Easily Explained: Chaos Theory in 13 Minutes

Рет қаралды 5,604

Жыл бұрын

A Chaotic System is very difficult to predict due to its sensitivity to initial conditions and its unpredictable nature. But, how does Chaos enter into the system? What makes a system Chaotic? And, for that matter, what makes any system sensitive to initial conditions, and what accounts for its unpredictable nature?
In the end of the video, I simulated the "Butterfly Effect" with 1 million particles depicting the Chaotic Nature of the Lorenz Attractor. It becomes quite evident such a small difference in the initial conditions leads to drastic difference in the system if it's evolved in time.
Chapters:
00:00 - Introduction to the Chaos Theory Series.
00:24 - What is this video about?
00:32 - Chaos is Everywhere!!
00:42 - Reasons why we call any system Chaotic.
00:50 - Why do Scientists have so much trouble in predicting the weather?
01:32 - Logo
01:38 - Can a Butterfly's flap in Brazil bring tornado in Texas?
02:34 - Tides and Weather are two different systems.
02:55 - Simulation of an Ideal Pendulum.
03:53 - Phase Relationship between two Simple Pendulums.
04:17 - Simulation of a Double Pendulum.
04:42 - Phase Relationship b/w two Double Pendulums.
05:20 - Simulation of Spiral Chaos System.
07:55 - Chaos implies Non-linearity.
08:06 - Story Continues... Why Unpredictability?
08:36 - Back to Spiral Chaos.
09:26 - How to minimize the error in system?
09:50 - Role of Lyapunov Exponent in Systems.
10:21 - Chaos is Deterministic??
10:50 - The Butterfly Effect.
11:30 - Simulation of the Lorenz Attractor.
11:50 - A million particles simulated.
12:30 - Thanks and Outro.
Some References I used to learn about Chaos Theory:
1. Some inspiring thoughts given by the paper "Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?" by Edward N. Lorenz
-- fermatslibrary.com/s/predictability-does-the-flap-of-a-butterflys-wings-in-brazil-set-off-a-tornado-in-texas
2. A great read "Chaos and Fractals" by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe: link.springer.com/book/10.1007/978-1-4757-4740-9
3. Numberphile Video for the idea of Spiral Chaos Simulation: kzfaq.info/get/bejne/obNynZWX1ZuWpHU.html
4. Double Pendulum Equations: www.myphysicslab.com/pendulum/double-pendulum-en.html
--------
I used a *no copyright* video to show the differences in two different weather conditions arise as a consequence of whether the initial conditions are effected by the flap of a butterfly. Link: kzfaq.info/get/bejne/pdiEga2CmKjWkpc.html
--------
**Correction: At approximately 11:20, it is *Lorenz* Attractor.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This video is animated using Grant Sanderson (@3blue1brown) library Manim (based on Python).
Manim is an engine for precise programmatic animations, designed for creating explanatory math videos.
Code for this video:
github.com/Varniex/manim_videos
Let's connect on Twitter:
Varniex
ManimGL GitHub Repository:
github.com/3b1b/manim
Support the Channel:
Become a Patreon: patreon.com/Varniex
Buy me a Book: buymeacoffee.com/Varniex
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#manim #python #animation

Пікірлер: 19

@InfiniteRegress Жыл бұрын

Very visually appealing, and such a fascinating topic! Thank you so much, and I look forward to what you make next! ^_^

@Varniex Жыл бұрын

Thanks a lot for such appreciation.

@OkAnup Жыл бұрын

Wow! Amazing video, and very clearly explained 👍

@Varniex Жыл бұрын

Thanks! Glad you liked it!!

@francoisiswatching 4 ай бұрын

Very interesting video!

@Varniex 2 ай бұрын

Glad you enjoyed it

@stevenhung8076 Ай бұрын

amazing. Thanks for your video. Wish your channel will get more sub and famuos like star of david. 😇😇😇

@Varniex Ай бұрын

Thank you very much for kind words. It means a lot!!

@JwalinBhatt Жыл бұрын

Super cool bro, your channel is gonna blow up soon :)

@Varniex Жыл бұрын

Thanks for supporting bro!!

@allinone0215 2 ай бұрын

manim_imports import * is not working how can i make it work?

@Varniex 2 ай бұрын

I used a python script 'manim_imports' where I am importing all the necessary modules. You need to use `from manim import *` or `from manimlib import *` depending on the version of manim you are using.

@allinone0215 2 ай бұрын

Okay I'll try

@stevenhung8076 Ай бұрын

music at 7:00 so amazing, remind me something familiar like dejavu, could i know it's song. Thanks

@Varniex Ай бұрын

I guess it's "Dreamland" by Akash Gandhi, from the KZfaq Library.

@stevenhung8076 Ай бұрын

@@Varniex got it, Thanks :)))

@donaldhobson8873 Жыл бұрын

So to illustrate "chaos", you use a "1 parameter" system that actually contains 4 parameters (x, y, angle, angle change) and that displays polynomial divergence not the exponential divergence characteristic of chaos. I mean it's largely a good video, and the visualizations are pretty, but that spiral thing just doesn't have the mathematical structure of chaos. It's not the exponentially sensitive dependence on initial conditions. It's non chaotic dependence on initial conditions, made to look sensitively dependent through choice of constants.

@Varniex Жыл бұрын

Two questions: 1. Since when coordinates are considered as parameters. If I take polar coordinates then, by the logic there are 3 parameters (r, theta, theta change). So, number of parameters got changed if I switch coordinate systems? There is *only* parameter as I am restricting everything else. Just like the "Random Walk" situation. 2. Lorenz Equations, and similarly other strange attractors have "polynomial divergence", not every attractor has "exponential divergence", then why are they called "Chaotic Systems"? I don't know any source which defines that if there's *only* exponential dependence then it's chaotic otherwise it's not. If there's non-linearity then the system is chaotic. Moreover, take Matt's (from Numberphile Video) word, he described that Euler Spiral "chaotic".

@donaldhobson8873 Жыл бұрын

@@Varniex By "parameter" I mean anything you would need to store as a variable not a constant in a computer program simulating the rules. If you used polar coordinates, you would need r (distance from origin to current position.) phi (angle between north and line from origin to current position) (these are needed to record current position, and so are a drop in replacement for x and y) theta (direction you are currently traveling in) theta change (are you on a straight line part, or doing a tight curve) The Lorentz equations have exponential divergance in the sense that if you pick two starting points within epsilon of each other, the distance between those points grows exponentially. (for a random choice of direction) A single pendulum is nonlinear (when you don't use small angle approximations), but not chaotic. Nonlinear systems can be strongly damped and converge to a point, and not be chaotic. Or they can explode to infinity. Or they can have periodic orbits. If M is the state space for the map f t f^{t}, then f t f^{t} displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d(. , .) such that 0 < d ( x , y ) < δ 0\mathrm {e} ^{a\tau }\,d(x,y) en.wikipedia.org/wiki/Butterfly_effect Clearly requires exponential divergance.