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12Discrete & Continuous Nonlinear DynamicsNonlinear dynamics is one of the success stories of computational science. It hasbeen explored by mathematicians, scientists, and engineers, with computers as anessential tool. The computations have led to the discovery of new phenomena such assolitons, chaos, and fractals, as you will discover on your own. In addition, becausebiological systems often have complex interactions and may not be in thermodynamicequilibrium states, models of them are often nonlinear, with properties similar to thoseof other complex systems.In Unit I we develop the logistic map as a model for how bug populations achievedynamic equilibrium. It is an example of a very simple but nonlinear equation producingsurprising complex behavior. In Unit II we explore chaos for a continuoussystem, the driven realistic pendulum. Our emphasis there is on using phase space asan example of the usefulness of an abstract space to display the simplicity underlyingcomplex behavior. In Unit III we extend the discrete logistic map to nonlinear differentialmodels of coupled predator–prey populations and their corresponding phasespace plots.12.1 Unit I. Bug Population Dynamics (Discrete)Problem: The populations of insects and the patterns of weather do not appear tofollow any simple laws. 1 At times they appear stable, at other times they vary periodically,and at other times they appear chaotic, only to settle down to somethingsimple again. Your problem is to deduce if a simple, discrete law can produce suchcomplicated behavior.12.2 The Logistic Map (Model)Imagine a bunch of insects reproducing generation after generation. We start withN 0 bugs, then in the next generation we have to live with N 1 of them, and afteri generations there are N i bugs to bug us. We want to define a model of howN n varies with the discrete generation number n. For guidance, we look to theradioactive decay simulation in Chapter 5, “Monte Carlo Simulations”, where the1 Except maybe in Oregon, where storm clouds come to spend their weekends.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 289