An Introduction to Catastrophe Theory
Almost every scientist has heard of catastrophe theory and knows that there has been a considerable amount of controversy surrounding it. Yet comparatively few know anything more about it than they may have read in an article written for the general public. The aim of this book is to make it possible for anyone with a comparatively modest background in mathematics - no more than is usually included in a first year university course for students not specialising in the subject - to understand the theory well enough to follow the arguments in papers in which it is used and, if the occasion arises, to use it. Over half the book is devoted to applications, partly because it is not possible yet for the mathematician applying catastrophe theory to separate the analysis from the original problem. Most of these examples are drawn from the biological sciences, partly because they are more easily understandable and partly because they give a better illustration of the distinctive nature of catastrophe theory. This controversial and intriguing book will find applications as a text and guide to theoretical biologists, and scientists generally who wish to learn more of a novel theory.
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amplitude analysis attractors axis Bazin & Saunders behaviour bifurcation set biology butterfly calculation canonical form catas catastrophe machine catastrophe theory caustics cell chapter codimension coefficients consider constant control space control trajectory control variables coordinate corank critical points curve cusp catastrophe D'Arcy Thompson delay convention derivatives diffeomorphism differential equations discontinuities Duffing equation dynamic eliminate elliptic umbilic equilibrium point equilibrium surface example frontier function geometry gradient happens Hence hyperbolic umbilic hysteresis implies limit cycle Lotka-Volterra equations mathematical mechanism minimum monomials morphogenesis move observed occur order terms origin oscillations parabolic umbilic phase space physics plane polynomial position possible Poston & Stewart potential predict primary wave problem real roots region result seven elementary catastrophes shown in Fig sketch smooth solutions sort specific growth rate splitting lemma structurally stable strut subset sudden jumps suppose swallowtail Taylor series Thom trophe theory unfolding parameters universal unfolding unstable vanish Zeeman zero