Lorenz, Edward

Lorenz, Edward. The Essence of Chaos. Reprint. St. Louis University of Washington Press, 1996. 

In the 1960 s the meteorologist Edward Lorenz worked on systems of differential equations describing weather patters, and found something utterly different. The smallest modification in the initial conditions can have a dramatic effect, resulting in a completely different outcome after a certain time. Such behaviour is called chaotic. The sets of differential equations initially were rather complex but later he developed a simpler set which shows the same effect. 

Although best known as an astronomer, Brahe was also a follower of Paracelsus, and a number of medicines that he concocted found their way into the official Danish pharmacopoeia. For years Tycho also kept a detailed weather diary, convinced that weather patterns held vital secrets. It was not until 1960 that Tycho s hunch was proved right — Edward Lorenz s studies of weather patterns led to the invention of a new science chaos theory. 

Example 13.1 Lorenz equations The strange attractor The Lorenz equations (published in 1963 by Edward N. Lorenz a meteorologist and mathematician) are derived to model some of the unpredictable behavior of weather. The Lorenz equations represent the convective motion of fluid cell that is warmed from below and cooled from above. Later, the Lorenz equations were used in studies of lasers and batteries. For certain settings and initial conditions, Lorenz found that the trajectories of such a system never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. Attractors in these systems are well-known strange attractors. 

Very similar conclusions to the above were reached by Reichman [146], Dare-Edwards et al. [147] and Lorenz and co-workers [148]. Dare-Edwards et al. [147] used a numerical integration procedure to show that this type of analysis was appropriate to the experimental results obtained on singlecrystal n-Fe203. Reichmann [146] obtained an expression similar to eqn. (456) his formula may be derived from eqn. (456) if we assume that Urz vs, so that... 

As observed in Figs.3.14-4, all patterns generated by the C3-C1, C3-C2 and C2-C1 representations, remind, in one way or another, a butterfly. The latter stands for a basic phenomenon in the chaos model known as the butterfly effect, after the title of a paper by Edward N.Lorenz Can the flap of a butterfly s wing stir up a tornado in Texas An additional point may be summarized as follows, i.e., How come that relatively simple mathematical models create very complicated dynamic behaviors, on the one hand, and how Order, followed by esthetics patterns, may be created by the specific representation of the transient behavior, on the other ... 

D.W. Hill, Cooperative research in industry (London, 1946) Ronald S. Edwards, Cooperative Industrial Research. A study of the economic aspects of the research assocications (London, 1949) Alfred D. Chandler Jr., "From industrial laboratories to departments of research and development," in Kim B. Clark, Robert H. Hayes, and Christopher Lorenz, eds.. The uneasy alliance. Managing the productivity-technology dilemma (Boston, 1985), 53-61 Edgerton and Horrocks (ref. 1). 

We may add that chaos, chaotic phenomena, and chaotic behavior are not so uncommon in science and have received the attention of mathematicians and scientists, people such as Henri Poincare, Jacques Hadamard, George David Birkoff, Andrei Nikolaevich Komogorov, John Edensor Littlewood, Stephen Smale, and Edward Lorenz. According to Lorenz, chaos can be defined as [51]... 

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