Please no butterfly analogies.
In: Mathematics
I’m seeing a lot of comments that have the basic idea, but wrong or misleading examples. Here is my go at it. But this is more line an ELI- 13 if only because I know my five year old wouldn’t read all this.
Chaos theory describes an area of mathematics that studies a specific kind of mathematical model. Math models are sometimes used to describe or predict events (like projectile motion — the path of a tossed ball, or weather). Sometimes mathematicians study models just because of interesting properties. They do not always have a physical application. In the case of the models studied as a part of chaos theory (aka chaotic dynamic systems), they have a special property: small differences in the starting conditions result in large differences in the results.
**Example 1)** Weather — As others have pointed out: this is why long term predictions are mostly useless. The problem is that we can never measure the exact conditions (temperature, humidity, wind direction and speed, etc.) everywhere at any moment. As mentioned by [Tejaansh_sara](https://www.reddit.com/r/explainlikeimfive/comments/mj70p9/eli5_what_is_chaos_theory_and_what_is_it_used_for/gt8n2e5/?utm_source=share&utm_medium=web2x&context=3), James Gleik’s “Chaos” is a good one. It has a really great story at the beginning regarding this particular problem.
**Non-Example 1)** Projectile motion is not a chaotic dynamic system. Differences in outcomes due to errors in initial conditions are predictable. This doesn’t mean there will be no error in our calculation. This means that we can meaningfully predict the end result of the system and calculate a reasonable error margin on the prediction. However, error is not chaos.
**The chaos in a chaotic dynamic systems is not due to difficulties or impossibility in measurement!** This is a really important point (if you couldn’t tell by my exclamation point and bold), and what most examples I am reading here are missing. Even if we know the exact beginning state of chaotic dynamic system (and we can in the case of the next example), changing one small tiny itsy bitsy part of it results in big changes.
**Example 2)** Pseudorandom number generator (PRNG). Randomness is a big deal behind the scenes in our society. It’s used to encrypt everything online as an example. This makes generating random numbers a really important problem in mathematics. It’s also a very hard problem because math is deterministic, meaning — if you know the input, you can calculate the output. That bit contradicts randomness. One solution is to use a purely numeric chaotic dynamic system, which we call a pseudorandom number generator, keeping the initial conditions (called a seed value) a secret. Note that the pseudo in pseudorandom means it is “fake” randomness. This is precisely because it is still a deterministic system. The same seed value will result in the same series. However, in a good PRNG, even if someone is able to view a sequence of output values (but not the seed), they still would not be able to predict the next values. (This is an oversimplification, different PRNGs have different properties depending on what you need them for. More on that in my [other comment](https://www.reddit.com/r/explainlikeimfive/comments/mj70p9/eli5_what_is_chaos_theory_and_what_is_it_used_for/gt8q3l0?utm_source=share&utm_medium=web2x&context=3).) This example is interesting because you can see that the chaos is not a result of errors in measurement, but instead a property of the system being modeled.
**Other cool resources:**
[Robert L. Devaney’s Chaos, Fractals and Dynamics](https://www.youtube.com/watch?v=6QIhaDvTHXk) – It’s long, but worth it, even if only for the (80s-90s?) graphics and his transparency slide “iterator”. This is more like ELI – high school. Devaney shows Fractals — striking visualizations of chaotic systems — and gives examples of applications of these systems. He also shows a very simple equation that can be used as a PRNG.
[RANDOM.ORG](https://www.random.org/) – Is creating randomness really that hard? This website does provide true randomness, lets you generate some randomness of your own and explains how and why this is a real problem. Super cool website.
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