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- The Birth of Chaos by Ethan James Clarke
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- Chaos theory

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 linear theory , the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple " noise " was considered by chaos theorists as a full component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration 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 was experimenting with analog computers and noticed, on November 27, , 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 Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course.

He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz 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. 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. In , Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. In , he published " How long is the coast of Britain?

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Statistical self-similarity and fractional dimension ", 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 infinitesimally small measuring device. In , Mandelbrot published The Fractal Geometry of Nature , which became a classic of chaos theory.

Yorke coiner of the term "chaos" as used in mathematics , Robert Shaw , and the meteorologist Edward Lorenz. The following year, independently Pierre Coullet and Charles Tresser with the article "Iterations d'endomorphismes et groupe de renormalisation" and Mitchell Feigenbaum with the article "Quantitative Universality for a Class of Nonlinear Transformations" described logistic maps. In , Albert J. Feigenbaum for their inspiring achievements. There, Bernardo Huberman presented a mathematical model of the eye tracking disorder among schizophrenics.

In , Per Bak , Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters [64] describing for the first time self-organized criticality SOC , considered one of the mechanisms by which complexity arises in nature. Alongside largely lab-based approaches such as the Bak—Tang—Wiesenfeld sandpile , many other investigations have focused on large-scale natural or social systems that are known or suspected to display scale-invariant 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 earthquakes , which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg—Richter law describing the statistical distribution of earthquake sizes, and the Omori law [65] describing the frequency of aftershocks , solar flares , fluctuations in economic systems such as financial markets references to SOC are common in econophysics , landscape formation, forest fires , landslides , epidemics , and biological evolution where SOC has been invoked, for example, as the dynamical mechanism behind the theory of " punctuated equilibria " put forward by Niles Eldredge and Stephen Jay Gould.

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 wars. In the same year, James Gleick published Chaos: Making a New Science , which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, though his history under-emphasized important Soviet contributions. Alluding to Thomas Kuhn 's concept of a paradigm shift exposed in The Structure of Scientific Revolutions , many "chaologists" as some described themselves claimed that this new theory was an example of such a shift, a thesis upheld by Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, [67] involving many different disciplines mathematics , topology , physics , [68] social systems , population modeling , biology , meteorology , astrophysics , information theory , computational neuroscience , etc. 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 geology , mathematics , microbiology , biology , computer science , economics , [70] [71] [72] engineering , [73] [74] finance , [75] [76] algorithmic trading , [77] [78] [79] meteorology , philosophy , anthropology , [11] physics , [80] [81] [82] politics , population dynamics , [83] psychology , [10] and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing. Chaos theory has been used for many years in cryptography.

In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of cryptographic primitives. These algorithms include image encryption algorithms , hash functions , secure pseudo-random number generators , stream ciphers , watermarking and steganography.

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 predictive model. For over a hundred years, biologists have been keeping track of populations of different species with population models.

Most models are continuous , but recently scientists have been able to implement chaotic models in certain populations. 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.

Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of fetal hypoxia can be obtained through chaotic modeling. In chemistry, predicting gas solubility is essential to manufacturing polymers , but models using particle swarm optimization 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 quantum physics and electrical engineering , the study of large arrays of Josephson junctions benefitted greatly from chaos theory.

Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately. 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.

Researchers have continued to apply chaos theory to psychology.

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## The Birth of Chaos by Ethan James Clarke

For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion '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. Redington and Reidbord 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 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 when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so. In their 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 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.

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 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. By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Aniundson and Bright found that better suggestions can be made to people struggling with career decisions.

For instance, team building and group development 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. 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.

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. 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. Traffic forecasting may benefit from applications of chaos theory.

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Better predictions of when traffic 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 prediction model see the plot of the BML traffic model at right.

Chaos theory has been applied to environmental water cycle data aka hydrological data , such as rainfall and streamflow. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics. From Wikipedia, the free encyclopedia. For other uses, see Chaos theory disambiguation and Chaos disambiguation. Main article: Supersymmetric theory of stochastic dynamics. Main article: Butterfly effect.

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Systems science portal Mathematics portal. Yorke George M. Retrieved University of Chicago Press. The British Journal for the Philosophy of Science. April Mathematics of Planet Earth Retrieved 12 June Journal of the Atmospheric Sciences. Bibcode : JAtS Ivancevic Complex nonlinearity: chaos, phase transitions, topology change, and path integrals. Bibcode : Chaos.. On the order of chaos. Social anthropology and the science of chaos. Oxford: Berghahn Books. Swiss Physical Society. Helvetica Physica Acta 62 : — Bibcode : Sci Cambridge University Press.

Discrete Chaos. Topology and its applications. The American Mathematical Monthly. Nonlinear Dynamics: A Primer. March Bibcode : Entrp.. Modern Physics Letters B. Bibcode : MPLB Global Warming and the Future of the Earth. American Mathematical Monthly. Bibcode : AmMM Archived from the original PDF on Bibcode : PhRvL.. Journal of Statistical Physics. Bibcode : JSP Soviet Journal of Quantum Electronics. Bibcode : QuEle.. Physics Letters A. Bibcode : PhLA.. Bibcode : Nonli.. The conservative case".

## The Chaos Born Series

Underdetermination of Science: Part I. Bulletin of the London Mathematical Society. Journal of Mathematical Physics. Bibcode : JMP Basov Ed. Optics and Spectroscopy. Bibcode : OptSp.. Chlouverakis and J. Acta Mathematica. Henri Popp, Bruce D. Cham, Switzerland: Springer International Publishing.

Princeton University Press. Birkhoff, Dynamical Systems, vol. Bibcode : DoSSR.. Reprinted in: Kolmogorov, A. Proceedings of the Royal Society A. Preservation of conditionally periodic movements with small change in the Hamiltonian function.

### Birth of Chaos

Lecture Notes in Physics. Bibcode : LNP Journal of the London Mathematical Society. Bulletin of the American Mathematical Society. Bibcode : BAMaS.. Chaos: Making a New Science. London: Cardinal. Journal of Business. The Fractal Geometry of Nature. New York: Freeman. New York: Basic Books. Statistical Self-Similarity and Fractional Dimension".

New York: Macmillan. In Bunde, Armin; Havlin, Shlomo eds. Fractals in Science. This time they had set up the printer not to make a graph, but simply to print out time stamps and the values of a few variables at each time. As Lorenz later recalled, they had re-run a previous weather simulation with what they thought were the same starting values, reading off the earlier numbers from the previous printout. The computer was keeping track of numbers to six decimal places, but the printer, to save space on the page, had rounded them to only the first three decimal places.

After the second run started, Lorenz went to get coffee. The new numbers that emerged from the LGP while he was gone looked at first like the ones from the previous run. This new run had started in a very similar place, after all. But the errors grew exponentially.

After about two months of imaginary weather, the two runs looked nothing alike. This system was still deterministic, with no random chance intruding between one moment and the next. Even so, its hair-trigger sensitivity to initial conditions made it unpredictable. This meant that in chaotic systems the smallest fluctuations get amplified. Weather predictions fail once they reach some point in the future because we can never measure the initial state of the atmosphere precisely enough.

Or, as Lorenz would later present the idea, even a seagull flapping its wings might eventually make a big difference to the weather. In , the seagull was deposed when a conference organizer, unable to check back about what Lorenz wanted to call an upcoming talk, wrote his own title that switched the metaphor to a butterfly.

But no one at the time registered it enough to scoop him. In the summer of , Hamilton moved on to another project, but not before training her replacement. Two years after Hamilton first stepped on campus, Ellen Fetter showed up at MIT in much the same fashion: a recent graduate of Mount Holyoke with a degree in math, seeking any sort of math-related job in the Boston area, eager and able to learn. She interviewed with a woman who ran the LGP in the nuclear engineering department, who recommended her to Hamilton, who hired her.

Once Fetter arrived in Building 24, Lorenz gave her a manual and a set of programming problems to practice, and before long she was up to speed. The project had progressed meanwhile. The 12 equations produced fickle weather, but even so, that weather seemed to prefer a narrow set of possibilities among all possible states, forming a mysterious cluster which Lorenz wanted to visualize.

Finding that difficult, he narrowed his focus even further. From a colleague named Barry Saltzman, he borrowed just three equations that would describe an even simpler nonperiodic system, a beaker of water heated from below and cooled from above. Here, again, the LGP chugged its way into chaos. Lorenz identified three properties of the system corresponding roughly to how fast convection was happening in the idealized beaker, how the temperature varied from side to side, and how the temperature varied from top to bottom.

The computer tracked these properties moment by moment. The properties could also be represented as a point in space. Lorenz and Fetter plotted the motion of this point.

## Chaos theory

They found that over time, the point would trace out a butterfly-shaped fractal structure now called the Lorenz attractor. The trajectory of the point — of the system — would never retrace its own path. And as before, two systems setting out from two minutely different starting points would soon be on totally different tracks. But just as profoundly, wherever you started the system, it would still head over to the attractor and start doing chaotic laps around it. Both were published in the landmark paper. But for a while only meteorologists noticed the result.

They stayed in touch with Lorenz and saw him at social events. Still, the notion of small differences leading to drastically different outcomes stayed in the back of her mind. She remembered the seagull, flapping its wings. Mission Control had to make a quick choice: land or abort.

She founded her own company in Cambridge in , and in recent years her legacy has been celebrated again and again. In she garnered arguably the greatest honor of all: a Margaret Hamilton Lego minifigure. After a few years, she left her job to raise her children. In the s, she took computer science classes at the University of Colorado, toying with the idea of returning to programming, but she eventually took a tax preparation job instead.

By the s, the demographics of programming had shifted. Chaos only reentered her life through her daughter, Sarah. As an undergraduate at Yale in the s, Sarah Gille sat in on a class about scientific programming. The case they studied? Later, Sarah studied physical oceanography as a graduate student at MIT, joining the same overarching department as both Lorenz and Rothman, who had arrived a few years earlier. Today, chaos theory is part of the scientific repertoire. In a study published just last month, researchers concluded that no amount of improvement in data gathering or in the science of weather forecasting will allow meteorologists to produce useful forecasts that stretch more than 15 days out.

Lorenz had suggested a similar two-week cap to weather forecasts in the mids.