Please no butterfly analogies.
In: Mathematics
When you’re running a complex computer simulation with many different factors, small scale changes in initial conditions can lead to large scale changes in final outcomes, making prediction possible, but limited in scope and fundamentally based on probability.
In chaos theory, you try to make your models better by accounting for the unaccountable. Your models are fallible, but honest. Because all models are fallible.
There’s a parallel phrase in Daoism: The Dao that can be spoken is not the eternal Dao.
Chaos theory is also closely related to the second law of thermodynamics and its application to information theory. All information is in flux, as is our ability to measure it.
Chaos Theory is essentially studying equations where the assumption that changing the equation a small amount will lead to a small change in the result is broken. The goal is to determine where this starts to happen.
If you do some sort of experiment, you’d expect that if you repeat the experiment with the starting point altered slightly, then you’d get a slightly different result. Chaotic systems don’t follow that.
The first sort of step is finding things called bifurcations. Imagine there’s an equation that tells you the elevation of your next step, based on the elevation of your current step, while walking on uneven ground. A bifurcation would occur if there was a cliff for you to step off of. One step back, and the difference between the elevation of your feet would be small. Step off the cliff, huge change.
An actual example of a bifurcated system would be a rocket trying to escape from Earth’s gravity. Below escape velocity, the rocket comes back down to earth. Above it, it goes off into space.
A chaotic system has a certain point where you start getting those sudden huge changes *everywhere*. It might not be immediate, but changing one of those parameters, or where you start out, will lead to a result that you *cannot* predict just by knowing what happened when you started out close to it.
Source:BScH in Mathematical Physics, MSc in Applied Math.
Does Dr. malcom even make any sort of sense in Jurassic Park or am I an idiot? Because that scene with the water droplet scene always seemed nonsensical to me
Life….uh finds a way
Chaos theory in ELI5 terms states that there is an order to seemingly random situations…every outcome is determined by a hell of a lot of variables
Great podcast episode here that made it easy for me to understand
[https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/how-chaos-theory-changed-the-universe-29467341/](https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/how-chaos-theory-changed-the-universe-29467341/)
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.
Paraphrased from chaos by James gleick – and amazing book!!
In complex turbulent simulations – like cold milk pouring into hot coffee it’s impossible to tell at any time what the temperature is at any point inside the liquid
But step back – wait an hour and you’ll be able to tell exactly the temp at every point – it will be room temperature
Apply this logic to all complex systems – stand back and take the wider view to make good long term decisions.
Consider the difference between dropping a bowling ball from the 10th floor and dropping a small feather from a the 10th floor.
In the first case you should be able to predict the landing spot to a meter even if you’re dropping the bowling ball by hand. You could quite easily improve the accuracy by dropping the ball with a robot arm or waiting till the wind is down. You can drop the ball from a higher place and the calculations won’t be much more complicated. If you control the conditions to a reasonable precision, your result will be reasonably precise.
In the second case, you’d be lucky if the feather landed at the feet of the building you’re dropping the feather from. You can’t even predict the landing spot with very complicated computer models. Even if you control the conditions to a really good precision – dropping the feather inside a big building to reduce wind, using a robot arm to drop the feather – your result will be wildly inaccurate. You can’t control the conditions well enough to get reasonably good precision.
The first system can be predicted with pretty simple models. The second one needs to be so precise that it’s practically impossible to predict.
I recently had to explain this to my 10 year-old as we received a chaos engine from Kiwicrate.
What I said was the idea behind chaos theory is the tiniest, unknowable variations at the start of a process can magnify until they have massive consequences at the end. Having the chaos engine I could demonstrate that even if we held the arms in what looked like exactly the same way we’d never be able to replicate the shapes it made thanks to tiny variations in weight distribution, even air currents and bearing heat etc. She loved it.
https://en.wikipedia.org/wiki/File:Demonstrating_Chaos_with_a_Double_Pendulum.gif
This gif probably shows it best. Three double pendulums, starting at almost exactly the same state, will have wildly different speeds and locations only a few seconds in.
Chaos theory, in a nutshell, is the study of how such systems behave, including but not limited to:
* When/how fast such systems diverge (for example, the pendulum example is mostly similar for the first few seconds)
* When they *don’t* diverge (because many chaotic systems only have regions where it is chaotic; going back to the pendulum, 3 pendulums starting close to the bottom obviously is not chaotic)
* How strongly such systems diverge, with respect to differences in the starting condition
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