Mathematical Surveys and Monographs Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications Cambridge Univ. Press, Cambridge. A regularity result for quasilinear stochastic partial differential equations of parabolic type. Available at arXiv You have access to this content. You have partial access to this content. You do not have access to this content. More like this Pointwise decay for the solutions of degenerate and singular parabolic equations Juutinen, Petri and Lindqvist, Peter, Advances in Differential Equations, Nonuniqueness for nonnegative solutions of parabolic stochastic partial differential equations Burdzy, K.

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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?

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.

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## Stochastic Partial Differential Equations: Six Perspectives

Yorke coiner of the term "chaos" as used in mathematics , Robert Shaw , and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Iterations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum 's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections. 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 [65] 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 [66] 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.

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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.

## Stochastic Partial Differential Equations: Six Perspectives

Currently, chaos theory remains an active area of research, [68] involving many different disciplines mathematics , topology , physics , [69] 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 , [71] [72] [73] engineering , [74] [75] finance , [76] [77] algorithmic trading , [78] [79] [80] meteorology , philosophy , anthropology , [11] physics , [81] [82] [83] politics , population dynamics , [84] 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.

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 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.

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. 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.

Systems science portal Mathematics portal. Yorke George M. Retrieved University of Chicago Press. Bibcode : arXivB. 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.

### Stochastic Partial Differential Equations: Six Perspectives

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. Check system status. Toggle navigation Menu. Name of resource. Problem URL. Describe the connection issue.

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