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Chaos Theory.doc (Size: 359 KB / Downloads: 100)
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B.E Mechanical (A)
This report is divided into two sections – The Theory and Applications. The theory starts with a basic approach involving fundamental concepts of chaos- nonlinearity, iteration, bifurcation and strange attractors. The more advanced mathematical concepts of Mandelbrot sets, Universality and Fractals are dealt with a simple overview without much detail. Due to the mathematical nature of the theory, an explanation has been attempted giving the example of the “Weather Model” by Lorenz.
The Applications of the theory are extremely widespread and it is difficult to explain all of them in detail. Hence, a brief introduction of each application is mentioned in order to understand the scope of this theory. Emphasis has been given on the applications which are directly related to Mechanical Engineering. This theory has the potential to explain the behaviour of a variety physical phenomena, ranging from the basic Pendulum, Turbulence, Convection, Vibrations, to the more complicated Managerial implications and Production techniques.Introduction
What exactly is chaos? The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data. It studies the behavior of the dynamical systems that are governed by non-linear differential equations. The behavior of such systems is predictable only to an extent. Chaotic behavior occurs even in those systems which are deterministic, meaning that their future dynamics are fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.
By the 1950s, the advancement in sciences had made Objects of everyday experience like fluids and mechanical systems seem so basic and so ordinary that physicists had a natural tendency to assume they were well understood. According to Newtonian Science, it could be claimed that: Given an ap¬proximate knowledge of a system's initial conditions and an un¬derstanding of natural law, one can calculate the approximate behavior of the system. It was not so.
The invention of chaos theory has lead to a paradigm shift in the way scientists comprehend systems. The simplest systems are now seen to create extraordinarily difficult problems of predictability. Yet order arises spontaneously in those systems-chaos and order together. Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does-one water molecule, one cell of heart tissue, one neuron-and what millions of them do.
Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side-these have been puzzles to science, or worse, monstrosities.
In the 1970s a few scientists in the United States and Europe began to find a way through disorder. They were mathe¬maticians, physicists, biologists, chemists, all seeking connections between different kinds of irregularity.