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I'm not the author, but I can happily provide some practical applications to his discussion of dynamical systems and chaos. One of the nice application of the logistic equation is its use as a simple model for population dynamics. In fact, it is in this context that Robert May started the study of the logistic maps [1]. Looking at the logistic map equation, the next value (population) is based on the previous value at the right-hand side. We can provide interpretation to this: with rx(1-x), the r*x-factor tells us that the population should change proportional to the current population as they reproduce. However, the environment may have limited resources and overpopulation may hinder its growth, thus the (1-x)-factor. The bifurcation diagram which the author showed should tell us some things about the population dynamics. At particular growth rates (0<r<1), the population will simply collapse to zero, which makes sense as the growth factor r rather tells that the population should decrease being a fraction of the current population. The arching region between 1 and 3 tells us that there should be a stable population (a fixed-point value) where, from the initial value, the population would always go to. The weird structure we see beyond r>3 shows how the population would oscillate from different values, which we call strange attractor: the population, after a long time, neither goes to a specific value nor increases uncontrollably: the population just goes around at different values. Another application would be pseudorandom number generators. For example, the most common implementation for quickly creating pseudorandom numbers is the Mersenne twister [2]. In its simplest explanation, the twister looks like a very complicated feedback system, which generates the next value using the previous values. Unlike the logistic map, the twister has a very long period. Plotting it on a bifurcation diagram would cover the entire range of values, which would be ideal as you would want to cover your range of numbers uniformly. You wouldn't want to get only two values as random numbers, unlike at around r=3.3 of the logistic map. Although you may say that you have weak background in mathematics, it is not impossible to appreciate and use these concepts. You can look for any video of Steven Strogatz on YouTube explaining these topics. His discussions are really accessible and easy to understand for any person. [1] May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459-467.
[2] Matsumoto, M., & Nishimura, T. (1998). Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation (TOMACS), 8(1), 3-30. (But I suggest reading the wiki page for the twister) |