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One of my favorite quotes is from a letter of Charles Darwin (1887): "I have long discovered that geologists never read each other's works, and that the only object in writing a book is proof of earnestness, and that you do not form your opinions without undergoing labour of some kind. " It is not clear if this private opinion of Darwin was one that he held to be absolutely true, or was one of those opinions that, as with most of us, coincides with our "bad days," but is replaced with a more optimistic view on our "good days. " I hold the sense of the statement to be true in general, but not with regard to scientists never reading each other's work. Even if that were true however, the present essay. would still have been written as a proof of earnestness. This essay outlines my personal view of how nonlinear mathematics may be of value in formulating models outside the physical sciences. This perspective has developed over a number of years during which time I have repeatedly been amazed at how an "accepted" model would fail to faithfully characterize the full range of avail able data because of its implicit or explicit dependence on linear concepts. This essay is intended to demonstrate how linear ideas have come to dominate and therefore limit a scientist's ability to understand any given class of phenomena.
Sommario
1. Introduction.- 1.1 The Five Stages of Model Building.- 1.2 Truth in Modeling,.- 2. Error Analysis, Statistics and Other Uncertainties Associated with Linearity.- 2.1 Distribution of Errors.- 2.2 Data Analysis.- 2.3 Langevin Equation.- 2.4 Chain Conditions and Fokker-Planck Equations.- 3. The Importance of Being Nonlinear.- 3.1 Physics; a Linear World View.- 3.2 Some Non-Gauss Statistics, Clustering and Fractals.- 3.3 Growth and Saturation.- 3.4 The Relaxation Oscillator.- 4. How to be Nonlinear.- 4.1 Deterministic Chaos.- 4.2 Growth, Competition and Avoidance.- 4.3 Stochastic Differential Equations.- 5. What is it that was Really Said?.- Appendix - Computer Programs.- References.
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Dr. Bruce J. West has been an active research scientist in industry, a university professor and department chair of physics, and he is presently a chief scientist in mathematics for the Army. Dr. West's career spans 30 years and he has over 310 research publications. His research specialization has been the application of nonlinear dynamics systems theory to a variety of physical and biomedical phenomena.
