Ebook: An Essay on the Importance of Being Nonlinear

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.

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1. Introduction.- 1.1 The Five Stages of Model Building.- 1.1.1 Linear Models.- 1.1.2 Nonlinear Models.- 1.2 Truth in Modeling,.- 2. Error Analysis, Statistics and Other Uncertainties Associated with Linearity.- 2.1 Distribution of Errors.- 2.2 Data Analysis.- 2.2.1 Discrete Data and Linear Regression.- 2.2.2 Time Series.- 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.- 3.1.1 Linear Superposition.- 3.1.2 Irreversibility.- 3.2 Some Non-Gauss Statistics, Clustering and Fractals.- 3.2.1 Levy Processes.- 3.2.2 Pareto-LeVy Tails.- 3.2.3 Fractal Random Walks.- 3.3 Growth and Saturation.- 3.3.1 Population and Prediction.- 3.3.2 Scaling and Saturation.- 3.4 The Relaxation Oscillator.- 3.4.1 Glossary of Nonlinear Terms.- 3.4.2 The van der Pol Oscillator.- 4. How to be Nonlinear.- 4.1 Deterministic Chaos.- 4.1.1 Continuum Systems (New Chaos).- 4.1.2 Discrete Systems (Another New Chaos).- 4.1.3 More Discrete Systems (New But Related Chaos).- 4.2 Growth, Competition and Avoidance.- 4.2.1 Prey-Predator Systems.- 4.2.2 Interacting Random Walkers.- (a) The “Big Brother” Weighting Function.- (b) An “Abhorrence of Congestion” Weighting Function.- 4.3 Stochastic Differential Equations.- 4.3.1 Multiplicative Fluctuations.- 5. What is it that was Really Said?.- Appendix — Computer Programs.- References.