By Alwyn C. Scott
Written in Alwyn Scott’s inimitable type – lucid and available – The Nonlinear Universe surveys and explores the explosion of job in nonlinear technological know-how that started within the Nineteen Seventies and Eighties and keeps at the present time. The e-book explains the wide-ranging implications of nonlinear phenomena for destiny advancements in lots of parts of contemporary technology, together with arithmetic, physics, engineering, chemistry, biology, and neuroscience. Arguably as very important as quantum conception, glossy nonlinear technological know-how – and an appreciation of its implications – is key for knowing clinical advancements of the twenty-first century.
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Additional info for The Nonlinear Universe: Chaos, Emergence, Life (The Frontiers Collection)
Example text
4a) and can be described in terms of action–angle variables – for example, fixed angular momenta and corresponding angles of rotation, which increase linearly with time. When the ratios of two angular frequencies – one (ω1 ) around the hole of the torus and the other (ω2 ) in the transverse direction – are given by an irrational number (not a ratio of integers), the corresponding torus is said to be non-resonant. As an energy-preserving perturbation is slowly turned on, most solution trajectories continue to lie on nested, non-resonant toroidal surfaces in the phase space, which is then governed by coupled dynamics of the action–angle variables.
In the H´enon–Heiles model, ω1 and ω2 would be the frequencies of the originally uncoupled oscillators. Nor was recognition of the difficulties with numerical weather prediction original with Lorenz; in the mid-1950s, Norbert Wiener had expressed his concerns about this problem [1037], and in 1898 physicist William Suddards Franklin had written [323]: Long range detailed weather prediction is therefore impossible, and the only detailed prediction which is possible is the inference of the ultimate trend and character of the storm from observations of its early stages; and the accuracy of this prediction is subject to the condition that the flight of a grasshopper in Montana may turn a storm aside from Philadelphia to New York!
24 2 Chaos z ? SP #3 SP #2 initial point SP #1 x Fig. 1. 1) with σ = 10, r = 28 and b = 8/3. The view is looking down from the y-axis onto the (x, z)-plane. There are three singular points. 48, 27). 1) which now bears his name. In these equations, σ is the Prandtl number (ratio of fluid viscosity to thermal conductivity), r represents a temperature difference driving the system, and b is a geometrical factor. With the parameter values indicated in Fig. 1, Lorenz found a region of phase space occupied by a solution trajectory having two wings, with the phase point jumping back and forth between nearly periodic oscillations on the wings at seemingly random intervals.