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Carl Jung: CW 8 “The Structure and Dynamics of the Psyche” – Quotations
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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 behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.
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. One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour; there is no chaotic system involving only two first-order, ordinary differential equations the system resulting in an equation of second order only. Here, A is an adjustable parameter. The output of op amp 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.
Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model , four conditions suffice to produce synchronization in a chaotic system. Examples include the coupled oscillation of Christiaan Huygens ' pendulums, fireflies, neurons , the London Millennium Bridge resonance, and large arrays of Josephson junctions. In the s, while studying the three-body problem , he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.
Chaos theory began in the field of ergodic 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.
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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.
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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. 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  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  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,  involving many different disciplines mathematics , topology , physics ,  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 ,    engineering ,   finance ,   algorithmic trading ,    meteorology , philosophy , anthropology ,  physics ,    politics , population dynamics ,  psychology ,  and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing.
Biophotonics and Coherent Systems in Biology | SpringerLink
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.
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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.
Related Psychicsphere: The Structure of Chaos
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