Easily Explained: Chaos Theory in 13 Minutes

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
-- https://fermatslibrary.com/s/predicta...

2. A great read "Chaos and Fractals" by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe: https://link.springer.com/book/10.100...

3. Numberphile Video for the idea of Spiral Chaos Simulation:    • The Plotting of Beautiful Curves (Eul...  

4. Double Pendulum Equations: https://www.myphysicslab.com/pendulum...

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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:    • Storm Stock Footage | Free HD Video -...  
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*Correction: At approximately 11:20, it is *Lorenz Attractor.

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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:
https://github.com/Varniex/manim_videos

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