13129: 12840: 12797: 12787: 12782: 12682: 12662: 12522: 12444: 12341: 12149: 11774: 11740: 11600: 11552: 11223: 10947: 10089: 9899: 9742: 9647: 9516: 9508: 9104: 8489: 8323: 7836: 7596: 7382: 7349: 7263: 6871: 6797: 6695: 6433: 6386: 6339: 6257: 6030: 5983: 5923: 5914: 5889: 5861: 5604: 5291: 4896: 4719: 4615: 4595: 4570: 4545: 4481: 4331: 4201: 4189: 4120: 4112: 4108: 4062: 3994: 3759: 3751: 3652: 3643: 3589: 3574: 3508: 3501: 3438:. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an 3431: 1712: 1579: 1573:

An easy way to visualize a chaotic attractor is to start with a point in the

1396:

is one of the simplest systems with density of periodic orbits. For example,

547: 311: 291: 270: 266: 195: 12993: 11994: 11105:. Dynamic Modeling and Econometrics in Economics and Finance. Vol. 29. 10952:

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

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Strelioff, Christopher; et., al. (2006). "Medium-Term Prediction of Chaos".

5544: 3027:{\displaystyle J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0} 12968: 12951: 12792: 12757: 12667: 12624: 12479: 12474: 12026: 12008: 11933: 11685: 11510: 10646: 10378: 10097: 10009: 9933: 9797: 8883: 8875: 8790: 8739: 7811: 7584: 7316:

Bellissard, J.; Bohigas, O.; Casati, G.; Shepelyansky, D.L. (1 July 1999).

7041: 6986: 6548:

Okulov, A Yu (2020). "Structured light entities, chaos and nonlocal maps".

6361:

Musielak, Dora E.; Musielak, Zdzislaw E.; Kennamer, Kenny S. (2005-03-01).

5807: 5702: 5299: 4704: 4669: 4664: 4659: 4506: 4496: 4441: 4367: 4307: 4080: 4046: 3923: 3863: 3824: 3747: 3627: 3597: 3570: 3512: 2912:{\displaystyle K({\vec {r}}-{\vec {r}}^{,},L)={\frac {ik\exp}{2\pi L}}\exp} 1716: 1708:

shows that a strange attractor can only arise in three or more dimensions.

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The Unity of Nature - Wholeness and Disintegration in Ecology and Science

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Invisible in the Storm: the role of mathematics in understanding weather

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Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change

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Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Zeng, Xiping (2024-07-16).

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European Journal of Obstetrics & Gynecology and Reproductive Biology

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the first important conference on chaos in biology and medicine. There,

1630: 1610:). Other discrete dynamical systems have a repelling structure called a 13110: 13105: 13023: 12926: 12835: 12732: 12179: 11500: 10689: 10433: 10168: 8686: 8191: 8004: 7981: 7824:

However, the conclusions of this article have been subject to dispute.

7676: 7488: 7255: 7080: 6940: 6837: 6820: 6526: 6022: 5757: 5497: 5337: 5282: 4042: 3812: 3804: 3788: 3755: 3660: 3635: 3562: 3547: 3543: 3435: 2359:

is propagator derived as Green function of a relevant physical system,

2078:

to a two-dimensional surface and therefore solutions are well behaved.

1318:. Intuitively, if a map is topologically transitive then given a point 794: 783: 766: 581: 222:. Chaos theory has applications in a variety of disciplines, including 183: 94: 11827:

Financial Time Series Analysis : Chaos and Neurodynamics Approach

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The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward

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Okulov, A Yu (2000). "Spatial soliton laser: geometry and stability".

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In chemistry, predicting gas solubility is essential to manufacturing

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Complexity : The Emerging Science at the Edge of Order and Chaos

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Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World

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describing the statistical distribution of earthquake sizes, and the

3770: 3631: 3536: 3527:). He applied this criterion to explain some experimental results on 3237:

is an adjustable parameter. This equation has a chaotic solution for

2525: 2480: 2067: 1697: 1611: 1598:, have great detail and complexity. Strange attractors occur in both 1567: 1371: 1355: 388: 247: 231: 11298: 10483: 10425: 9819: 5749: 5489: 4119:, which is modeled well by chaos theory. Another type of computing, 2939:, with respect to time. As such, differential equations of the form 1205:

is said to be topologically transitive if for any pair of non-empty

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Morbidelli, A. (2001). "Chaotic diffusion in celestial mechanics".

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On the order of chaos. Social anthropology and the science of chaos

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Gerig, A. (2007). "Chaos in a one-dimensional compressible flow".

8151:"The problem of deducing the climate from the governing equations" 7921: 7781: 5376:

A First Course in Dynamics: With a Panorama of Recent Developments

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organized the first symposium on chaos, attended by David Ruelle,

3309:, and all capacitors are of equal size. The dominant frequency is 1535: 13067: 13062: 13050: 13028: 13013: 12921: 12154: 12106: 11533:

Chaos Theory in the Social Sciences: Foundations and Applications

11398: 8265:"A Popular but Inaccurate Analogy for Chaos and Butterfly Effect" 8190:

Shen, Bo-Wen; Pielke Sr., Roger; Zeng, Xubin, eds. (2023-10-11).

7997:

Chaos Theory in the Social Sciences: Foundations and Applications

6637:"Secure Image Encryption Based On a Chua Chaotic Noise Generator" 6446: 4050: 4034: 4018: 4010: 3998: 3915: 3903: 3766: 3667:, which is infinitely long yet encloses a finite space and has a 3648: 2928: 1615: 360:

values diverge markedly over time from a tiny initial difference.

352:

displays sensitivity to initial x positions. Here, two series of

251: 210:. This behavior can be studied through the analysis of a chaotic 164: 11382:

The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory

11053: 10942: 8592:

Trading Chaos: Maximize Profits with Proven Technical Techniques

8520: 7065:

Stochastic Behavior in Classical and Quantum Hamiltonian Systems

6491: 5475: 3734:

In 1986, the New York Academy of Sciences co-organized with the

3244: 2624:{\displaystyle \psi _{n+1}=A+B\psi _{n}e^{i(|\psi _{n}|^{2}+C)}} 1676:

Each vertical slice shows the attractor for a specific value of

12727: 12042: 9821:

The Feasibility of a Global Observation and Analysis Experiment

7858:

Journal of the College of Science, Imperial University of Tokyo

6615:, J. C. Sprott, IEEE Transactions on Circuits and Systems,2011. 4243: 3989: 3608: 3381: 3251:

In the above circuit, all resistors are of equal value, except

1578:

attractor results from a simple three-dimensional model of the

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Chaotic dynamics: An introduction based on classical mechanics

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Motter, Adilson E.; Campbell, David K. (2013). "Chaos at 50".

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In the Wake of Chaos: Unpredictable Order in Dynamical Systems

3515:. What had been attributed to measure imprecision and simple " 13100: 12036:

Three Kinds of Butterfly Effects within Lorenz Models

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Chaos and Life: Complexity and Order in Evolution and Thought

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Gardini, Laura; Grebogi, Celso; Lenci, Stefano (2020-10-01).

6058:

13th Chaotic Modeling and Simulation International Conference

5154: 4906:

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

4902: 4350:

traffic jams and some areas where cars can move at top speed.

4196:. Adding variables exaggerates this: Chaos is more common in 3957: 3353:

Similar circuits only require one diode or no diodes at all.

542:

Sensitivity to initial conditions is popularly known as the "

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Day, R.H.; Pavlov, O.V. (2004). "Computing Economic Chaos".

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Communications in Nonlinear Science and Numerical Simulation

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Shen, Bo-Wen; Pielke, Sr., Roger; Zeng, Xubin (2023-08-12).

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cannot be chaotic, two-dimensional continuous systems with

1594:, the attractors that arise from chaotic systems, known as 1354:, then topological transitivity implies the existence of a 620:, the two trajectories end up diverging at a rate given by 135:, there are underlying patterns, interconnection, constant 11910:

Society for Chaos Theory in Psychology & Life Sciences

10873:

Dynamical systems: Stability, symbolic dynamics, and chaos

10808:

Complexity: hierarchical structures and scaling in physics

10237:

Global Analysis of Dynamic Models in Economics and Finance

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Orlando, Giuseppe; Zimatore, Giovanna (18 December 2017).

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Sprott, J.C. (1997). "Simplest dissipative chaotic flow".

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Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin (2022-05-07).

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dynamical systems (such as the Lorenz system) and in some

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Chaos in Wonderland: Visual Adventures in a Fractal World

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An experimental approach to nonlinear dynamics and chaos

9770:

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Celestial Encounters: The Origins of Chaos and Stability

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

588:

that are infinitesimally close, with initial separation

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Orlando, Giuseppe; Zimatore, Giovanna (1 August 2020).

9711:"Recurrence quantification analysis of business cycles" 9039: 8907: 7856:

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Journal of Statistical Mechanics: Theory and Experiment

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Chaos Under Control: The Art and Science of Complexity

11095: 11008: 10199: 9678:"RQA correlations on real business cycles time series" 6486:(in Russian). Vol. 187. Nauka. pp. 202–222. 6051: 5950:"The Dual Nature of Chaos and Order in the Atmosphere" 5448: 4908:

Creative Commons Attribution 4.0 International License

2631:

may serve. When wave propagation problems at distance

2098:

with heating and dissipations, on solution stability.

1486: 1445: 1404: 546:", so-called because of the title of a paper given by 11229:

Ordinary Differential Equations and Dynamical Systems

10262:

Transportation Research Part C: Emerging Technologies

10069: 9621:"Chaos in foreign exchange markets: a sceptical view" 9418: 8903: 8901: 8861: 8553: 8438: 6818: 5947: 3945:

Lorenz's pioneering contributions to chaotic modeling

3692:(coiner of the term "chaos" as used in mathematics), 3626:, discovered that noise in many phenomena (including 3315: 3257: 3082: 3059: 2950: 2729: 2703: 2666: 2637: 2533: 2488: 2425: 2365: 2299: 2112: 2052: 2032: 2012: 1992: 1968: 1948: 1928: 1739: 1484: 1443: 1402: 1272: 1246: 1214: 1179: 1038: 992: 933: 913: 842: 804: 740: 720: 629: 594: 504: 484: 464: 11906:

An advanced graduate textbook on chaos (no fractals)

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Orlando, Giuseppe; Zimatore, Giovanna (1 May 2018).

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Ivancevic, Vladimir G.; Tijana T. Ivancevic (2008).

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Introduction to Applied Dynamical Systems and Chaos

9178: 8060: 6759: 5423: 4853:. University of Washington Press. pp. 181–206. 3794:Alongside largely lab-based approaches such as the 552:

American Association for the Advancement of Science

276:Chaos theory differs from numerous fields, such as 11728: 11656: 11354: 11015:Time Reversibility, Computer Simulation, and Chaos 10987: 10832:Iterated Maps on the Interval as Dynamical Systems 10804: 10764: 9871: 8969: 8898: 8590:Gregory-Williams, Justine; Williams, Bill (2004). 8376: 8189: 7959: 7519: 7518:Mandelbrot, Benoît B.; Hudson, Richard L. (2004). 5887: 5690: 5007: 5005: 4921: 4236:Chaos could be found in economics by the means of 3338: 3301: 3222: 3065: 3026: 2911: 2709: 2689: 2652: 2623: 2516: 2467: 2411: 2351: 2282: 2058: 2038: 2018: 1998: 1974: 1954: 1934: 1911: 1511: 1470: 1429: 1314:. Topological transitivity is a weaker version of 1306: 1258: 1232: 1197: 1089: 1024: 978: 919: 899: 822: 746: 726: 703: 612: 510: 490: 470: 11928:Nonlinear dynamics: how science comprehends chaos 11379:Abraham, Ralph H.; Ueda, Yoshisuke, eds. (2000). 10110: 10078:Nonlinear Dynamics, Psychology, and Life Sciences 9912: 8760: 8644:(2nd ed.). New York: John Wiley & Sons. 6541: 6498: 6475: 6440: 5085: 5014:The British Journal for the Philosophy of Science 5011: 4868: 4252: 4192:co-evolution sometimes shows chaotic dynamics in 3787:(SOC), considered one of the mechanisms by which 1547:Some dynamical systems, like the one-dimensional 13230: 11956:Systems Analysis, Modelling and Prediction Group 11156: 10234: 10075: 8504:"Applying Chaos Theory to Embedded Applications" 8288:Saiki, Yoshitaka; Yorke, James A. (2023-05-02). 7638: 7517: 6602:http://sprott.physics.wisc.edu/pubs/paper297.htm 5617: 5398: 3715:, presented his experimental observation of the 3671:of circa 1.2619). In 1982, Mandelbrot published 3480:. Later studies, also on the topic of nonlinear 1696:, can exhibit strange attractors whatever their 11806:Sync: The emerging science of spontaneous order 11655:Peitgen, Heinz-Otto; Richter, Peter H. (1986). 11273: 11254: 11036:Chaos: The Science of Predictable Random Motion 10765:Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). 10609: 9998:Sync: The Emerging Science of Spontaneous Order 9763: 9708: 9675: 9546:. pp. xvi + 460 + 26b/will. + 33colorill. 8336: 8104:"A Review of Lorenz's Models from 1960 to 2008" 7999:. Ann Arbor, MI: University of Michigan Press. 6650:Sync: The Emerging Science of Spontaneous Order 6481: 5997:Yorke, James A.; Yorke, Ellen D. (1979-09-01). 5894:Bulletin of the American Meteorological Society 5321:Sync: The Emerging Science of Spontaneous Order 5002: 4362:Chaos theory has been applied to environmental 11900:. An introductory primer on chaos and fractals 11796:Does God Play Dice?: The Mathematics of Chaos 11654: 11099:; Pisarchick, Alexander; Stoop, Ruedi (2021). 11032: 10893:Chaos and Fractals: An Elementary Introduction 10889: 10710: 9682:Indian Academy of Sciences – Conference Series 9538:Jones, Gareth (1998). Jones, D. Gareth (ed.). 9472: 9239:The International Journal of Robotics Research 8108:International Journal of Bifurcation and Chaos 7902: 7731: 7617: 7501: 7178: 5832:International Journal of Bifurcation and Chaos 5568: 3719:cascade that leads to chaos and turbulence in 2352:{\displaystyle K({\vec {r}}-{\vec {r}}^{,},t)} 12877: 12058: 11893:The Chaos group at the University of Maryland 11535:, University of Michigan Press, 1997, 360 pp. 11367:, World Scientific Publishing Company, 2012, 11074: 10968: 10533: 10259: 9531: 7185:Bulletin of the American Mathematical Society 5251: 5081: 5079: 4079:Chaos theory has been used for many years in 1365: 1307:{\displaystyle f^{k}(U)\cap V\neq \emptyset } 1161:or fluids is an example of a chaotic system. 12025: This article incorporates text from a 10854:An Introduction to Chaotic Dynamical Systems 10785: 10153: 9414: 9412: 9410: 9314:Computational Statistics & Data Analysis 8664: 8619:(2. print. ed.). New York u.a.: Wiley. 8339:"The Geometry and Pigmentation of Seashells" 6906:Journal de Mathématiques Pures et Appliquées 6484:Proceedings of the Lebedev Physics Institute 5777: 5196: 5194: 5168: 5166: 4779:"What is Chaos Theory? – Fractal Foundation" 4244:Finite Predictability in Weather and Climate 3401:supersymmetric theory of stochastic dynamics 3243: 2101: 11378: 11135: 11054:Kiel, L. Douglas; Elliott, Euel W. (1997). 10788:Chaos, Scattering and Statistical Mechanics 10768:Chaos: an introduction to dynamical systems 10661:Hübler, A.; Foster, G.; Phelps, K. (2007). 10361: 10060: 8521:Hristu-Varsakelis, D.; Kyrtsou, C. (2008). 7994: 7395: 7214:"РЕЗОНАНСНЫЕ ПРОЦЕССЫ В МАГНИТНЫХ ЛОВУШКАХ" 7181:"Morse inequalities for a dynamical system" 6661: 6399: 6273:Bulletin of the London Mathematical Society 4541:Chaos as topological supersymmetry breaking 3974:For want of a battle, the kingdom was lost. 3819:, fluctuations in economic systems such as 3711:, during a symposium organized in Aspen by 3409:stochastic (partial) differential equations 1512:{\displaystyle {\tfrac {5-{\sqrt {5}}}{8}}} 1471:{\displaystyle {\tfrac {5+{\sqrt {5}}}{8}}} 1430:{\displaystyle {\tfrac {5-{\sqrt {5}}}{8}}} 1168: 12884: 12870: 12065: 12051: 12005:Chaos: The Science of the Butterfly Effect 11505:From Clocks to Chaos: The Rhythms of Life, 10318: 10063:The impact of chaos on science and society 10048:The impact of chaos on science and society 9969: 9179:Nehmzow, Ulrich; Keith Walker (Dec 2005). 9125: 8287: 7737:Annals of the New York Academy of Sciences 7644: 7550: 7150:Journal of the London Mathematical Society 7062: 7019: 7007: 6964: 6935: 6737:"James Clerk Maxwell: a force for physics" 6228:Bollettino dell'Unione Matematica Italiana 5996: 5664: 5662: 5622:Global Warming and the Future of the Earth 5374:Hasselblatt, Boris; Anatole Katok (2003). 5200: 5076: 5059:"Chaos in an Atmosphere Hanging on a Wall" 4172:can be obtained through chaotic modeling. 4136:. Chaotic dynamics have been exhibited by 3976:And all for the want of a horseshoe nail. 3958:A popular but inaccurate analogy for chaos 3815:describing the frequency of aftershocks), 3677:, which became a classic of chaos theory. 11967:High Anxieties — The Mathematics of Chaos 11915:Nonlinear Dynamics Research Group at CSDC 11700:Newton's Clock: Chaos in the Solar System 11337:Hamiltonian Chaos and Fractional Dynamics 10724: 10663:"Managing Chaos: Thinking out of the Box" 10576: 10562: 10415: 10169:Pryor, Robert G. L.; Norman E. Amundson; 9918: 9889: 9693: 9600: 9544:Springer Science+Business Media Dordrecht 9429:Springer Science+Business Media Dordrecht 9407: 9339: 9285:SIAM Journal on Applied Dynamical Systems 9250: 9199: 9139: 8836: 8821: 8713: 8538: 8479: 8313: 8201: 8174: 8086: 7920: 7855: 7666: 7288:; Dima, Shepelyansky (27 February 1998). 7196: 6836: 6787: 6777: 6708: 6561: 6239: 5973: 5913: 5851: 5739: 5594: 5281: 5191: 5163: 5111: 5025: 4886: 3972:For want of a rider, the battle was lost. 214:or through analytical techniques such as 11731:Chaotic Evolution and Strange Attractors 11562: 11522: 11507:Princeton University Press 1988, 272 pp. 11415: 10991:Chaos in Classical and Quantum Mechanics 10870: 8803: 7370: 7284: 7237: 6903: 6851: 6709:Hunt, Brian R.; Yorke, James A. (1993). 6221: 6178: 6133: 5679: 5056: 3988: 3970:For want of a horse, the rider was lost. 3604:were using a simple digital computer, a 3542: 3426: 1647: 1629: 1625: 1534: 1100: 793: 429: 310: 116:. It focuses on underlying patterns and 80: 36: 11930:, talk presented by Sunny Auyang, 1998. 11559:, University of Washington Press, 1996. 11527:. Penguin Press Science. Penguin Books. 11315: 10848: 10377: 9468: 9466: 9464: 9425:Population and Community Biology Series 8527:Discrete Dynamics in Nature and Society 7903:Motter, A. E.; Campbell, D. K. (2013). 7477:IBM Journal of Research and Development 7240:"Resonance processes in magnetic traps" 6734: 6400:Roy, D.; Musielak, Z. E. (2007-05-01). 5701: 5668: 5659: 5403:. Chapman & Hall/CRC. p. 137. 5057:Danforth, Christopher M. (April 2013). 4996: 4984: 4966:The Stanford Encyclopedia of Philosophy 4928:. University of Chicago Press. p. 4919: 4750:"chaos theory | Definition & Facts" 3968:For want of a shoe, the horse was lost. 3887:The Structure of Scientific Revolutions 2468:{\displaystyle \psi \rightarrow G\psi } 900:{\displaystyle x_{k+1}=4x_{k}(1-x_{k})} 613:{\displaystyle \delta \mathbf {Z} _{0}} 561:As suggested in Lorenz's book entitled 14: 13231: 12891: 11962:A page about the Mackey-Glass equation 11726: 11531:L Douglas Kiel, Euel W Elliott (ed.), 11222: 11180: 9615: 9582: 9376: 9004: 8639: 8614: 8229: 8225: 8223: 8148: 7995:Kiel, L.; Elliott, Euel, eds. (1996). 7877: 7447: 7407: 6885:Diacu, Florin; Holmes, Philip (1996). 6547: 6504: 6096: 5519:. Cambridge University Press. p. 5351:"Definition of chaos | Dictionary.com" 5172: 5155:Mosko M.S., Damon F.H. (Eds.) (2005). 4959: 4848: 4822: 4355:prediction model (see the plot of the 4097:secure pseudo-random number generators 3966:For want of a nail, the shoe was lost. 3746:presented a mathematical model of the 3577:later predicted, against Landau, that 2717:may have a form of Green function for 1692:Discrete chaotic systems, such as the 1644:Minimum complexity of a chaotic system 12865: 12046: 11764: 10239:. Springer-Verlag. pp. 185–204. 10045: 9537: 8823:10.30759/1728-9718-2019-2(63)-127-136 8699: 7873: 7871: 7422: 5943: 5941: 5883: 5881: 5879: 5821: 5819: 5817: 5640: 5511:Medio, Alfredo; Lines, Marji (2001). 5150: 5148: 4797: 3596:in 1961. Lorenz and his collaborator 1530: 1090:{\displaystyle y_{k+1}=x_{k}+y_{k}-1} 789: 12029:work. Licensed under CC-BY ( 11706:Ian Roulstone; John Norbury (2013). 11439: 10922:Gollub, J. P.; Baker, G. L. (1996). 9461: 8262: 8101: 8035: 7211: 6625:"Simple Autonomous Chaotic Circuits" 6305: 5825: 4955: 4953: 4951: 4949: 4864: 4862: 4860: 4844: 4842: 4840: 4773: 4771: 4769: 4299:can be predicted fairly accurately. 3419:embodiment of the butterfly effect. 3363: 299:be made, the system appears random. 13170:The Chemical Basis of Morphogenesis 12205:Measure-preserving dynamical system 12087: 12007:(2019) an explanation presented by 11969:(2008) BBC documentary directed by 11549:, Princeton University Press, 1991. 11057:Chaos Theory in the Social Sciences 10202:European Sport Management Quarterly 9310: 8972:Commun. Nonlinear Sci. Numer. Simul 8763:The Journal of Physical Chemistry A 8220: 7708:"The Wolf Prize in Physics in 1986" 7244:The Soviet Journal of Atomic Energy 6312:Journal of the Atmospheric Sciences 5611: 5092:Journal of the Atmospheric Sciences 4825:"Chaos Theory and the Logistic Map" 4260:Butterfly Effect in Popular Culture 3736:National Institute of Mental Health 3476:Chaos theory began in the field of 3457:. In the 1880s, while studying the 2479:. For examples of complex maps the 979:{\displaystyle y_{k+1}=x_{k}+y_{k}} 306: 93:would result in a vastly different 24: 11692:, Princeton University Press 1993. 11519:, New York: Penguin, 1988. 368 pp. 11202:Tél, Tamás; Gruiz, Márton (2006). 10610:Strelioff, C.; Hübler, A. (2006). 10587:10.1023/B:CSEM.0000026787.81469.1f 10350: 10187:10.1002/j.2161-0045.2008.tb00096.x 10065:. United Nations University Press. 10050:. United Nations University Press. 9540:The Epidemiology of Plant Diseases 9074: 7868: 7753:10.1111/j.1749-6632.1987.tb48737.x 7022:Proceedings of the Royal Society A 6967:Proceedings of the Royal Society A 5938: 5876: 5814: 5643:"Lyapunov Characteristic Exponent" 5145: 4344: 4238:recurrence quantification analysis 3731:for their inspiring achievements. 3182: 3172: 3148: 3132: 3104: 3088: 1870: 1860: 1811: 1801: 1758: 1748: 1301: 798:Six iterations of a set of states 776: 25: 13260: 12773:Oleksandr Mykolayovych Sharkovsky 12014: 11887:Nonlinear Dynamics Research Group 11833: 10612:"Medium-Term Prediction of Chaos" 10523:10.1038/scientificamerican1286-46 10509:(6): 38–49 (bibliography p.136). 8910:Journal of the Franklin Institute 8594:(2nd ed.). New York: Wiley. 8337:Stephen Coombes (February 2009). 8263:Shen, Bo-Wen (20 February 2023). 7878:Gleick, James (August 26, 2008). 7551:Mandelbrot, Benoît (5 May 1967). 7526:. New York: Basic Books. p. 7506:. 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"Chaos". 10312: 10288: 10253: 10228: 10193: 10175:The Career Development Quarterly 10162: 10147: 10104: 10054: 10039: 10003: 9990: 9963: 9927: 9906: 9865: 9852: 9846: 9812: 9757: 9702: 9669: 9665:from the original on 2017-08-09. 9609: 9576: 9370: 9333: 9304: 9275: 9230: 9172: 9119: 9068: 9033: 8998: 8963: 8928: 8864:Bulletin of Mathematical Biology 8855: 8830: 8797: 8754: 8693: 8658: 8633: 8608: 8583: 8547: 8514: 8496: 8447: 8432: 8397: 8370: 8330: 8281: 8256: 8183: 8142: 8095: 7454:. London: Cardinal. p. 17. 6662:Ovchinnikov, I.V. (2024-02-15). 5691:Alligood, Sauer & Yorke 1997 5063:Mathematics of Planet Earth 2013 4901: 4395: 4381: 3997:shell, similar in appearance to 3567:Landau-Hopf theory of turbulence 2690:{\displaystyle \lambda =2\pi /k} 1025:{\displaystyle x_{k}+y_{k}<1} 683: 639: 600: 576:In more mathematical terms, the 11988:Chaos, A Mathematical Adventure 11759:Fractals, Chaos, and Power Laws 11355:Semitechnical and popular works 11181:Sprott, Julien Clinton (2003). 10910:from the original on 2019-12-31 9984:10.1070/rd2001v006n04abeh000182 9188:Robotics and Autonomous Systems 9158:10.1016/j.optlastec.2014.01.015 9062:10.1016/j.optlastec.2017.04.022 8922:10.1016/j.jfranklin.2011.05.001 8359:from the original on 2013-11-05 8230:Lorenz, E. N. (December 2008). 8054: 8029: 7988: 7953: 7896: 7849: 7775: 7725: 7700: 7691: 7611: 7544: 7510: 7495: 7468: 7441: 7416: 7401: 7389: 7364: 7309: 7278: 7231: 7205: 7198:10.1090/S0002-9904-1960-10386-2 7179:Smale, Stephen (January 1960). 7172: 7105: 7101:Kolmogorov–Arnold–Moser theorem 7056: 7001: 6929: 6916: 6897: 6878: 6845: 6812: 6753: 6735:Everitt, Francis (2006-12-01). 6728: 6702: 6655: 6642: 6630: 6618: 6606: 6594: 6449:Journal of Mathematical Physics 6393: 6354: 6299: 6264: 6215: 6172: 6127: 6090: 6045: 5990: 5771: 5695: 5684: 5673: 5634: 5562: 5537: 5504: 5469: 5442: 5417: 5392: 5367: 5343: 5326: 5314: 5245: 5221: 5120: 5050: 4990: 4823:Boeing, Geoff (26 March 2015). 4586:Kolmogorov–Arnold–Moser theorem 4294:, the study of large arrays of 4068: 3984: 3634:circuits) was patterned like a 3388:resonance, and large arrays of 2922: 375:sensitive to initial conditions 284: 194:. The theory was summarized by 12538:Rabinovich–Fabrikant equations 11712:. Princeton University Press. 11589:David Peak and Michael Frame, 11208:. Cambridge University Press. 11184:Chaos and Time-Series Analysis 11142:. Cambridge University Press. 10928:. Cambridge University Press. 10811:. Cambridge University Press. 10790:. Cambridge University Press. 9972:Regular & Chaotic Dynamics 9715:Chaos, Solitons & Fractals 8937:Chaos, Solitons & Fractals 8203:10.3390/books978-3-0365-8911-4 7647:Journal of Statistical Physics 7621:The Fractal Geometry of Nature 7504:The Fractal Geometry of Nature 7425:"The Hidden Heroines of Chaos" 7423:Sokol, Joshua (May 20, 2019). 7322:Physica D: Nonlinear Phenomena 7238:Chirikov, B. V. (1960-12-01). 6937:Kolmogorov, Andrey Nikolaevich 6819:Poincaré, Jules Henri (1890). 6668:Chaos, Solitons & Fractals 6550:Chaos, Solitons & Fractals 6406:Chaos, Solitons & Fractals 6003:Journal of Statistical Physics 5463:10.1080/00029890.1992.11995856 5378:. Cambridge University Press. 4978: 4913: 4816: 4791: 4742: 4265: 4253:AI-Extended Modeling Framework 3783:describing for the first time 3750:dysfunction among people with 3674:The Fractal Geometry of Nature 3302:{\displaystyle R_{A}=R/A=5R/3} 3204: 3196: 2906: 2885: 2871: 2855: 2845: 2832: 2809: 2797: 2776: 2758: 2742: 2733: 2616: 2600: 2584: 2580: 2498: 2492: 2462: 2459: 2453: 2438: 2429: 2406: 2403: 2391: 2382: 2369: 2346: 2328: 2312: 2303: 2268: 2255: 2252: 2234: 2224: 2211: 2205: 2187: 2171: 2162: 2150: 2138: 2129: 1289: 1283: 1189: 894: 875: 817: 805: 694: 674: 653: 649: 643: 631: 554:in Washington, D.C., entitled 412:If attention is restricted to 13: 1: 13249:Computational fields of study 11976:The chaos theory of evolution 11822:, Simon & Schuster, 1992. 11798:, Blackwell Publishers, 1990. 11605:The Science of Fractal Images 11446:. Columbia University Press. 11335:Zaslavsky, George M. (2005). 11238:American Mathematical Society 10805:Badii, R.; Politi A. (1997). 10639:10.1103/PhysRevLett.96.044101 10395:American Mathematical Monthly 10341:10.1016/s0309-1708(99)00008-1 10214:10.1080/16184742.2018.1439984 10013:Mining Science and Technology 9393:10.1016/s0301-2115(96)02628-0 9364:10.1016/S0022-1694(99)00186-9 9128:Optics & Laser Technology 9042:Optics & Laser Technology 7342:10.1016/s0167-2789(99)90007-6 6224:"Slow chaos in surface flows" 6121:10.1016/S0375-9601(97)00088-1 5800:10.1103/PhysRevLett.96.044101 5719:American Mathematical Monthly 5426:Topology and its applications 4964:, in Zalta, Edward N. (ed.), 4736: 3490:Andrey Nikolaevich Kolmogorov 1370:For a chaotic system to have 127:that are highly sensitive to 32:Chaos theory (disambiguation) 12031:license statement/permission 11859:Resources in other libraries 11542:, National Book Trust, 2003. 11480:George Washington Law Review 11470:John Briggs and David Peat, 11257:Nonlinear Dynamics And Chaos 11163:Nonlinear Dynamics and Chaos 11078:Chaotic and Fractal Dynamics 10758: 10387:"Period Three Implies Chaos" 10125:10.1016/0006-3223(92)90093-F 9327:10.1016/0167-9473(95)00056-9 8804:Borodkin, Leonid I. (2019). 8441:New Trends in Macroeconomics 8391:10.1016/j.jmacro.2005.10.019 8038:"Butterfly Effect and Chaos" 7577:10.1126/science.156.3775.636 6613:"A New Chaotic Jerk Circuit" 6066:10.1007/978-3-030-70795-8_57 5711:"Period Three Implies Chaos" 5515:Nonlinear Dynamics: A Primer 5424:Basener, William F. (2006). 4920:Kellert, Stephen H. (1993). 4227: 3796:Bak–Tang–Wiesenfeld sandpile 3723:systems. He was awarded the 3682:New York Academy of Sciences 3521:Boris Valerianovich Chirikov 2419:might be logistic map alike 1622:can be calculated for them. 1233:{\displaystyle U,V\subset X} 769:theory) and properties of a 7: 12273:Poincaré recurrence theorem 11958:at the University of Oxford 11875:Encyclopedia of Mathematics 11516:Chaos: Making a New Science 11497:, Adam Hilger 1989, 648 pp. 11339:. Oxford University Press. 11259:. John Wiley and Sons Ltd. 11187:. Oxford University Press. 11102:Nonlinearities in Economics 11039:. Oxford University Press. 10988:Gutzwiller, Martin (1990). 10896:. Oxford University Press. 10477:10.1016/j.cnsns.2016.09.010 10355: 10321:Advances in Water Resources 9957:10.1016/j.fluid.2013.07.017 9735:10.1016/j.chaos.2018.02.032 9602:10.1016/j.procs.2010.12.161 9210:10.1016/j.robot.2005.09.009 8992:10.1016/j.cnsns.2010.02.014 8957:10.1016/j.chaos.2006.05.011 8640:Peters, / Edgar E. (1996). 8426:10.1016/j.physa.2006.11.002 8102:Shen, Bo-Wen (2023-09-04). 7880:Chaos: Making a New Science 7451:Chaos: Making a New Science 7374:Chaos at Fifty Four in 2013 6852:Poincaré, J. Henri (2017). 6688:10.1016/j.chaos.2024.114611 6580:10.1016/j.chaos.2020.109638 6426:10.1016/j.chaos.2006.02.013 6306:Shen, Bo-Wen (2014-05-01). 5826:Shen, Bo-Wen (2019-03-01). 5274:10.1126/science.257.5066.56 4888:10.3390/encyclopedia2030084 4410:Examples of chaotic systems 4374: 4276:particle swarm optimization 4126: 3869:Chaos: Making a New Science 3754:. This led to a renewal of 2935:is the third derivative of 2517:{\displaystyle f=\psi ^{2}} 1334:whose orbit passes through 10: 13265: 12268:Poincaré–Bendixson theorem 11617:, Páginas de Espuma, 2021. 11139:Chaos in Dynamical Systems 10156:Chaos theory in psychology 10033:10.1016/j.mstc.2011.05.010 9261:10.1177/027836499801701202 8732:10.1103/PhysRevE.75.045202 8346:www.maths.nottingham.ac.uk 8176:10.3402/tellusa.v16i1.8893 8036:Chen, G.-R. (2020-01-01). 7804:10.1103/PhysRevLett.59.381 7097:Doklady Akademii Nauk SSSR 7010:Doklady Akademii Nauk SSSR 6946:Doklady Akademii Nauk SSSR 6891:Princeton University Press 6779:10.1007/s11071-020-05903-0 6250:10.1007/s40574-020-00267-0 6222:Ulcigrai, Corinna (2021). 6201:10.1088/0951-7715/12/3/012 6158:10.1088/0951-7715/10/5/014 5201:Weinberger, David (2019). 5086:Lorenz, Edward N. (1963). 4143: 4072: 3932:computational neuroscience 3785:self-organized criticality 3721:Rayleigh–Bénard convection 3422: 2697:are considered the kernel 1724:Poincaré–Bendixson theorem 1706:Poincaré–Bendixson theorem 1366:Density of periodic orbits 423: 29: 13200: 13150:D'Arcy Wentworth Thompson 13093: 13001: 12990: 12899: 12821: 12638: 12620:Swinging Atwood's machine 12565: 12503: 12373: 12360: 12312: 12293: 12263:Krylov–Bogolyubov theorem 12243: 12140: 12080: 11854:Resources in your library 11669:10.1007/978-3-642-61717-1 11440:Bird, Richard J. (2003). 11316:Wiggins, Stephen (2003). 11111:10.1007/978-3-030-70982-2 10548:10.1007/s11253-005-0055-4 10282:10.1016/j.trc.2012.08.004 9695:10.29195/iascs.01.01.0009 9589:Procedia Computer Science 9583:Juárez, Fernando (2011). 9552:10.1007/978-94-017-3302-1 9437:10.1007/978-94-011-4010-2 9097:10.1007/s13319-017-0126-y 9019:10.1007/s11047-012-9334-9 8615:Peters, Edgar E. (1994). 8379:Journal of Macroeconomics 8128:10.1142/S0218127423300240 7099:(1954) 98: 527. See also 6379:10.1142/S0218348X0500274X 5853:10.1142/S0218127419500378 5618:Watts, Robert G. (2007). 5399:Elaydi, Saber N. (1999). 5159:. Oxford: Berghahn Books. 4502:Swinging Atwood's machine 4220:oscillations in pathogen 3405:topological supersymmetry 3339:{\displaystyle 1/2\pi RC} 2102:Infinite dimensional maps 1719:or infinite-dimensional. 263:complex dynamical systems 12528:Lotka–Volterra equations 12352:Synchronization of chaos 12155:axiom A dynamical system 11889:with Animations in Flash 11775:10.1017/CBO9780511554544 11741:10.1017/CBO9780511608773 11607:, Springer 1988, 312 pp. 10871:Robinson, Clark (1995). 10649:. 044101. Archived from 8481:10.1088/1367-2630/aaad41 8350:University of Nottingham 7162:10.1112/jlms/s1-20.3.180 5915:10.1175/BAMS-D-19-0165.1 4631:Synchronization of chaos 4328:complex adaptive systems 4085:cryptographic primitives 3827:), landscape formation, 3740:Office of Naval Research 3581:could develop through a 3440:iterated function system 3411:, and the corresponding 3386:London Millennium Bridge 3380:' pendulums, fireflies, 3356:See also the well-known 1362:that have dense orbits. 1198:{\displaystyle f:X\to X} 1169:Topological transitivity 747:{\displaystyle \lambda } 382:topologically transitive 12513:Double scroll attractor 12278:Stable manifold theorem 12185:False nearest neighbors 12033:). Text taken from 11898:The Chaos Hypertextbook 11563:Marshall, Alan (2002). 11503:and Michael C. Mackey, 11033:Kautz, Richard (2011). 10890:Feldman, D. P. (2012). 10565:Computational Economics 9855:Bull. Am. Meteorol. Soc 9640:10.1023/A:1008650024944 9628:Computational Economics 9485:Nature Publishing Group 8810:Ural Historical Journal 8569:10.1023/A:1023939610962 8557:Computational Economics 8114:(10): 2330024–2330220. 7784:Physical Review Letters 7624:. New York: Macmillan. 7618:Mandelbrot, B. (1982). 7502:Mandelbrot, B. (1977). 7396:Abraham & Ueda 2000 6718:Nonlinear Science Today 6507:Optics and Spectroscopy 6332:10.1175/JAS-D-13-0223.1 4960:Bishop, Robert (2017), 4849:Lorenz, Edward (1993). 4754:Encyclopedia Britannica 4286:rotate chaotically. In 4188:Perry & Wall 1984. 4117:diffusion and confusion 3780:Physical Review Letters 3498:John Edensor Littlewood 3407:which is hidden in all 2019:{\displaystyle \sigma } 1704:dynamical systems, the 1680:. The diagram displays 1326:, there exists a point 491:{\displaystyle \sigma } 133:chaotic complex systems 13244:Complex systems theory 12553:Van der Pol oscillator 12533:Mackey–Glass equations 12165:Box-counting dimension 11659:The Beauty of Fractals 11166:. Perseus Publishing. 11075:Moon, Francis (1990). 11060:. Perseus Publishing. 11010:Hoover, William Graham 10969:Gulick, Denny (1992). 10300:pasternack.ucdavis.edu 9937:Fluid Phase Equilibria 8876:10.1006/bulm.2000.0213 8460:New Journal of Physics 8149:Lorenz, E. N. (1964). 8013:2027/fulcrum.d504rm03n 7448:Gleick, James (1987). 7398:, See Chapters 3 and 4 7298:Annales Henri Poincaré 7168:Van der Pol oscillator 7042:10.1098/rspa.1991.0076 6987:10.1098/rspa.1991.0075 5838:(3): 1950037–1950091. 4636:Unintended consequence 4432:Bouncing ball dynamics 4403:Systems science portal 4351: 4292:electrical engineering 4006: 4005:with chaotic behaviour 3876:analysis. Alluding to 3729:Mitchell J. Feigenbaum 3680:In December 1977, the 3651:(examples include the 3586: 3484:, were carried out by 3482:differential equations 3449:first emphasized the " 3443: 3340: 3303: 3248: 3224: 3067: 3028: 2913: 2711: 2691: 2654: 2625: 2518: 2469: 2413: 2353: 2284: 2087:non-Euclidean geometry 2060: 2059:{\displaystyle \beta } 2040: 2020: 2000: 1976: 1956: 1936: 1913: 1728:differential equations 1689: 1636: 1588:fixed-point attractors 1544: 1513: 1472: 1431: 1308: 1260: 1259:{\displaystyle k>0} 1234: 1199: 1143: 1098: 1091: 1026: 980: 921: 901: 824: 748: 728: 705: 614: 532: 512: 511:{\displaystyle \beta } 492: 472: 361: 204: 98: 78: 12984:Widmanstätten pattern 12703:Svetlana Jitomirskaya 12610:Multiscroll attractor 12455:Interval exchange map 12408:Dyadic transformation 12393:Complex quadratic map 12235:Topological conjugacy 12170:Correlation dimension 12145:Anosov diffeomorphism 11985:et Aurélien Alvarez, 11818:M. Mitchell Waldrop, 11765:Smith, Peter (1998). 11637:, St Martins Pr 1994. 11627:, St Martins Pr 1991. 11573:10.1142/9781860949548 11495:Universality in Chaos 10971:Encounters with Chaos 10786:Baker, G. L. (1996). 10113:Biological Psychiatry 9891:10.3390/atmos15070837 9853:GARP. "GARP topics". 8315:10.3390/atmos14050821 8088:10.3390/atmos14081279 6285:10.1112/blms/33.2.196 5975:10.3390/atmos13111892 5647:mathworld.wolfram.com 5596:10.3390/atmos13050753 5428:. Wiley. p. 42. 4804:mathworld.wolfram.com 4526:Anosov diffeomorphism 4348: 4175:As Perry points out, 4089:encryption algorithms 3992: 3845:punctuated equilibria 3809:Gutenberg–Richter law 3777:published a paper in 3725:Wolf Prize in Physics 3546: 3486:George David Birkhoff 3430: 3397:spontaneous breakdown 3341: 3304: 3247: 3225: 3068: 3039:are sometimes called 3029: 2914: 2712: 2692: 2655: 2626: 2519: 2470: 2414: 2354: 2285: 2061: 2041: 2039:{\displaystyle \rho } 2021: 2001: 1977: 1957: 1937: 1914: 1651: 1633: 1626:Coexisting attractors 1606:systems (such as the 1538: 1521:Sharkovskii's theorem 1514: 1473: 1432: 1352:complete metric space 1309: 1261: 1235: 1200: 1104: 1092: 1027: 981: 922: 902: 825: 797: 749: 729: 706: 615: 513: 493: 473: 471:{\displaystyle \rho } 446:were changed between 433: 314: 236:environmental science 200: 176:numerical computation 84: 40: 12713:Edward Norton Lorenz 11631:Clifford A. Pickover 11621:Clifford A. Pickover 11557:The Essence of Chaos 11417:Barnsley, Michael F. 11361:Christophe Letellier 11136:Ott, Edward (2002). 11018:. World Scientific. 10828:Eckmann, Jean-Pierre 10487:; Tucker; Morrison; 9431:. pp. xii+226. 9343:Journal of Hydrology 7371:Shepelyansky, Dima. 7318:"A pioneer of chaos" 6922:George D. Birkhoff, 4851:The Essence of Chaos 4514:Other related topics 4487:List of chaotic maps 3841:biological evolution 3565:, who developed the 3494:Mary Lucy Cartwright 3467:Hadamard's billiards 3313: 3255: 3080: 3057: 2948: 2727: 2719:Schrödinger equation 2701: 2664: 2653:{\displaystyle L=ct} 2635: 2531: 2486: 2423: 2363: 2297: 2110: 2050: 2030: 2010: 1990: 1966: 1946: 1926: 1737: 1482: 1441: 1400: 1270: 1244: 1212: 1177: 1036: 990: 931: 911: 840: 802: 738: 718: 627: 592: 563:The Essence of Chaos 502: 482: 462: 278:structural stability 30:For other uses, see 13218:Mathematics and art 13208:Pattern recognition 13178:Aristid Lindenmayer 12673:Mitchell Feigenbaum 12615:Population dynamics 12600:Hénon–Heiles system 12460:Irrational rotation 12413:Dynamical billiards 12398:Coupled map lattice 12258:Liouville's theorem 12190:Hausdorff dimension 12175:Conservative system 12160:Bifurcation diagram 11757:Manfred Schroeder, 11727:Ruelle, D. (1989). 11425:. Morgan Kaufmann. 11422:Fractals Everywhere 11391:2000cagm.book.....A 11291:1993AmJPh..61..958T 11081:. Springer-Verlag. 10994:. Springer-Verlag. 10954:. Springer-Verlag. 10771:. Springer-Verlag. 10735:2013PhT....66e..27M 10682:2007Cmplx..12c..10H 10631:2006PhRvL..96d4101S 10515:1986SciAm.255d..38T 10502:Scientific American 10469:2017CNSNS..44..495A 10408:1975AmMM...82..985L 10333:1999AdWR...23..253P 10274:2013TRPC...27..219W 10025:2011MiSTC..21..439L 9949:2013FlPEq.356...11L 9782:2020Chaos..30h3129O 9727:2018CSF...110...82O 9493:1992Natur.360..121T 9356:2000JHyd..227....1S 9150:2014OptLT..60..111L 9089:2017TDR.....8..126X 9054:2017OptLT..95...94A 8984:2010CNSNS..15.4052W 8949:2008CSF....35..408B 8775:2009JPCA..113...19W 8724:2007PhRvE..75d5202G 8679:2007Cmplx..13b..62W 8540:10.1155/2008/138547 8472:2018NJPh...20b3048H 8418:2007PhyA..377..227K 8306:2023Atmos..14..821S 8167:1964Tell...16....1L 8120:2023IJBC...3330024S 8079:2023Atmos..14.1279S 7974:2007Cmplx..12c..10H 7931:2013PhT....66e..27M 7796:1987PhRvL..59..381B 7745:1987NYASA.504..260H 7659:1978JSP....19...25F 7569:1967Sci...156..636M 7334:1999PhyD..131D...8B 7073:1979LNP....93...51K 7034:1991RSPSA.434...15K 6979:1991RSPSA.434....9K 6959:1941DoSSR..30..301K 6680:2024CSF...18114611O 6572:2020CSF...13309638O 6519:2000OptSp..89..131O 6461:1988JMP....29...63A 6418:2007CSF....32.1038R 6324:2014JAtS...71.1701S 6193:1999Nonli..12..617H 6150:1997Nonli..10.1289F 6113:1997PhLA..228..271S 6015:1979JSP....21..263Y 5966:2022Atmos..13.1892S 5906:2021BAMS..102E.148S 5844:2019IJBC...2950037S 5792:2006PhRvL..96d4101S 5732:1975AmMM...82..985L 5641:Weisstein, Eric W. 5587:2022Atmos..13..753S 5266:1992Sci...257...56S 5233:www.philarchive.org 5173:Piotrowski, Chris. 5104:1963JAtS...20..130L 5036:10.1093/bjps/axn053 4798:Weisstein, Eric W. 4730:George M. Zaslavsky 4472:Gaspard-Rice system 4462:Dynamical billiards 4447:Coupled map lattice 4296:Josephson junctions 4280:celestial mechanics 4274:, but models using 4059:population dynamics 3912:population modeling 3727:in 1986 along with 3709:Albert J. Libchaber 3698:Mitchell Feigenbaum 3447:James Clerk Maxwell 3390:Josephson junctions 3374:coupled oscillation 2092:functional analysis 1700:. In contrast, for 1653:Bifurcation diagram 1618:structure, and the 1575:basin of attraction 1105:The map defined by 760:measure-theoretical 188:deterministic chaos 87:double-rod pendulum 13156:On Growth and Form 13056:Logarithmic spiral 12893:Patterns in nature 12851:Santa Fe Institute 12718:Aleksandr Lyapunov 12548:Three-body problem 12435:Gingerbreadman map 12322:Bifurcation theory 12200:Lyapunov stability 11949:2007-02-02 at the 11934:Nonlinear Dynamics 11815:, Aerial Pr, 1993. 11649:Order Out of Chaos 11597:Heinz-Otto Peitgen 11491:Predrag Cvitanović 10944:Guckenheimer, John 10850:Devaney, Robert L. 10690:10.1002/cplx.20159 10158:. Greenwood Press. 8687:10.1002/cplx.20204 8443:. Springer Verlag. 8005:10.3998/mpub.14623 7982:10.1002/cplx.20159 7835:. See especially: 7677:10.1007/BF01020332 7599:on 19 October 2021 7489:10.1147/rd.73.0224 7256:10.1007/BF01483352 7081:10.1007/BFb0021737 6924:Dynamical Systems, 6838:10.1007/BF02392506 6766:Nonlinear Dynamics 6711:"Maxwell on Chaos" 6527:10.1134/BF03356001 6023:10.1007/BF01011469 5355:www.dictionary.com 5229:"Structural Chaos" 4690:Aleksandr Lyapunov 4621:Santa Fe Institute 4606:Patterns in nature 4556:Chaotic scattering 4531:Catastrophe theory 4427:Bifurcation theory 4389:Mathematics portal 4352: 4194:allele frequencies 4152:. Most models are 4075:Chaotic cryptology 4007: 4003:cellular automaton 3928:information theory 3791:arises in nature. 3624:information theory 3594:weather prediction 3587: 3529:plasma confinement 3525:Chirikov criterion 3459:three-body problem 3444: 3434:created using the 3378:Christiaan Huygens 3336: 3299: 3249: 3220: 3063: 3024: 2909: 2707: 2687: 2650: 2621: 2514: 2465: 2409: 2349: 2280: 2056: 2036: 2016: 1996: 1972: 1952: 1932: 1909: 1907: 1710:Finite-dimensional 1690: 1637: 1596:strange attractors 1545: 1531:Strange attractors 1509: 1507: 1468: 1466: 1427: 1425: 1316:topological mixing 1304: 1256: 1230: 1195: 1147:Topological mixing 1144: 1140:topological mixing 1099: 1087: 1022: 976: 917: 897: 820: 790:Topological mixing 744: 724: 701: 610: 533: 508: 488: 468: 362: 212:mathematical model 129:initial conditions 99: 85:An animation of a 79: 13226: 13225: 13183:Benoît Mandelbrot 13083:Self-organization 13019:Natural selection 13009:Pattern formation 12859: 12858: 12723:Benoît Mandelbrot 12688:Martin Gutzwiller 12678:Peter Grassberger 12561: 12560: 12543:Rössler attractor 12291: 12290: 12195:Invariant measure 12117:Lyapunov exponent 11840:Library resources 11813:The Road To Chaos 11808:, Hyperion, 2003. 11678:978-3-642-61719-5 11645:Isabelle Stengers 11453:978-0-231-12662-5 11432:978-0-12-079069-2 11408:978-981-238-647-2 11373:978-981-4374-42-2 11346:978-0-19-852604-9 11327:978-0-387-00177-7 11308:978-0-201-55441-0 11277:; Reilly (1992). 11266:978-0-471-87645-8 11247:978-0-8218-8328-0 11215:978-0-521-83912-9 11194:978-0-19-850840-3 11173:978-0-7382-0453-6 11149:978-0-521-01084-9 11120:978-3-030-70981-5 11097:Orlando, Giuseppe 11088:978-0-471-54571-2 11067:978-0-472-08472-2 11046:978-0-19-959458-0 11025:978-981-02-4073-8 11001:978-0-387-97173-5 10980:978-0-07-025203-5 10961:978-0-387-90819-9 10935:978-0-521-47685-0 10903:978-0-19-956644-0 10863:978-0-8133-4085-2 10841:978-0-8176-4926-5 10818:978-0-521-66385-4 10797:978-0-521-39511-3 10778:978-0-387-94677-1 10743:10.1063/PT.3.1977 10536:Ukrainian Math. J 10367:Ukrainian Math. J 10363:Sharkovskii, A.N. 10246:978-3-642-29503-4 9996:Steven Strogatz, 9839:978-0-309-35922-1 9790:10.1063/5.0015916 9561:978-94-017-3302-1 9446:978-94-010-5772-1 9298:10.1137/120868980 9245:(12): 1282–1301. 9007:Natural Computing 8978:(12): 4052–4057. 8848:978-3-030-27672-0 8783:10.1021/jp804420g 8702:Physical Review E 8510:on 9 August 2011. 8213:978-3-0365-8911-4 7939:10.1063/pt.3.1977 7882:. Penguin Books. 7461:978-0-434-29554-8 7212:Chirikov, Boris. 7090:978-3-540-09120-2 6652:, Hyperion, 2003. 6648:Steven Strogatz, 6100:Physics Letters A 6075:978-3-030-70795-8 5530:978-0-521-55874-7 5435:978-0-471-68755-9 5410:978-1-58488-002-8 5385:978-0-521-58750-1 5138:978-3-540-79356-4 4939:978-0-226-42976-2 4675:Martin Gutzwiller 4492:Rössler attractor 4417:Advected contours 4357:BML traffic model 4336:group development 4150:population models 3939:crisis management 3874:nonlinear systems 3853:Stephen Jay Gould 3847:" put forward by 3821:financial markets 3744:Bernardo Huberman 3669:fractal dimension 3657:Sierpiński gasket 3620:Benoit Mandelbrot 3613:Lorenz attractors 3602:Margaret Hamilton 3583:strange attractor 3471:Lyapunov exponent 3364:Spontaneous order 3190: 3163: 3119: 3066:{\displaystyle x} 3004: 2989: 2975: 2904: 2874: 2858: 2824: 2761: 2745: 2710:{\displaystyle K} 2412:{\displaystyle f} 2394: 2331: 2315: 2271: 2237: 2190: 2174: 2141: 2072:Rössler equations 1999:{\displaystyle t} 1975:{\displaystyle z} 1955:{\displaystyle y} 1935:{\displaystyle x} 1878: 1819: 1766: 1620:fractal dimension 1506: 1500: 1465: 1459: 1424: 1418: 927:, was created as 920:{\displaystyle y} 765:(as discussed in 727:{\displaystyle t} 578:Lyapunov exponent 458:. The values for 367:Robert L. Devaney 259:crisis management 149:self-organization 125:dynamical systems 106:interdisciplinary 91:initial condition 16:(Redirected from 13256: 13034:Sexual selection 12996: 12886: 12879: 12872: 12863: 12862: 12831:Butterfly effect 12743:Itamar Procaccia 12693:Brosl Hasslacher 12590:Elastic pendulum 12518:Duffing equation 12465:Kaplan–Yorke map 12383:Arnold's cat map 12371: 12370: 12347:Stability theory 12332:Dynamical system 12327:Control of chaos 12307: 12299: 12283:Takens's theorem 12215:Poincaré section 12085: 12084: 12067: 12060: 12053: 12044: 12043: 12024: 11883: 11829:, Lambert, 2012. 11825:Antonio Sawaya, 11811:Yoshisuke Ueda, 11788: 11767:Explaining Chaos 11761:, Freeman, 1991. 11754: 11734: 11723: 11702:, Freeman, 1993. 11690:Chance and Chaos 11682: 11662: 11593:, Freeman, 1994. 11586: 11545:Hans Lauwerier, 11528: 11487: 11463:and David Peat, 11457: 11436: 11412: 11350: 11331: 11312: 11284: 11270: 11251: 11219: 11198: 11177: 11158:Strogatz, Steven 11153: 11132: 11092: 11071: 11050: 11029: 11005: 10984: 10965: 10939: 10925:Chaotic dynamics 10918: 10916: 10915: 10886: 10867: 10845: 10826:Collet, Pierre; 10822: 10801: 10782: 10754: 10728: 10707: 10705: 10704: 10698: 10692:. Archived from 10667: 10657: 10655: 10616: 10606: 10580: 10559: 10526: 10480: 10451: 10449: 10448: 10442: 10436:. Archived from 10419: 10391: 10374: 10345: 10344: 10316: 10310: 10309: 10307: 10306: 10292: 10286: 10285: 10257: 10251: 10250: 10232: 10226: 10225: 10197: 10191: 10190: 10171:Jim E. H. Bright 10166: 10160: 10159: 10151: 10145: 10144: 10119:(10): 993–1007. 10108: 10102: 10101: 10073: 10067: 10066: 10058: 10052: 10051: 10043: 10037: 10036: 10007: 10001: 10000:, Hyperion, 2003 9994: 9988: 9987: 9967: 9961: 9960: 9931: 9925: 9924: 9922: 9910: 9904: 9903: 9893: 9869: 9863: 9862: 9850: 9844: 9843: 9816: 9810: 9809: 9761: 9755: 9754: 9706: 9700: 9699: 9697: 9673: 9667: 9666: 9664: 9625: 9613: 9607: 9606: 9604: 9580: 9574: 9573: 9535: 9529: 9528: 9501:10.1038/360121a0 9470: 9459: 9458: 9416: 9405: 9404: 9374: 9368: 9367: 9337: 9331: 9330: 9308: 9302: 9301: 9292:(4): 1200–1214. 9279: 9273: 9272: 9254: 9234: 9228: 9227: 9225: 9224: 9218: 9212:. Archived from 9203: 9194:(3–4): 177–193. 9185: 9176: 9170: 9169: 9143: 9123: 9117: 9116: 9072: 9066: 9065: 9037: 9031: 9030: 9002: 8996: 8995: 8967: 8961: 8960: 8932: 8926: 8925: 8916:(8): 1797–1813. 8905: 8896: 8895: 8859: 8853: 8852: 8834: 8828: 8827: 8825: 8801: 8795: 8794: 8758: 8752: 8751: 8717: 8697: 8691: 8690: 8662: 8656: 8655: 8637: 8631: 8630: 8612: 8606: 8605: 8587: 8581: 8580: 8551: 8545: 8544: 8542: 8518: 8512: 8511: 8506:. Archived from 8500: 8494: 8493: 8483: 8451: 8445: 8444: 8436: 8430: 8429: 8401: 8395: 8394: 8374: 8368: 8367: 8365: 8364: 8358: 8343: 8334: 8328: 8327: 8317: 8285: 8279: 8278: 8276: 8275: 8260: 8254: 8253: 8251: 8249: 8244:on June 10, 2023 8243: 8237:. Archived from 8236: 8227: 8218: 8217: 8205: 8187: 8181: 8180: 8178: 8146: 8140: 8139: 8099: 8093: 8092: 8090: 8058: 8052: 8051: 8049: 8047: 8042: 8033: 8027: 8026: 7992: 7986: 7985: 7957: 7951: 7950: 7924: 7905:"Chaos at fifty" 7900: 7894: 7893: 7875: 7866: 7865: 7853: 7847: 7846: 7834: 7829:. Archived from 7823: 7779: 7773: 7772: 7729: 7723: 7722: 7720: 7719: 7710:. Archived from 7704: 7698: 7695: 7689: 7688: 7670: 7642: 7636: 7635: 7615: 7609: 7608: 7606: 7604: 7595:. Archived from 7548: 7542: 7541: 7525: 7514: 7508: 7507: 7499: 7493: 7492: 7472: 7466: 7465: 7445: 7439: 7438: 7436: 7435: 7420: 7414: 7405: 7399: 7393: 7387: 7386: 7368: 7362: 7361: 7328:(1–4): viii–xv. 7313: 7307: 7306: 7294: 7286:Jean, Bellissard 7282: 7276: 7275: 7235: 7229: 7228: 7218: 7209: 7203: 7202: 7200: 7176: 7170: 7165: 7109: 7103: 7094: 7060: 7054: 7053: 7017: 7005: 6999: 6998: 6962: 6933: 6927: 6920: 6914: 6913: 6901: 6895: 6894: 6882: 6876: 6875: 6849: 6843: 6842: 6840: 6825:Acta Mathematica 6816: 6810: 6809: 6791: 6781: 6757: 6751: 6750: 6748: 6747: 6732: 6726: 6725: 6715: 6706: 6700: 6699: 6659: 6653: 6646: 6640: 6634: 6628: 6622: 6616: 6610: 6604: 6598: 6592: 6591: 6565: 6545: 6539: 6538: 6502: 6496: 6495: 6479: 6473: 6472: 6469:10.1063/1.528136 6444: 6438: 6437: 6412:(3): 1038–1052. 6397: 6391: 6390: 6358: 6352: 6351: 6318:(5): 1701–1723. 6303: 6297: 6296: 6268: 6262: 6261: 6243: 6219: 6213: 6212: 6176: 6170: 6169: 6144:(5): 1289–1303. 6131: 6125: 6124: 6107:(4–5): 271–274. 6094: 6088: 6087: 6049: 6043: 6042: 5994: 5988: 5987: 5977: 5945: 5936: 5935: 5917: 5900:(1): E148–E158. 5885: 5874: 5873: 5855: 5823: 5812: 5811: 5775: 5769: 5768: 5766: 5760:. Archived from 5743: 5715: 5699: 5693: 5688: 5682: 5677: 5671: 5666: 5657: 5656: 5654: 5653: 5638: 5632: 5631: 5625: 5615: 5609: 5608: 5598: 5566: 5560: 5559: 5557: 5556: 5541: 5535: 5534: 5518: 5508: 5502: 5501: 5473: 5467: 5466: 5446: 5440: 5439: 5421: 5415: 5414: 5396: 5390: 5389: 5371: 5365: 5364: 5362: 5361: 5347: 5341: 5330: 5324: 5318: 5312: 5311: 5285: 5249: 5243: 5242: 5240: 5239: 5225: 5219: 5218: 5198: 5189: 5188: 5186: 5185: 5179:researchgate.net 5170: 5161: 5160: 5152: 5143: 5142: 5124: 5118: 5117: 5115: 5083: 5074: 5073: 5071: 5069: 5054: 5048: 5047: 5029: 5009: 5000: 4994: 4988: 4982: 4976: 4975: 4974: 4973: 4957: 4944: 4943: 4927: 4917: 4911: 4905: 4900: 4890: 4881:(3): 1250–1259. 4866: 4855: 4854: 4846: 4835: 4834: 4832: 4831: 4820: 4814: 4813: 4811: 4810: 4795: 4789: 4788: 4786: 4785: 4775: 4764: 4763: 4761: 4760: 4746: 4680:Brosl Hasslacher 4601:Nonlinear system 4591:Ill-conditioning 4561:Control of chaos 4457:Duffing equation 4422:Arnold's cat map 4405: 4400: 4399: 4398: 4391: 4386: 4385: 4166:cardiotocography 4134:predictive model 4023:computer science 3880:'s concept of a 3713:Pierre Hohenberg 3579:fluid turbulence 3560:Soviet physicist 3463:Jacques Hadamard 3451:butterfly effect 3346:. The output of 3345: 3343: 3342: 3337: 3323: 3308: 3306: 3305: 3300: 3295: 3278: 3267: 3266: 3229: 3227: 3226: 3221: 3207: 3199: 3191: 3189: 3185: 3179: 3175: 3169: 3164: 3162: 3161: 3160: 3151: 3145: 3141: 3140: 3135: 3128: 3120: 3118: 3117: 3116: 3107: 3101: 3097: 3096: 3091: 3084: 3072: 3070: 3069: 3064: 3033: 3031: 3030: 3025: 3017: 3013: 3006: 3005: 2997: 2991: 2990: 2982: 2976: 2974: 2960: 2918: 2916: 2915: 2910: 2905: 2903: 2895: 2894: 2893: 2888: 2882: 2881: 2876: 2875: 2867: 2860: 2859: 2851: 2848: 2836: 2825: 2823: 2812: 2783: 2769: 2768: 2763: 2762: 2754: 2747: 2746: 2738: 2716: 2714: 2713: 2708: 2696: 2694: 2693: 2688: 2683: 2660:with wavelength 2659: 2657: 2656: 2651: 2630: 2628: 2627: 2622: 2620: 2619: 2609: 2608: 2603: 2597: 2596: 2587: 2571: 2570: 2549: 2548: 2523: 2521: 2520: 2515: 2513: 2512: 2474: 2472: 2471: 2466: 2418: 2416: 2415: 2410: 2396: 2395: 2387: 2381: 2380: 2358: 2356: 2355: 2350: 2339: 2338: 2333: 2332: 2324: 2317: 2316: 2308: 2289: 2287: 2286: 2281: 2279: 2278: 2273: 2272: 2264: 2245: 2244: 2239: 2238: 2230: 2223: 2222: 2198: 2197: 2192: 2191: 2183: 2176: 2175: 2167: 2143: 2142: 2134: 2128: 2127: 2065: 2063: 2062: 2057: 2045: 2043: 2042: 2037: 2025: 2023: 2022: 2017: 2005: 2003: 2002: 1997: 1981: 1979: 1978: 1973: 1961: 1959: 1958: 1953: 1941: 1939: 1938: 1933: 1918: 1916: 1915: 1910: 1908: 1879: 1877: 1873: 1867: 1863: 1857: 1820: 1818: 1814: 1808: 1804: 1798: 1767: 1765: 1761: 1755: 1751: 1745: 1675: 1565: 1541:Lorenz attractor 1518: 1516: 1515: 1510: 1508: 1502: 1501: 1496: 1487: 1477: 1475: 1474: 1469: 1467: 1461: 1460: 1455: 1446: 1436: 1434: 1433: 1428: 1426: 1420: 1419: 1414: 1405: 1395: 1348:second countable 1313: 1311: 1310: 1305: 1282: 1281: 1265: 1263: 1262: 1257: 1239: 1237: 1236: 1231: 1204: 1202: 1201: 1196: 1137: 1119: 1096: 1094: 1093: 1088: 1080: 1079: 1067: 1066: 1054: 1053: 1031: 1029: 1028: 1023: 1015: 1014: 1002: 1001: 985: 983: 982: 977: 975: 974: 962: 961: 949: 948: 926: 924: 923: 918: 906: 904: 903: 898: 893: 892: 874: 873: 858: 857: 829: 827: 826: 823:{\displaystyle } 821: 753: 751: 750: 745: 734:is the time and 733: 731: 730: 725: 710: 708: 707: 702: 697: 692: 691: 686: 677: 672: 671: 656: 642: 634: 619: 617: 616: 611: 609: 608: 603: 544:butterfly effect 517: 515: 514: 509: 497: 495: 494: 489: 477: 475: 474: 469: 426:Butterfly effect 351: 333: 307:Chaotic dynamics 240:computer science 216:recurrence plots 157:nonlinear system 153:butterfly effect 110:scientific study 77: 76: 74: 73: 70: 67: 55: 51: 43:Lorenz attractor 21: 13264: 13263: 13259: 13258: 13257: 13255: 13254: 13253: 13229: 13228: 13227: 13222: 13196: 13089: 12997: 12988: 12895: 12890: 12860: 12855: 12823: 12817: 12763:Caroline Series 12658:Mary Cartwright 12640: 12634: 12585:Double pendulum 12567: 12557: 12506: 12499: 12425:Exponential map 12376: 12362: 12356: 12314: 12308: 12301: 12287: 12253:Ergodic theorem 12246: 12239: 12230:Stable manifold 12220:Recurrence plot 12136: 12090: 12076: 12071: 12017: 12001:, May 16, 2002) 11951:Wayback Machine 11868: 11865: 11864: 11863: 11848: 11847: 11843: 11836: 11802:Steven Strogatz 11785: 11751: 11720: 11679: 11583: 11525:Deep Simplicity 11454: 11433: 11409: 11365:Chaos in Nature 11357: 11347: 11328: 11309: 11299:10.1119/1.17380 11267: 11248: 11216: 11195: 11174: 11150: 11121: 11089: 11068: 11047: 11026: 11002: 10981: 10973:. McGraw-Hill. 10962: 10936: 10913: 10911: 10904: 10883: 10864: 10842: 10819: 10798: 10779: 10761: 10702: 10700: 10696: 10665: 10653: 10619:Phys. Rev. Lett 10614: 10446: 10444: 10440: 10426:10.2307/2318254 10417:10.1.1.329.5038 10389: 10358: 10353: 10351:Further reading 10348: 10317: 10313: 10304: 10302: 10294: 10293: 10289: 10258: 10254: 10247: 10233: 10229: 10198: 10194: 10167: 10163: 10152: 10148: 10109: 10105: 10074: 10070: 10059: 10055: 10044: 10040: 10008: 10004: 9995: 9991: 9968: 9964: 9932: 9928: 9911: 9907: 9870: 9866: 9851: 9847: 9840: 9818: 9817: 9813: 9762: 9758: 9707: 9703: 9674: 9670: 9662: 9623: 9614: 9610: 9581: 9577: 9562: 9536: 9532: 9471: 9462: 9447: 9417: 9408: 9375: 9371: 9338: 9334: 9309: 9305: 9280: 9276: 9235: 9231: 9222: 9220: 9216: 9201:10.1.1.105.9178 9183: 9177: 9173: 9124: 9120: 9073: 9069: 9038: 9034: 9003: 8999: 8968: 8964: 8933: 8929: 8906: 8899: 8860: 8856: 8849: 8835: 8831: 8802: 8798: 8759: 8755: 8698: 8694: 8663: 8659: 8652: 8638: 8634: 8627: 8613: 8609: 8602: 8588: 8584: 8552: 8548: 8519: 8515: 8502: 8501: 8497: 8452: 8448: 8437: 8433: 8402: 8398: 8375: 8371: 8362: 8360: 8356: 8341: 8335: 8331: 8286: 8282: 8273: 8271: 8261: 8257: 8247: 8245: 8241: 8234: 8228: 8221: 8214: 8188: 8184: 8147: 8143: 8100: 8096: 8059: 8055: 8045: 8043: 8040: 8034: 8030: 8023: 7993: 7989: 7958: 7954: 7901: 7897: 7890: 7876: 7869: 7854: 7850: 7825: 7780: 7776: 7730: 7726: 7717: 7715: 7706: 7705: 7701: 7696: 7692: 7668:10.1.1.418.9339 7643: 7639: 7632: 7616: 7612: 7602: 7600: 7563:(3775): 636–8. 7549: 7545: 7538: 7515: 7511: 7500: 7496: 7473: 7469: 7462: 7446: 7442: 7433: 7431: 7429:Quanta Magazine 7421: 7417: 7406: 7402: 7394: 7390: 7369: 7365: 7314: 7310: 7292: 7283: 7279: 7236: 7232: 7221:Атомная энергия 7216: 7210: 7206: 7177: 7173: 7110: 7106: 7095:Translation of 7091: 7061: 7057: 7028:(1890): 15–17. 7006: 7002: 6934: 6930: 6921: 6917: 6902: 6898: 6883: 6879: 6864: 6850: 6846: 6817: 6813: 6758: 6754: 6745: 6743: 6733: 6729: 6713: 6707: 6703: 6660: 6656: 6647: 6643: 6635: 6631: 6623: 6619: 6611: 6607: 6599: 6595: 6546: 6542: 6503: 6499: 6480: 6476: 6445: 6441: 6398: 6394: 6359: 6355: 6304: 6300: 6269: 6265: 6220: 6216: 6177: 6173: 6132: 6128: 6095: 6091: 6076: 6050: 6046: 5995: 5991: 5946: 5939: 5886: 5877: 5824: 5815: 5780:Phys. Rev. Lett 5776: 5772: 5764: 5750:10.2307/2318254 5741:10.1.1.329.5038 5713: 5700: 5696: 5689: 5685: 5678: 5674: 5667: 5660: 5651: 5649: 5639: 5635: 5616: 5612: 5567: 5563: 5554: 5552: 5551:. 16 April 2008 5543: 5542: 5538: 5531: 5509: 5505: 5490:10.2307/2975629 5474: 5470: 5447: 5443: 5436: 5422: 5418: 5411: 5397: 5393: 5386: 5372: 5368: 5359: 5357: 5349: 5348: 5344: 5331: 5327: 5319: 5315: 5260:(5066): 56–62. 5250: 5246: 5237: 5235: 5227: 5226: 5222: 5215: 5199: 5192: 5183: 5181: 5171: 5164: 5153: 5146: 5139: 5125: 5121: 5084: 5077: 5067: 5065: 5055: 5051: 5010: 5003: 4995: 4991: 4983: 4979: 4971: 4969: 4958: 4947: 4940: 4918: 4914: 4867: 4858: 4847: 4838: 4829: 4827: 4821: 4817: 4808: 4806: 4796: 4792: 4783: 4781: 4777: 4776: 4767: 4758: 4756: 4748: 4747: 4743: 4739: 4734: 4640: 4626:Shadowing lemma 4521:Amplitude death 4511: 4467:Economic bubble 4452:Double pendulum 4401: 4396: 4394: 4387: 4380: 4377: 4288:quantum physics 4268: 4255: 4246: 4230: 4214:quasi-periodic- 4146: 4138:passive walking 4129: 4077: 4071: 3987: 3978: 3975: 3973: 3971: 3969: 3967: 3960: 3947: 3800:scale-invariant 3775:Kurt Wiesenfeld 3644:infinitesimally 3425: 3417:field-theoretic 3413:order parameter 3366: 3319: 3314: 3311: 3310: 3291: 3274: 3262: 3258: 3256: 3253: 3252: 3203: 3195: 3181: 3180: 3171: 3170: 3168: 3156: 3152: 3147: 3146: 3136: 3131: 3130: 3129: 3127: 3112: 3108: 3103: 3102: 3092: 3087: 3086: 3085: 3083: 3081: 3078: 3077: 3058: 3055: 3054: 2996: 2995: 2981: 2980: 2964: 2959: 2958: 2954: 2949: 2946: 2945: 2925: 2896: 2889: 2884: 2883: 2877: 2866: 2865: 2864: 2850: 2849: 2844: 2837: 2835: 2813: 2784: 2782: 2764: 2753: 2752: 2751: 2737: 2736: 2728: 2725: 2724: 2702: 2699: 2698: 2679: 2665: 2662: 2661: 2636: 2633: 2632: 2604: 2599: 2598: 2592: 2588: 2583: 2576: 2572: 2566: 2562: 2538: 2534: 2532: 2529: 2528: 2508: 2504: 2487: 2484: 2483: 2424: 2421: 2420: 2386: 2385: 2376: 2372: 2364: 2361: 2360: 2334: 2323: 2322: 2321: 2307: 2306: 2298: 2295: 2294: 2274: 2263: 2262: 2261: 2240: 2229: 2228: 2227: 2218: 2214: 2193: 2182: 2181: 2180: 2166: 2165: 2133: 2132: 2117: 2113: 2111: 2108: 2107: 2104: 2066:are the system 2051: 2048: 2047: 2031: 2028: 2027: 2011: 2008: 2007: 1991: 1988: 1987: 1967: 1964: 1963: 1947: 1944: 1943: 1927: 1924: 1923: 1906: 1905: 1880: 1869: 1868: 1859: 1858: 1856: 1853: 1852: 1821: 1810: 1809: 1800: 1799: 1797: 1794: 1793: 1768: 1757: 1756: 1747: 1746: 1744: 1740: 1738: 1735: 1734: 1682:period-doubling 1659: 1646: 1628: 1552: 1533: 1495: 1488: 1485: 1483: 1480: 1479: 1454: 1447: 1444: 1442: 1439: 1438: 1413: 1406: 1403: 1401: 1398: 1397: 1382: 1375:periodic orbits 1368: 1277: 1273: 1271: 1268: 1267: 1245: 1242: 1241: 1240:, there exists 1213: 1210: 1209: 1178: 1175: 1174: 1171: 1121: 1106: 1075: 1071: 1062: 1058: 1043: 1039: 1037: 1034: 1033: 1010: 1006: 997: 993: 991: 988: 987: 970: 966: 957: 953: 938: 934: 932: 929: 928: 912: 909: 908: 888: 884: 869: 865: 847: 843: 841: 838: 837: 803: 800: 799: 792: 779: 777:Non-periodicity 739: 736: 735: 719: 716: 715: 693: 687: 682: 681: 673: 664: 660: 652: 638: 630: 628: 625: 624: 604: 599: 598: 593: 590: 589: 550:in 1972 to the 503: 500: 499: 483: 480: 479: 463: 460: 459: 428: 422: 392:periodic orbits 335: 320: 309: 287: 141:self-similarity 71: 68: 65: 64: 62: 57: 53: 46: 35: 28: 23: 22: 15: 12: 11: 5: 13262: 13252: 13251: 13246: 13241: 13224: 13223: 13221: 13220: 13215: 13210: 13204: 13202: 13198: 13197: 13195: 13194: 13193: 13192: 13180: 13175: 13174: 13173: 13161: 13160: 13159: 13147: 13145:Wilson Bentley 13142: 13140:Joseph Plateau 13137: 13132: 13127: 13126: 13125: 13113: 13108: 13103: 13097: 13095: 13091: 13090: 13088: 13087: 13086: 13085: 13080: 13078:Plateau's laws 13075: 13073:Fluid dynamics 13070: 13060: 13059: 13058: 13053: 13048: 13038: 13037: 13036: 13031: 13026: 13021: 13011: 13005: 13003: 12999: 12998: 12991: 12989: 12987: 12986: 12981: 12976: 12971: 12966: 12965: 12964: 12959: 12954: 12949: 12939: 12934: 12929: 12924: 12919: 12914: 12909: 12903: 12901: 12897: 12896: 12889: 12888: 12881: 12874: 12866: 12857: 12856: 12854: 12853: 12848: 12846:Predictability 12843: 12838: 12833: 12827: 12825: 12819: 12818: 12816: 12815: 12813:Lai-Sang Young 12810: 12808:James A. Yorke 12805: 12803:Amie Wilkinson 12800: 12795: 12790: 12785: 12780: 12775: 12770: 12765: 12760: 12755: 12750: 12745: 12740: 12738:Henri Poincaré 12735: 12730: 12725: 12720: 12715: 12710: 12705: 12700: 12695: 12690: 12685: 12680: 12675: 12670: 12665: 12660: 12655: 12650: 12644: 12642: 12636: 12635: 12633: 12632: 12627: 12622: 12617: 12612: 12607: 12605:Kicked rotator 12602: 12597: 12592: 12587: 12582: 12577: 12575:Chua's circuit 12571: 12569: 12563: 12562: 12559: 12558: 12556: 12555: 12550: 12545: 12540: 12535: 12530: 12525: 12520: 12515: 12509: 12507: 12504: 12501: 12500: 12498: 12497: 12495:Zaslavskii map 12492: 12490:Tinkerbell map 12487: 12482: 12477: 12472: 12467: 12462: 12457: 12452: 12447: 12442: 12437: 12432: 12427: 12422: 12421: 12420: 12410: 12405: 12400: 12395: 12390: 12385: 12379: 12377: 12374: 12368: 12358: 12357: 12355: 12354: 12349: 12344: 12339: 12337:Ergodic theory 12334: 12329: 12324: 12318: 12316: 12310: 12309: 12294: 12292: 12289: 12288: 12286: 12285: 12280: 12275: 12270: 12265: 12260: 12255: 12249: 12247: 12244: 12241: 12240: 12238: 12237: 12232: 12227: 12222: 12217: 12212: 12207: 12202: 12197: 12192: 12187: 12182: 12177: 12172: 12167: 12162: 12157: 12152: 12147: 12141: 12138: 12137: 12135: 12134: 12129: 12127:Periodic point 12124: 12119: 12114: 12109: 12104: 12099: 12093: 12091: 12088: 12082: 12078: 12077: 12070: 12069: 12062: 12055: 12047: 12041: 12040: 12016: 12015:Copyright note 12013: 12012: 12011: 12002: 11995:"Chaos Theory" 11992: 11979: 11973: 11964: 11959: 11953: 11937: 11931: 11925: 11912: 11907: 11901: 11895: 11890: 11884: 11862: 11861: 11856: 11850: 11849: 11838: 11837: 11835: 11834:External links 11832: 11831: 11830: 11823: 11816: 11809: 11799: 11789: 11783: 11762: 11755: 11749: 11724: 11719:978-0691152721 11718: 11703: 11696:Ivars Peterson 11693: 11683: 11677: 11652: 11651:, Bantam 1984. 11641:Ilya Prigogine 11638: 11628: 11618: 11611:Nuria Perpinya 11608: 11594: 11587: 11581: 11560: 11550: 11543: 11538:Arvind Kumar, 11536: 11529: 11523:John Gribbin. 11520: 11508: 11498: 11488: 11475: 11468: 11458: 11452: 11437: 11431: 11413: 11407: 11376: 11356: 11353: 11352: 11351: 11345: 11332: 11326: 11313: 11307: 11271: 11265: 11252: 11246: 11224:Teschl, Gerald 11220: 11214: 11199: 11193: 11178: 11172: 11154: 11148: 11133: 11119: 11093: 11087: 11072: 11066: 11051: 11045: 11030: 11024: 11006: 11000: 10985: 10979: 10966: 10960: 10948:Holmes, Philip 10940: 10934: 10919: 10902: 10887: 10881: 10868: 10862: 10846: 10840: 10834:. Birkhauser. 10823: 10817: 10802: 10796: 10783: 10777: 10760: 10757: 10756: 10755: 10708: 10658: 10656:on 2013-04-26. 10607: 10571:(4): 289–301. 10560: 10542:(8): 1242–57. 10531: 10528:Online version 10481: 10452: 10402:(10): 985–92. 10375: 10357: 10354: 10352: 10349: 10347: 10346: 10327:(3): 253–260. 10311: 10287: 10252: 10245: 10227: 10208:(5): 544–562. 10192: 10181:(4): 309–318. 10161: 10146: 10103: 10084:(2): 295–315. 10068: 10053: 10038: 10019:(3): 439–443. 10002: 9989: 9978:(4): 339–353. 9962: 9926: 9905: 9864: 9845: 9838: 9830:10.17226/21272 9824:. 1966-01-01. 9811: 9756: 9701: 9668: 9634:(3): 265–281. 9608: 9575: 9560: 9542:(1 ed.). 9530: 9460: 9445: 9427:(1 ed.). 9406: 9387:(2): 163–168. 9369: 9332: 9321:(4): 409–423. 9303: 9274: 9252:10.1.1.17.4861 9229: 9171: 9118: 9067: 9032: 9013:(1): 101–107. 8997: 8962: 8943:(2): 408–419. 8927: 8897: 8870:(2): 207–230. 8854: 8847: 8829: 8816:(2): 127–136. 8796: 8753: 8692: 8657: 8651:978-0471139386 8650: 8632: 8626:978-0471585244 8625: 8607: 8600: 8582: 8563:(3): 257–276. 8546: 8513: 8495: 8446: 8431: 8412:(1): 227–229. 8396: 8385:(1): 256–266. 8369: 8329: 8280: 8255: 8219: 8212: 8182: 8141: 8094: 8053: 8028: 8021: 7987: 7952: 7895: 7889:978-0143113454 7888: 7867: 7848: 7833:on 2007-12-14. 7774: 7733:Huberman, B.A. 7724: 7699: 7690: 7637: 7631:978-0716711865 7630: 7610: 7543: 7536: 7509: 7494: 7483:(3): 224–236. 7467: 7460: 7440: 7415: 7400: 7388: 7363: 7308: 7277: 7250:(6): 464–470. 7230: 7204: 7171: 7104: 7089: 7055: 7018:Reprinted in: 7000: 6973:(1890): 9–13. 6963:Reprinted in: 6928: 6915: 6896: 6877: 6862: 6844: 6831:(1–2): 1–270. 6811: 6772:(2): 643–644. 6752: 6727: 6701: 6654: 6641: 6629: 6617: 6605: 6593: 6540: 6513:(1): 145–147. 6497: 6474: 6439: 6392: 6353: 6298: 6263: 6234:(1): 231–255. 6214: 6187:(3): 617–633. 6171: 6126: 6089: 6074: 6044: 6009:(3): 263–277. 5989: 5937: 5875: 5813: 5770: 5767:on 2009-12-29. 5726:(10): 985–92. 5694: 5683: 5672: 5658: 5633: 5610: 5561: 5536: 5529: 5503: 5468: 5457:(4): 332–334. 5441: 5434: 5416: 5409: 5401:Discrete Chaos 5391: 5384: 5366: 5342: 5332:Definition of 5325: 5313: 5244: 5220: 5213: 5190: 5162: 5144: 5137: 5119: 5098:(2): 130–141. 5075: 5049: 5020:(1): 195–220. 5001: 4989: 4977: 4945: 4938: 4912: 4856: 4836: 4815: 4790: 4765: 4740: 4738: 4735: 4733: 4732: 4727: 4725:James A. Yorke 4722: 4717: 4712: 4707: 4702: 4697: 4695:Norman Packard 4692: 4687: 4682: 4677: 4672: 4667: 4662: 4657: 4652: 4646: 4639: 4638: 4633: 4628: 4623: 4618: 4613: 4611:Predictability 4608: 4603: 4598: 4593: 4588: 4583: 4581:Mandelbrot set 4578: 4573: 4568: 4563: 4558: 4553: 4551:Chaotic mixing 4548: 4543: 4538: 4533: 4528: 4523: 4517: 4510: 4509: 4504: 4499: 4494: 4489: 4484: 4479: 4474: 4469: 4464: 4459: 4454: 4449: 4444: 4439: 4437:Chua's circuit 4434: 4429: 4424: 4419: 4413: 4407: 4406: 4392: 4376: 4373: 4284:moons of Pluto 4267: 4264: 4254: 4251: 4245: 4242: 4229: 4226: 4212:may result in 4145: 4142: 4140:biped robots. 4128: 4125: 4101:stream ciphers 4093:hash functions 4073:Main article: 4070: 4067: 3986: 3983: 3964: 3959: 3956: 3946: 3943: 3908:social systems 3882:paradigm shift 3849:Niles Eldredge 3760:cardiac cycles 3690:James A. Yorke 3478:ergodic theory 3455:Henri Poincaré 3424: 3421: 3370:Kuramoto model 3365: 3362: 3358:Chua's circuit 3335: 3332: 3329: 3326: 3322: 3318: 3298: 3294: 3290: 3287: 3284: 3281: 3277: 3273: 3270: 3265: 3261: 3231: 3230: 3219: 3216: 3213: 3210: 3206: 3202: 3198: 3194: 3188: 3184: 3178: 3174: 3167: 3159: 3155: 3150: 3144: 3139: 3134: 3126: 3123: 3115: 3111: 3106: 3100: 3095: 3090: 3062: 3041:jerk equations 3037: 3036: 3035: 3034: 3023: 3020: 3016: 3012: 3009: 3003: 3000: 2994: 2988: 2985: 2979: 2973: 2970: 2967: 2963: 2957: 2953: 2924: 2921: 2908: 2902: 2899: 2892: 2887: 2880: 2873: 2870: 2863: 2857: 2854: 2847: 2843: 2840: 2834: 2831: 2828: 2822: 2819: 2816: 2811: 2808: 2805: 2802: 2799: 2796: 2793: 2790: 2787: 2781: 2778: 2775: 2772: 2767: 2760: 2757: 2750: 2744: 2741: 2735: 2732: 2706: 2686: 2682: 2678: 2675: 2672: 2669: 2649: 2646: 2643: 2640: 2618: 2615: 2612: 2607: 2602: 2595: 2591: 2586: 2582: 2579: 2575: 2569: 2565: 2561: 2558: 2555: 2552: 2547: 2544: 2541: 2537: 2511: 2507: 2503: 2500: 2497: 2494: 2491: 2464: 2461: 2458: 2455: 2452: 2449: 2446: 2443: 2440: 2437: 2434: 2431: 2428: 2408: 2405: 2402: 2399: 2393: 2390: 2384: 2379: 2375: 2371: 2368: 2348: 2345: 2342: 2337: 2330: 2327: 2320: 2314: 2311: 2305: 2302: 2277: 2270: 2267: 2260: 2257: 2254: 2251: 2248: 2243: 2236: 2233: 2226: 2221: 2217: 2213: 2210: 2207: 2204: 2201: 2196: 2189: 2186: 2179: 2173: 2170: 2164: 2161: 2158: 2155: 2152: 2149: 2146: 2140: 2137: 2131: 2126: 2123: 2120: 2116: 2103: 2100: 2055: 2035: 2015: 1995: 1971: 1951: 1931: 1920: 1919: 1904: 1901: 1898: 1895: 1892: 1889: 1886: 1883: 1881: 1876: 1872: 1866: 1862: 1855: 1854: 1851: 1848: 1845: 1842: 1839: 1836: 1833: 1830: 1827: 1824: 1822: 1817: 1813: 1807: 1803: 1796: 1795: 1792: 1789: 1786: 1783: 1780: 1777: 1774: 1771: 1769: 1764: 1760: 1754: 1750: 1743: 1742: 1713:linear systems 1698:dimensionality 1645: 1642: 1627: 1624: 1532: 1529: 1505: 1499: 1494: 1491: 1464: 1458: 1453: 1450: 1423: 1417: 1412: 1409: 1367: 1364: 1303: 1300: 1297: 1294: 1291: 1288: 1285: 1280: 1276: 1255: 1252: 1249: 1229: 1226: 1223: 1220: 1217: 1194: 1191: 1188: 1185: 1182: 1170: 1167: 1138:also displays 1086: 1083: 1078: 1074: 1070: 1065: 1061: 1057: 1052: 1049: 1046: 1042: 1021: 1018: 1013: 1009: 1005: 1000: 996: 973: 969: 965: 960: 956: 952: 947: 944: 941: 937: 916: 896: 891: 887: 883: 880: 877: 872: 868: 864: 861: 856: 853: 850: 846: 819: 816: 813: 810: 807: 791: 788: 778: 775: 743: 723: 712: 711: 700: 696: 690: 685: 680: 676: 670: 667: 663: 659: 655: 651: 648: 645: 641: 637: 633: 607: 602: 597: 507: 487: 467: 424:Main article: 421: 418: 396: 395: 385: 378: 308: 305: 286: 283: 139:, repetition, 137:feedback loops 112:and branch of 41:A plot of the 26: 9: 6: 4: 3: 2: 13261: 13250: 13247: 13245: 13242: 13240: 13237: 13236: 13234: 13219: 13216: 13214: 13211: 13209: 13206: 13205: 13203: 13199: 13191: 13190: 13186: 13185: 13184: 13181: 13179: 13176: 13172: 13171: 13167: 13166: 13165: 13162: 13158: 13157: 13153: 13152: 13151: 13148: 13146: 13143: 13141: 13138: 13136: 13135:Ernst Haeckel 13133: 13131: 13130:Adolf Zeising 13128: 13124: 13123: 13119: 13118: 13117: 13114: 13112: 13109: 13107: 13104: 13102: 13099: 13098: 13096: 13092: 13084: 13081: 13079: 13076: 13074: 13071: 13069: 13066: 13065: 13064: 13061: 13057: 13054: 13052: 13049: 13047: 13044: 13043: 13042: 13039: 13035: 13032: 13030: 13027: 13025: 13022: 13020: 13017: 13016: 13015: 13012: 13010: 13007: 13006: 13004: 13000: 12995: 12985: 12982: 12980: 12977: 12975: 12974:Vortex street 12972: 12970: 12967: 12963: 12960: 12958: 12955: 12953: 12952:Quasicrystals 12950: 12948: 12945: 12944: 12943: 12940: 12938: 12935: 12933: 12930: 12928: 12925: 12923: 12920: 12918: 12915: 12913: 12910: 12908: 12905: 12904: 12902: 12898: 12894: 12887: 12882: 12880: 12875: 12873: 12868: 12867: 12864: 12852: 12849: 12847: 12844: 12842: 12841:Edge of chaos 12839: 12837: 12834: 12832: 12829: 12828: 12826: 12820: 12814: 12811: 12809: 12806: 12804: 12801: 12799: 12798:Marcelo Viana 12796: 12794: 12791: 12789: 12788:Audrey Terras 12786: 12784: 12783:Floris Takens 12781: 12779: 12776: 12774: 12771: 12769: 12766: 12764: 12761: 12759: 12756: 12754: 12751: 12749: 12746: 12744: 12741: 12739: 12736: 12734: 12731: 12729: 12726: 12724: 12721: 12719: 12716: 12714: 12711: 12709: 12706: 12704: 12701: 12699: 12696: 12694: 12691: 12689: 12686: 12684: 12683:Celso Grebogi 12681: 12679: 12676: 12674: 12671: 12669: 12666: 12664: 12663:Chen Guanrong 12661: 12659: 12656: 12654: 12651: 12649: 12648:Michael Berry 12646: 12645: 12643: 12637: 12631: 12628: 12626: 12623: 12621: 12618: 12616: 12613: 12611: 12608: 12606: 12603: 12601: 12598: 12596: 12593: 12591: 12588: 12586: 12583: 12581: 12578: 12576: 12573: 12572: 12570: 12564: 12554: 12551: 12549: 12546: 12544: 12541: 12539: 12536: 12534: 12531: 12529: 12526: 12524: 12523:Lorenz system 12521: 12519: 12516: 12514: 12511: 12510: 12508: 12502: 12496: 12493: 12491: 12488: 12486: 12483: 12481: 12478: 12476: 12473: 12471: 12470:Langton's ant 12468: 12466: 12463: 12461: 12458: 12456: 12453: 12451: 12448: 12446: 12445:Horseshoe map 12443: 12441: 12438: 12436: 12433: 12431: 12428: 12426: 12423: 12419: 12416: 12415: 12414: 12411: 12409: 12406: 12404: 12401: 12399: 12396: 12394: 12391: 12389: 12386: 12384: 12381: 12380: 12378: 12372: 12369: 12366: 12359: 12353: 12350: 12348: 12345: 12343: 12342:Quantum chaos 12340: 12338: 12335: 12333: 12330: 12328: 12325: 12323: 12320: 12319: 12317: 12311: 12306: 12302: 12298: 12284: 12281: 12279: 12276: 12274: 12271: 12269: 12266: 12264: 12261: 12259: 12256: 12254: 12251: 12250: 12248: 12242: 12236: 12233: 12231: 12228: 12226: 12223: 12221: 12218: 12216: 12213: 12211: 12208: 12206: 12203: 12201: 12198: 12196: 12193: 12191: 12188: 12186: 12183: 12181: 12178: 12176: 12173: 12171: 12168: 12166: 12163: 12161: 12158: 12156: 12153: 12151: 12150:Arnold tongue 12148: 12146: 12143: 12142: 12139: 12133: 12130: 12128: 12125: 12123: 12120: 12118: 12115: 12113: 12110: 12108: 12105: 12103: 12100: 12098: 12095: 12094: 12092: 12086: 12083: 12079: 12075: 12068: 12063: 12061: 12056: 12054: 12049: 12048: 12045: 12038: 12037: 12032: 12028: 12023: 12019: 12018: 12010: 12006: 12003: 12000: 11996: 11993: 11990: 11989: 11984: 11980: 11977: 11974: 11972: 11968: 11965: 11963: 11960: 11957: 11954: 11952: 11948: 11945: 11943: 11938: 11935: 11932: 11929: 11926: 11924: 11920: 11916: 11913: 11911: 11908: 11905: 11904:ChaosBook.org 11902: 11899: 11896: 11894: 11891: 11888: 11885: 11881: 11877: 11876: 11871: 11867: 11866: 11860: 11857: 11855: 11852: 11851: 11846: 11841: 11828: 11824: 11821: 11817: 11814: 11810: 11807: 11803: 11800: 11797: 11793: 11790: 11786: 11784:9780511554544 11780: 11776: 11772: 11768: 11763: 11760: 11756: 11752: 11750:9780521362726 11746: 11742: 11738: 11733: 11732: 11725: 11721: 11715: 11711: 11710: 11704: 11701: 11697: 11694: 11691: 11687: 11684: 11680: 11674: 11670: 11666: 11661: 11660: 11653: 11650: 11646: 11642: 11639: 11636: 11632: 11629: 11626: 11622: 11619: 11616: 11612: 11609: 11606: 11602: 11601:Dietmar Saupe 11598: 11595: 11592: 11588: 11584: 11582:9781860949548 11578: 11574: 11570: 11566: 11561: 11558: 11554: 11553:Edward Lorenz 11551: 11548: 11544: 11541: 11537: 11534: 11530: 11526: 11521: 11518: 11517: 11512: 11509: 11506: 11502: 11499: 11496: 11492: 11489: 11485: 11481: 11476: 11473: 11469: 11466: 11462: 11459: 11455: 11449: 11445: 11444: 11438: 11434: 11428: 11424: 11423: 11418: 11414: 11410: 11404: 11400: 11396: 11392: 11388: 11384: 11383: 11377: 11374: 11370: 11366: 11362: 11359: 11358: 11348: 11342: 11338: 11333: 11329: 11323: 11319: 11314: 11310: 11304: 11300: 11296: 11292: 11288: 11283: 11282: 11276: 11272: 11268: 11262: 11258: 11253: 11249: 11243: 11239: 11235: 11231: 11230: 11225: 11221: 11217: 11211: 11207: 11206: 11200: 11196: 11190: 11186: 11185: 11179: 11175: 11169: 11165: 11164: 11159: 11155: 11151: 11145: 11141: 11140: 11134: 11130: 11126: 11122: 11116: 11112: 11108: 11104: 11103: 11098: 11094: 11090: 11084: 11080: 11079: 11073: 11069: 11063: 11059: 11058: 11052: 11048: 11042: 11038: 11037: 11031: 11027: 11021: 11017: 11016: 11011: 11007: 11003: 10997: 10993: 10992: 10986: 10982: 10976: 10972: 10967: 10963: 10957: 10953: 10949: 10945: 10941: 10937: 10931: 10927: 10926: 10920: 10909: 10905: 10899: 10895: 10894: 10888: 10884: 10882:0-8493-8493-1 10878: 10875:. CRC Press. 10874: 10869: 10865: 10859: 10855: 10851: 10847: 10843: 10837: 10833: 10829: 10824: 10820: 10814: 10810: 10809: 10803: 10799: 10793: 10789: 10784: 10780: 10774: 10770: 10769: 10763: 10762: 10752: 10748: 10744: 10740: 10736: 10732: 10727: 10722: 10718: 10714: 10713:Physics Today 10709: 10699:on 2012-10-30 10695: 10691: 10687: 10683: 10679: 10675: 10671: 10664: 10659: 10652: 10648: 10644: 10640: 10636: 10632: 10628: 10625:(4): 044101. 10624: 10620: 10613: 10608: 10604: 10600: 10596: 10592: 10588: 10584: 10579: 10574: 10570: 10566: 10561: 10557: 10553: 10549: 10545: 10541: 10537: 10532: 10529: 10524: 10520: 10516: 10512: 10508: 10504: 10503: 10498: 10494: 10490: 10486: 10482: 10478: 10474: 10470: 10466: 10462: 10458: 10453: 10443:on 2009-12-29 10439: 10435: 10431: 10427: 10423: 10418: 10413: 10409: 10405: 10401: 10397: 10396: 10388: 10384: 10380: 10376: 10372: 10368: 10364: 10360: 10359: 10342: 10338: 10334: 10330: 10326: 10322: 10315: 10301: 10297: 10291: 10283: 10279: 10275: 10271: 10267: 10263: 10256: 10248: 10242: 10238: 10231: 10223: 10219: 10215: 10211: 10207: 10203: 10196: 10188: 10184: 10180: 10176: 10172: 10165: 10157: 10150: 10142: 10138: 10134: 10130: 10126: 10122: 10118: 10114: 10107: 10099: 10095: 10091: 10087: 10083: 10079: 10072: 10064: 10057: 10049: 10042: 10034: 10030: 10026: 10022: 10018: 10014: 10006: 9999: 9993: 9985: 9981: 9977: 9973: 9966: 9958: 9954: 9950: 9946: 9942: 9938: 9930: 9921: 9916: 9909: 9901: 9897: 9892: 9887: 9883: 9879: 9875: 9868: 9860: 9856: 9849: 9841: 9835: 9831: 9827: 9823: 9822: 9815: 9807: 9803: 9799: 9795: 9791: 9787: 9783: 9779: 9776:(8): 083129. 9775: 9771: 9767: 9760: 9752: 9748: 9744: 9740: 9736: 9732: 9728: 9724: 9720: 9716: 9712: 9705: 9696: 9691: 9687: 9683: 9679: 9672: 9661: 9657: 9653: 9649: 9645: 9641: 9637: 9633: 9629: 9622: 9618: 9617:Brooks, Chris 9612: 9603: 9598: 9594: 9590: 9586: 9579: 9571: 9567: 9563: 9557: 9553: 9549: 9545: 9541: 9534: 9526: 9522: 9518: 9514: 9510: 9506: 9502: 9498: 9494: 9490: 9486: 9482: 9478: 9477: 9469: 9467: 9465: 9456: 9452: 9448: 9442: 9438: 9434: 9430: 9426: 9422: 9415: 9413: 9411: 9402: 9398: 9394: 9390: 9386: 9382: 9381: 9373: 9365: 9361: 9357: 9353: 9350:(1–4): 1–20. 9349: 9345: 9344: 9336: 9328: 9324: 9320: 9316: 9315: 9307: 9299: 9295: 9291: 9287: 9286: 9278: 9270: 9266: 9262: 9258: 9253: 9248: 9244: 9240: 9233: 9219:on 2017-08-12 9215: 9211: 9207: 9202: 9197: 9193: 9189: 9182: 9175: 9167: 9163: 9159: 9155: 9151: 9147: 9142: 9137: 9133: 9129: 9122: 9114: 9110: 9106: 9102: 9098: 9094: 9090: 9086: 9082: 9078: 9071: 9063: 9059: 9055: 9051: 9047: 9043: 9036: 9028: 9024: 9020: 9016: 9012: 9008: 9001: 8993: 8989: 8985: 8981: 8977: 8973: 8966: 8958: 8954: 8950: 8946: 8942: 8938: 8931: 8923: 8919: 8915: 8911: 8904: 8902: 8893: 8889: 8885: 8881: 8877: 8873: 8869: 8865: 8858: 8850: 8844: 8840: 8833: 8824: 8819: 8815: 8811: 8807: 8800: 8792: 8788: 8784: 8780: 8776: 8772: 8768: 8764: 8757: 8749: 8745: 8741: 8737: 8733: 8729: 8725: 8721: 8716: 8711: 8708:(4): 045202. 8707: 8703: 8696: 8688: 8684: 8680: 8676: 8672: 8668: 8661: 8653: 8647: 8643: 8636: 8628: 8622: 8618: 8611: 8603: 8601:9780471463085 8597: 8593: 8586: 8578: 8574: 8570: 8566: 8562: 8558: 8550: 8541: 8536: 8532: 8528: 8524: 8517: 8509: 8505: 8499: 8491: 8487: 8482: 8477: 8473: 8469: 8466:(2): 023048. 8465: 8461: 8457: 8450: 8442: 8435: 8427: 8423: 8419: 8415: 8411: 8407: 8400: 8392: 8388: 8384: 8380: 8373: 8355: 8351: 8347: 8340: 8333: 8325: 8321: 8316: 8311: 8307: 8303: 8299: 8295: 8291: 8284: 8270: 8266: 8259: 8240: 8233: 8226: 8224: 8215: 8209: 8204: 8199: 8195: 8194: 8186: 8177: 8172: 8168: 8164: 8160: 8156: 8152: 8145: 8137: 8133: 8129: 8125: 8121: 8117: 8113: 8109: 8105: 8098: 8089: 8084: 8080: 8076: 8072: 8068: 8064: 8057: 8039: 8032: 8024: 8022:9780472106387 8018: 8014: 8010: 8006: 8002: 7998: 7991: 7983: 7979: 7975: 7971: 7967: 7963: 7956: 7948: 7944: 7940: 7936: 7932: 7928: 7923: 7918: 7914: 7910: 7906: 7899: 7891: 7885: 7881: 7874: 7872: 7863: 7859: 7852: 7844: 7840: 7832: 7828: 7821: 7817: 7813: 7809: 7805: 7801: 7797: 7793: 7789: 7785: 7778: 7770: 7766: 7762: 7758: 7754: 7750: 7746: 7742: 7738: 7734: 7728: 7714:on 2024-05-25 7713: 7709: 7703: 7694: 7686: 7682: 7678: 7674: 7669: 7664: 7660: 7656: 7652: 7648: 7641: 7633: 7627: 7623: 7622: 7614: 7598: 7594: 7590: 7586: 7582: 7578: 7574: 7570: 7566: 7562: 7558: 7554: 7547: 7539: 7537:9780465043552 7533: 7529: 7524: 7523: 7513: 7505: 7498: 7490: 7486: 7482: 7478: 7471: 7463: 7457: 7453: 7452: 7444: 7430: 7426: 7419: 7413: 7409: 7404: 7397: 7392: 7384: 7380: 7376: 7375: 7367: 7359: 7355: 7351: 7347: 7343: 7339: 7335: 7331: 7327: 7323: 7319: 7312: 7304: 7300: 7299: 7291: 7287: 7281: 7273: 7269: 7265: 7261: 7257: 7253: 7249: 7245: 7241: 7234: 7226: 7222: 7215: 7208: 7199: 7194: 7190: 7186: 7182: 7175: 7169: 7163: 7159: 7155: 7151: 7147: 7143: 7139: 7135: 7131: 7127: 7123: 7119: 7115: 7108: 7102: 7098: 7092: 7086: 7082: 7078: 7074: 7070: 7066: 7059: 7051: 7047: 7043: 7039: 7035: 7031: 7027: 7023: 7016:(6): 538–540. 7015: 7011: 7004: 6996: 6992: 6988: 6984: 6980: 6976: 6972: 6968: 6960: 6956: 6952: 6948: 6947: 6942: 6938: 6932: 6925: 6919: 6911: 6907: 6900: 6892: 6888: 6881: 6873: 6869: 6865: 6863:9783319528984 6859: 6855: 6848: 6839: 6834: 6830: 6826: 6822: 6815: 6807: 6803: 6799: 6795: 6790: 6785: 6780: 6775: 6771: 6767: 6763: 6756: 6742: 6741:Physics World 6738: 6731: 6723: 6719: 6712: 6705: 6697: 6693: 6689: 6685: 6681: 6677: 6674:(5): 114611. 6673: 6669: 6665: 6658: 6651: 6645: 6638: 6633: 6626: 6621: 6614: 6609: 6603: 6597: 6589: 6585: 6581: 6577: 6573: 6569: 6564: 6559: 6556:(4): 109638. 6555: 6551: 6544: 6536: 6532: 6528: 6524: 6520: 6516: 6512: 6508: 6501: 6493: 6489: 6485: 6478: 6470: 6466: 6462: 6458: 6454: 6450: 6443: 6435: 6431: 6427: 6423: 6419: 6415: 6411: 6407: 6403: 6396: 6388: 6384: 6380: 6376: 6372: 6368: 6364: 6357: 6349: 6345: 6341: 6337: 6333: 6329: 6325: 6321: 6317: 6313: 6309: 6302: 6294: 6290: 6286: 6282: 6278: 6274: 6267: 6259: 6255: 6251: 6247: 6242: 6237: 6233: 6229: 6225: 6218: 6210: 6206: 6202: 6198: 6194: 6190: 6186: 6182: 6175: 6167: 6163: 6159: 6155: 6151: 6147: 6143: 6139: 6138: 6130: 6122: 6118: 6114: 6110: 6106: 6102: 6101: 6093: 6085: 6081: 6077: 6071: 6067: 6063: 6059: 6055: 6048: 6040: 6036: 6032: 6028: 6024: 6020: 6016: 6012: 6008: 6004: 6000: 5993: 5985: 5981: 5976: 5971: 5967: 5963: 5959: 5955: 5951: 5944: 5942: 5933: 5929: 5925: 5921: 5916: 5911: 5907: 5903: 5899: 5895: 5891: 5884: 5882: 5880: 5871: 5867: 5863: 5859: 5854: 5849: 5845: 5841: 5837: 5833: 5829: 5822: 5820: 5818: 5809: 5805: 5801: 5797: 5793: 5789: 5786:(4): 044101. 5785: 5781: 5774: 5763: 5759: 5755: 5751: 5747: 5742: 5737: 5733: 5729: 5725: 5721: 5720: 5712: 5708: 5704: 5698: 5692: 5687: 5681: 5680:Robinson 1995 5676: 5670: 5665: 5663: 5648: 5644: 5637: 5629: 5624: 5623: 5614: 5606: 5602: 5597: 5592: 5588: 5584: 5580: 5576: 5572: 5565: 5550: 5546: 5540: 5532: 5526: 5522: 5517: 5516: 5507: 5499: 5495: 5491: 5487: 5483: 5479: 5472: 5464: 5460: 5456: 5452: 5445: 5437: 5431: 5427: 5420: 5412: 5406: 5402: 5395: 5387: 5381: 5377: 5370: 5356: 5352: 5346: 5339: 5335: 5329: 5322: 5317: 5309: 5305: 5301: 5297: 5293: 5289: 5284: 5279: 5275: 5271: 5267: 5263: 5259: 5255: 5248: 5234: 5230: 5224: 5216: 5214:9781633693968 5210: 5206: 5205: 5197: 5195: 5180: 5176: 5169: 5167: 5158: 5151: 5149: 5140: 5134: 5130: 5123: 5114: 5109: 5105: 5101: 5097: 5093: 5089: 5082: 5080: 5064: 5060: 5053: 5045: 5041: 5037: 5033: 5028: 5023: 5019: 5015: 5008: 5006: 4998: 4993: 4986: 4981: 4967: 4963: 4956: 4954: 4952: 4950: 4941: 4935: 4931: 4926: 4925: 4916: 4909: 4904: 4898: 4894: 4889: 4884: 4880: 4876: 4872: 4865: 4863: 4861: 4852: 4845: 4843: 4841: 4826: 4819: 4805: 4801: 4794: 4780: 4774: 4772: 4770: 4755: 4751: 4745: 4741: 4731: 4728: 4726: 4723: 4721: 4720:Floris Takens 4718: 4716: 4713: 4711: 4708: 4706: 4703: 4701: 4698: 4696: 4693: 4691: 4688: 4686: 4683: 4681: 4678: 4676: 4673: 4671: 4668: 4666: 4663: 4661: 4658: 4656: 4655:Michael Berry 4653: 4651: 4650:Ralph Abraham 4648: 4647: 4645: 4644: 4637: 4634: 4632: 4629: 4627: 4624: 4622: 4619: 4617: 4616:Quantum chaos 4614: 4612: 4609: 4607: 4604: 4602: 4599: 4597: 4596:Ill-posedness 4594: 4592: 4589: 4587: 4584: 4582: 4579: 4577: 4574: 4572: 4571:Edge of chaos 4569: 4567: 4564: 4562: 4559: 4557: 4554: 4552: 4549: 4547: 4546:Chaos machine 4544: 4542: 4539: 4537: 4534: 4532: 4529: 4527: 4524: 4522: 4519: 4518: 4516: 4515: 4508: 4505: 4503: 4500: 4498: 4495: 4493: 4490: 4488: 4485: 4483: 4482:Horseshoe map 4480: 4478: 4475: 4473: 4470: 4468: 4465: 4463: 4460: 4458: 4455: 4453: 4450: 4448: 4445: 4443: 4440: 4438: 4435: 4433: 4430: 4428: 4425: 4423: 4420: 4418: 4415: 4414: 4412: 4411: 4404: 4393: 4390: 4384: 4379: 4372: 4369: 4365: 4360: 4358: 4347: 4343: 4339: 4337: 4333: 4332:team building 4329: 4323: 4319: 4315: 4311: 4309: 4304: 4300: 4297: 4293: 4289: 4285: 4281: 4277: 4273: 4263: 4261: 4250: 4241: 4239: 4234: 4225: 4223: 4219: 4215: 4211: 4207: 4203: 4202:Robert M. May 4199: 4195: 4191: 4190:Gene-for-gene 4186: 4182: 4178: 4173: 4171: 4170:fetal hypoxia 4167: 4163: 4159: 4158:Canadian lynx 4155: 4151: 4141: 4139: 4135: 4124: 4122: 4121:DNA computing 4118: 4114: 4113:symmetric key 4110: 4109:steganography 4106: 4102: 4098: 4094: 4090: 4086: 4082: 4076: 4066: 4064: 4060: 4056: 4052: 4048: 4044: 4040: 4036: 4032: 4028: 4024: 4020: 4016: 4012: 4004: 4000: 3996: 3995:conus textile 3991: 3982: 3977: 3963: 3955: 3951: 3942: 3940: 3937: 3933: 3929: 3925: 3921: 3917: 3913: 3909: 3905: 3901: 3897: 3891: 3889: 3888: 3883: 3879: 3875: 3871: 3870: 3865: 3862:Also in 1987 3860: 3858: 3854: 3850: 3846: 3842: 3838: 3834: 3830: 3826: 3822: 3818: 3814: 3810: 3806: 3801: 3797: 3792: 3790: 3786: 3782: 3781: 3776: 3772: 3768: 3763: 3761: 3757: 3753: 3752:schizophrenia 3749: 3745: 3741: 3737: 3732: 3730: 3726: 3722: 3718: 3714: 3710: 3705: 3703: 3699: 3695: 3691: 3687: 3683: 3678: 3676: 3675: 3670: 3666: 3662: 3658: 3654: 3653:Menger sponge 3650: 3645: 3641: 3637: 3633: 3629: 3625: 3621: 3616: 3614: 3610: 3607: 3603: 3599: 3595: 3591: 3590:Edward Lorenz 3584: 3580: 3576: 3575:Floris Takens 3572: 3568: 3564: 3561: 3557: 3553: 3549: 3545: 3541: 3538: 3533: 3530: 3526: 3522: 3518: 3514: 3510: 3509:linear theory 3505: 3503: 3502:Stephen Smale 3499: 3495: 3491: 3487: 3483: 3479: 3474: 3472: 3468: 3464: 3460: 3456: 3452: 3448: 3441: 3437: 3433: 3432:Barnsley fern 3429: 3420: 3418: 3414: 3410: 3406: 3402: 3398: 3393: 3391: 3387: 3383: 3379: 3375: 3371: 3361: 3359: 3354: 3351: 3349: 3333: 3330: 3327: 3324: 3320: 3316: 3296: 3292: 3288: 3285: 3282: 3279: 3275: 3271: 3268: 3263: 3259: 3246: 3242: 3240: 3236: 3217: 3214: 3211: 3208: 3200: 3192: 3186: 3176: 3165: 3157: 3153: 3142: 3137: 3124: 3121: 3113: 3109: 3098: 3093: 3076: 3075: 3074: 3060: 3052: 3047: 3044: 3042: 3021: 3018: 3014: 3010: 3007: 3001: 2998: 2992: 2986: 2983: 2977: 2971: 2968: 2965: 2961: 2955: 2951: 2944: 2943: 2942: 2941: 2940: 2938: 2934: 2930: 2920: 2900: 2897: 2890: 2878: 2868: 2861: 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1731: 1729: 1725: 1720: 1718: 1714: 1711: 1707: 1703: 1699: 1695: 1687: 1683: 1679: 1673: 1669: 1666: 1662: 1658: 1654: 1650: 1641: 1632: 1623: 1621: 1617: 1613: 1609: 1605: 1601: 1597: 1593: 1589: 1584: 1581: 1576: 1571: 1569: 1563: 1559: 1555: 1550: 1542: 1537: 1528: 1524: 1522: 1503: 1497: 1492: 1489: 1462: 1456: 1451: 1448: 1421: 1415: 1410: 1407: 1393: 1389: 1385: 1380: 1376: 1373: 1363: 1361: 1358:of points in 1357: 1353: 1349: 1345: 1339: 1337: 1333: 1329: 1325: 1322:and a region 1321: 1317: 1298: 1295: 1292: 1286: 1278: 1274: 1253: 1250: 1247: 1227: 1224: 1221: 1218: 1215: 1208: 1192: 1186: 1183: 1180: 1166: 1162: 1160: 1156: 1152: 1148: 1141: 1135: 1132: 1128: 1124: 1117: 1113: 1109: 1103: 1084: 1081: 1076: 1072: 1068: 1063: 1059: 1055: 1050: 1047: 1044: 1040: 1019: 1016: 1011: 1007: 1003: 998: 994: 971: 967: 963: 958: 954: 950: 945: 942: 939: 935: 914: 889: 885: 881: 878: 870: 866: 862: 859: 854: 851: 848: 844: 834: 814: 811: 808: 796: 787: 785: 774: 772: 768: 764: 761: 756: 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Chua 12625:Tilt-A-Whirl 12595:FPUT problem 12480:Standard map 12475:Logistic map 12300: 12074:Chaos theory 12073: 12035: 12027:free content 12009:Derek Muller 11998: 11987: 11983:Étienne Ghys 11971:David Malone 11941: 11873: 11845:Chaos theory 11844: 11826: 11819: 11812: 11805: 11795: 11766: 11758: 11730: 11708: 11699: 11689: 11686:David Ruelle 11658: 11648: 11634: 11624: 11614: 11604: 11590: 11564: 11556: 11546: 11539: 11532: 11524: 11514: 11511:James Gleick 11504: 11494: 11483: 11479: 11471: 11464: 11442: 11421: 11399:10.1142/4510 11381: 11364: 11336: 11320:. Springer. 11317: 11280: 11256: 11228: 11204: 11183: 11162: 11138: 11101: 11077: 11056: 11035: 11014: 10990: 10970: 10951: 10924: 10912:. Retrieved 10892: 10872: 10853: 10831: 10807: 10787: 10767: 10716: 10712: 10701:. Retrieved 10694:the original 10676:(3): 10–13. 10673: 10669: 10651:the original 10622: 10618: 10568: 10564: 10539: 10535: 10506: 10500: 10460: 10456: 10445:. 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Chua 4642: 4641: 4513: 4512: 4507:Tilt A Whirl 4497:Standard map 4442:Cliodynamics 4409: 4408: 4368:hydrological 4361: 4353: 4340: 4324: 4320: 4316: 4312: 4308:Wilfred Bion 4305: 4301: 4269: 4256: 4247: 4235: 4231: 4174: 4147: 4130: 4115:, relies on 4105:watermarking 4081:cryptography 4078: 4069:Cryptography 4047:anthropology 4008: 3985:Applications 3979: 3965: 3961: 3952: 3948: 3924:astrophysics 3892: 3885: 3867: 3864:James Gleick 3861: 3829:forest fires 3825:econophysics 3817:solar flares 3793: 3778: 3764: 3748:eye tracking 3733: 3706: 3702:universality 3679: 3672: 3664: 3628:stock prices 3617: 3598:Ellen Fetter 3588: 3571:David Ruelle 3534: 3513:logistic map 3506: 3475: 3445: 3394: 3367: 3355: 3352: 3250: 3238: 3234: 3232: 3048: 3045: 3040: 3038: 2926: 2923:Jerk systems 2723: 2292: 2105: 2096: 2080: 1984:system state 1982:make up the 1921: 1721: 1694:logistic map 1691: 1685: 1677: 1671: 1667: 1664: 1660: 1657:logistic map 1638: 1592:limit cycles 1585: 1572: 1561: 1557: 1553: 1549:logistic map 1546: 1525: 1391: 1387: 1383: 1379:logistic map 1369: 1359: 1343: 1340: 1335: 1331: 1327: 1323: 1319: 1172: 1163: 1145: 1130: 1126: 1122: 1115: 1111: 1107: 832: 780: 757: 713: 582:trajectories 575: 571:geologic era 567: 562: 560: 555: 541: 535: 534: 527: 523: 519: 455: 451: 447: 443: 439: 435: 411: 397: 363: 357: 353: 344: 340: 336: 329: 325: 321: 301: 288: 285:Introduction 275: 228:anthropology 208:road traffic 205: 201: 191: 190:, or simply 187: 173: 163:can cause a 102:Chaos theory 101: 100: 58: 47: 18:Chaos Theory 13164:Alan Turing 13122:Liber Abaci 13041:Mathematics 12947:in crystals 12937:Soap bubble 12932:Phyllotaxis 12778:Nina Snaith 12768:Yakov Sinai 12653:Rufus Bowen 12403:Duffing map 12388:Baker's map 12313:Theoretical 12225:SRB measure 12132:Phase space 12102:Bifurcation 11999:In Our Time 11792:Ian Stewart 11461:John Briggs 10489:J.D. Farmer 10485:Crutchfield 10463:: 495–505. 10383:Yorke, J.A. 10268:: 219–232. 9595:: 982–986. 9487:: 121–125. 9134:: 111–115. 9077:3D Research 8248:January 29, 8161:(1): 1–11. 8073:(8): 1279. 7909:Phys. Today 7408:Sprott 2003 5707:Yorke, J.A. 5283:1721.1/5961 4715:Robert Shaw 4566:Determinism 4366:data (also 4364:water cycle 4359:at right). 4266:Other areas 4181:time series 4179:of chaotic 4039:meteorology 4031:engineering 4015:mathematics 3920:meteorology 3896:mathematics 3884:exposed in 3878:Thomas Kuhn 3805:earthquakes 3717:bifurcation 3694:Robert Shaw 3622:, studying 3606:Royal McBee 3051:Rössler map 2477:complex map 1583:butterfly. 1551:defined by 1381:defined by 1155:phase space 586:phase space 531:conditions. 380:it must be 373:it must be 319:defined by 273:processes. 269:theory and 244:engineering 224:meteorology 114:mathematics 45:for values 13233:Categories 13111:Empedocles 13106:Pythagoras 13024:Camouflage 12962:in biology 12957:in flowers 12927:Parastichy 12836:Complexity 12733:Edward Ott 12580:Convection 12505:Continuous 12180:Ergodicity 11981:Jos Leys, 11501:Leon Glass 11234:Providence 10914:2016-12-29 10703:2011-07-17 10670:Complexity 10578:2211.02441 10447:2009-08-12 10305:2017-06-12 9920:2401.03729 9884:(7): 837. 9878:Atmosphere 9861:: 136–141. 9223:2017-10-25 8667:Complexity 8363:2013-04-10 8300:(5): 821. 8294:Atmosphere 8274:2023-02-21 8067:Atmosphere 7962:Complexity 7864:: 111–200. 7718:2008-01-17 7603:31 January 7516:See also: 7434:2022-11-09 7410:, p. 7166:See also: 6789:2164/17003 6746:2023-11-03 6563:1901.09274 6241:2010.06231 5954:Atmosphere 5652:2019-11-24 5581:(5): 753. 5575:Atmosphere 5555:2019-11-24 5360:2019-11-24 5338:Wiktionary 5238:2024-08-18 5184:2020-05-13 4972:2019-11-24 4830:2020-05-17 4809:2019-11-24 4784:2019-11-24 4759:2019-11-24 4737:References 4222:population 4154:continuous 4043:philosophy 3866:published 3833:landslides 3789:complexity 3756:physiology 3686:Robert May 3661:Koch curve 3659:, and the 3636:Cantor set 3563:Lev Landau 3552:tip vortex 3548:Turbulence 3436:chaos game 2076:asymptotic 2068:parameters 1702:continuous 1600:continuous 1266:such that 1097:otherwise. 784:almost all 400:continuous 95:trajectory 13213:Emergence 13116:Fibonacci 12748:Mary Rees 12708:Bryna Kra 12641:theorists 12450:Ikeda map 12440:Hénon map 12430:Gauss map 12112:Limit set 12097:Attractor 11944:(excerpt) 11940:Gleick's 11880:EMS Press 11275:Tufillaro 11129:239756912 11012:(2001) . 10759:Textbooks 10726:1306.5777 10719:(5): 27. 10595:119972392 10556:207251437 10412:CiteSeerX 10222:149352680 10090:1090-0578 9943:: 11–17. 9900:2073-4433 9806:235909725 9743:0960-0779 9721:: 82–94. 9656:118329463 9648:1572-9974 9517:0028-0836 9509:1476-4687 9247:CiteSeerX 9196:CiteSeerX 9141:1307.4279 9113:125169427 9105:2092-6731 9083:(2): 15. 9048:: 94–99. 8673:(2): 62. 8577:154202123 8543:. 138547. 8490:1367-2630 8406:Physica A 8324:2073-4433 8136:261548506 7968:(3): 10. 7922:1306.5777 7685:124498882 7663:CiteSeerX 7383:859751750 7358:119107150 7350:0167-2789 7305:(4): 379. 7264:1573-8205 7191:: 43–49. 7050:122060992 6995:123612939 6872:987302273 6806:225246631 6798:1573-269X 6696:0960-0779 6588:118828500 6535:122790937 6455:(1): 63. 6434:0960-0779 6387:0218-348X 6348:123683839 6340:0022-4928 6293:121429354 6258:1972-6724 6209:250853499 6166:250757113 6084:245197840 6031:1572-9613 5984:2073-4433 5932:208369617 5924:0003-0007 5870:132494234 5862:0218-1274 5736:CiteSeerX 5605:2073-4433 5292:1095-9203 5027:1310.1576 4897:2673-8392 4576:Emergence 4536:Causality 4477:Hénon map 4228:Economics 4162:hydrology 4027:economics 3837:epidemics 3813:Omori law 3771:Chao Tang 3765:In 1987, 3707:In 1979, 3665:snowflake 3632:telephone 3618:In 1963, 3537:iteration 3328:π 3193:− 3002:˙ 2987:¨ 2872:→ 2862:− 2856:→ 2830:⁡ 2818:π 2795:⁡ 2759:→ 2749:− 2743:→ 2677:π 2668:λ 2590:ψ 2564:ψ 2536:ψ 2526:Ikeda map 2506:ψ 2496:ψ 2481:Julia set 2457:ψ 2451:⁡ 2445:− 2436:ψ 2430:→ 2427:ψ 2392:→ 2374:ψ 2329:→ 2319:− 2313:→ 2269:→ 2235:→ 2216:ψ 2188:→ 2178:− 2172:→ 2157:∫ 2139:→ 2115:ψ 2054:β 2034:ρ 2014:σ 1897:β 1894:− 1844:− 1835:− 1829:ρ 1785:σ 1782:− 1776:σ 1730:such as: 1717:nonlinear 1612:Julia set 1608:Hénon map 1568:attractor 1493:− 1411:− 1356:dense set 1302:∅ 1299:≠ 1293:∩ 1225:⊂ 1207:open sets 1190:→ 1082:− 882:− 742:λ 679:δ 666:λ 658:≈ 636:δ 596:δ 506:β 486:σ 466:ρ 414:intervals 248:economics 232:sociology 12942:Symmetry 12900:Patterns 12824:articles 12566:Physical 12485:Tent map 12375:Discrete 12315:branches 12245:Theorems 12081:Concepts 11947:Archived 11919:Florence 11603:(Eds.), 11547:Fractals 11419:(2000). 11226:(2012). 11160:(2000). 10950:(1983). 10908:Archived 10852:(2003). 10830:(1980). 10751:54005470 10647:16486826 10495:; N.H.; 10385:(1975). 10379:Li, T.Y. 10373:: 61–71. 10356:Articles 10098:23517610 9798:32872798 9751:85526993 9660:Archived 9619:(1998). 9483:(6400). 9455:37855255 9166:18740000 9027:18407251 8884:11276524 8791:19072712 8748:45804559 8740:17500951 8354:Archived 8196:. 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