What is chaos anyways?

We think of a lot of systems as "chaotic". The weather, the stock market, the motion of a double pendulum - all these systems have structures which to the human eye do not have an obvious pattern. However, the lack of a discernable pattern does not mean that a system is necessarily chaotic. Consider, for example, a gas of non-interacting molecules trapped in a cubical box. At any moment in time, a snapshot of the molecules will show them all moving around in different directions and at different positions. Hence there will be no obvious pattern. However, if I start each molecule at t=0t=0 with a known initial position and velocity to within some experimental accuracy, then I can predict the position of each molecule at a later time t=Tt=T to the same accuracy. Hence the system is deterministic, even though it might appear random to our eyes. The randomness comes from the variation in the initial conditions, not from an inherent randomness in the evolution of the system itself.

In contrast, a chaotic system possesses a randomness in the evolution of the system itself. This set of problems will help illuminate the difference between a deterministic system and a chaotic one.

Note by David Mattingly
7 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

I'm quite intrigued by chaos theory and things like the butterfly effect. Very interesting!

Vishnuram Leonardodavinci - 7 years, 1 month ago

A simple but deeply interwoven theory... Its implications would be amazing to see.

Tejas Menon - 7 years, 1 month ago

Absolutely stunning.... theory... What ultimately translates is that there is predictability in a chaotic situation. What about unexpected results despite knowing the exact parameters, !!!

samarth Shukla - 7 years ago
×

Problem Loading...

Note Loading...

Set Loading...