Understanding Nonlinear Dynamics
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics ( TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. About the Authors Daniel Kaplan specializes in the analysis of data using techniques motivated by nonlinear dynamics. His primary interest is in the interpretation of irregular physiological rhythms, but the methods he has developed have been used in geo physics, economics, marine ecology, and other fields. He joined McGill in 1991, after receiving his Ph.D from Harvard University and working at MIT. His un dergraduate studies were completed at Swarthmore College. He has worked with several instrumentation companies to develop novel types of medical monitors.
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action potential algebraic amplitude analysis approach approximately assume attractor autocorrelation function automaton behavior bifurcations biological Boolean functions Boolean networks calculate called Cantor set cells cellular automata chaos chaotic dynamics Chapter consider correlation correlation integral cycle of period density derivative described determine deterministic Dynamics in Action eigenvalues elements embedding dimension example exponential decay exponential growth finite-difference equation fractal frequency gene geometry given gives graph inflection points initial condition input integral iteration length linear dynamics mathematical mean measurement noise molecules node null hypothesis one-dimensional ordinary differential equations original oscillations output parameters period-doubling bifurcations periodic cycles phase plane plot population possible prediction problem random number result saddle point self-similar sensitive dependence shown in Figure shows sigmoidal function sine wave slope solution standard deviation steady steps surrogate data trajectory truth table two-dimensional unstable variable versus white noise y-isocline zero
Page 402 - Transfer function analysis of autonomic regulation I. Canine atrial rate response,