
Chaos Theory and the Logistic Map - alexcasalboni
http://geoffboeing.com/2015/03/chaos-theory-logistic-map/
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tunesmith
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?

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streptomycin
_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?_

Kind of... see
[http://en.wikipedia.org/wiki/Control_of_chaos](http://en.wikipedia.org/wiki/Control_of_chaos)

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.

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streptomycin
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?

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krylon
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)?

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oakridge
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)

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krylon
Thank you very much for the explanation!

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Rainymood
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.

