I've found it fun to think about the subjects of complexity theory and chaos theory as a layperson, and there's something really enticing about it (from a naive perspective) in pop culture in general.
I think there's a general base belief that it can still be tamed, that if you zoom out enough, all these crazy variations in dynamic systems can still be bounded within predictable parameters.
Silly examples - the Emperor's big plans in the Star Wars saga, or the second Foundation in Asimov's Foundation series. In the latter, he did allow for the fact that chaos theory meant that the psychohistorians couldn't predict small outcomes, but that they could certainly predict big ones. In both of those, the authors were enjoying a fantasy that you could plug in some key inputs, put up with a ton of variation, but still get the general output that you originally planned for. And then closer to home, there's public policy planning or even bigger planning like the introduction of the Euro.
Is that true in chaos theory though? Are there mathematical ways to create and predict dynamic systems with behavior that is truly bound within certain parameters?
Is that true in chaos theory though? Are there mathematical ways to create and predict dynamic systems with behavior that is truly bound within certain parameters?
There was some hype about this in the 90s, but not much seems to have come from it in terms of practical applications. However it is really cool from a theoretical perspective, and some of the experiments (IIRC precisely controlling chaotic contraction patterns in heart tissue from rabbits in vitro) were also really interesting.
If you are a lay person interested in this subject in general I can recommend John Gribbon's book Deep Simplicity (it's publish with two different subtitles)
I love that such a simple equation has such incredibly beautiful and complex behavior - and that nobody noticed until like 50 years ago. What else is out there, right under our noses?
Very interesting article! As somebody with a rather weak background in mathematics (to put it mildly), I really like the author's style, it is very accessible.
It would have been even nicer, though, if the author had given one or two examples of practical applications. Or would those have been to complex for the intended audience (which very much includes me)?
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)
Bifurcationtheory is so interesting. I recently followed a course on it. It was one of the hardest, but best classes I've ever had. It's amazing how a system like f(x,a) = x^2 + a can have such a deep complexity already.
I think there's a general base belief that it can still be tamed, that if you zoom out enough, all these crazy variations in dynamic systems can still be bounded within predictable parameters.
Silly examples - the Emperor's big plans in the Star Wars saga, or the second Foundation in Asimov's Foundation series. In the latter, he did allow for the fact that chaos theory meant that the psychohistorians couldn't predict small outcomes, but that they could certainly predict big ones. In both of those, the authors were enjoying a fantasy that you could plug in some key inputs, put up with a ton of variation, but still get the general output that you originally planned for. And then closer to home, there's public policy planning or even bigger planning like the introduction of the Euro.
Is that true in chaos theory though? Are there mathematical ways to create and predict dynamic systems with behavior that is truly bound within certain parameters?