Applied Chaos Theory: A Paradigm for Complexity
These are exciting times for mathematics, science, and technology. One of the fields that has been receiving great attention is Chaos Theory. Actually, this is not a single discipline, but a potpourri of nonlinear dynamics, nonequilibrium thermodynamics, information theory, and fractal geometry. In the less than two decades that Chaos Theory has become a major part of mathematics and physics, it has become evident that the old paradigm of determinism is insufficient if we are to understand - and perhaps solve - real life problems. Curiously, many of these problems are deterministic, but they are intertwined with randomness and chance. Thus the deterministic laws of physics coexist with the laws of probability. Consequently, uncertainty arises and unpredictability occurs, characteristic of complex systems. In its short lifetime Chaos Theory has already helped us gain insights into problems that in the past we found intractable. Examples of such problems include weather, turbulence, cardiological and neurophysiological episodes, economic restructuring, financial transactions, policy analysis, and decision making. Admittedly, we can as yet solve only relatively simple problems, but much progress has been made and we are now able to observe complex problems from new vantage points that provide us with numerous benefits. One such benefit is the universality of Chaos Theory in its applicability to different situations, which enables us to look at communal problems in an interdisciplinary manner, so that persons of different backgrounds can communicate with one another. Chaos Theory also enables us to reason in a holistic manner, rather than being constrained by simplistic reductionism.Finally, it is gratifying that the mathematics is not intimidating, and one can accomplish much with a personal computer or even a handheld calculator.
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1 Living with Complexity
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algorithm analysis applications approach behavior bifurcation Boltzmann Cambel Cambridge Cantor set chaos theory chapter Clausius complex systems concept considered denotes depends determine deterministic differential discrete logistic equation discuss dissipative systems dynamical systems energy equilibrium ergodicity example exponential Fibonacci Figure flow follows fractal dimensions fractal geometry function growth hence Henon map increase initial conditions irreversible iterative Julia set Koch snowflake Kolmogorov entropy limit cycle linear logistic curve Lyapunov exponent macroscopic Mandelbrot set mathematical measure mechanics molecules namely noise nonequilibrium Nonlinear Dynamics occur oscillator parameter particles pendulum phase space phenomena physics Poincare points population predator-prey Press Prigogine problems random Reynolds number scale Schaffer Scientific American self-organization Shannon shown in Fig Sierpinsky triangle software program stable statistical entropy strange attractors structure technologies temperature term thermodynamics trajectories uncertainty Univ University unstable variable volume W. H. Freeman York