If you'd like to support these videos:
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Resources:
1. Strogaz's Nonlinear Dynamics and Chaos is the go-to for anything in this field.
I also highly recommend Dr. Steve Brunton and Dr. Nathan Kutz's youtube channels. They have really good educational videos from their labs and how they combine these techniques with machine learning to important real world problems like hydrodynamics and seizure detection.
2. Audio from the channel ClaireCats. If you need some cat purring ASMR to ease your day/night please check her channel out    / @clairecats6852  
3. Cited paper is: A practical test for noisy chaotic dynamics by Ahmed BenSaida


Chapters:
0:00 I need a hobby lol
1:39 What is Chaos?
5:05 Shadow Manifold
8:17 Multi-dimensional Cat Purring
12:11 Bring it All Together Now

Music (in order):
RPG Store - Chris Doerksen https://chrisdoerksen.bandcamp.com/al...
Atlas - HOME https://open.spotify.com/artist/2exeb...
Road Trip - Chris Doerksen (see previous link)
Intermission - Stux.io and Vaporwavez https://stuxio.bandcamp.com/album/qua...
Broken Drum Machine - Godmode (youtube audio library)
It's nice to be home - Not David (not posted yet)
Killer Vacation - Chris Doerksen https://chrisdoerksen.bandcamp.com/al...
Fiesta de la Vida - Aaron Kenny (youtube audio library)

Notes (roughly in order):

Someone requested a blender copy of the caterfly so here y'all go:
https://github.com/notDavidsGit/Cater...

Petting Analysis: For each cat then I had 2 categories (pet and not pet) with multiple segments in each category. I extracted the audio in each segment and used Fourier analysis. I found the average spectrum for each cat (N = 6) using zero-padding and a hamming window. From these 6 average spectrum, I found the average shift for 5 of the cats was 3 Hz, and the last cat had no shift at all.

A hand wavey explanation for the time delay coordinates: If you want to go deeper into this method, its also often called 'Takens Embedding Theorem'. The reason this method works roughly speaking is because the past of one variable influences the present state of another variable. By plotting one variable against the past of itself, it is bringing out the influence of these other variables. However, this puts some conditions as well. First, the variable of interest needs to be a function of all the other variables. In the Lorenz system this means we cannot use what is traditionally labeled the z-coordinate (e.g., as seen on the wikipedia page for 'Lorenz system'). The z variable is only a function of x (and z itself). This means the shadow manifold would not accurately recover the Lorenz system because the past of z does not contain information about the y-coordinate. I actually would like to perhaps make a video on this because there was originally a 5ish minute long section in the video just on this topic, but I cut it for time. If you've read this and are interested let me know. If I have time I'll make it and put it on the secondary channel.
   / @notdavidsecondary9958  

A small mistake(ish): The way I define the stretch factor is a bit loose. At 13:01 the new size is equal to the stretch factor times the initial size. This is fine, but then I say the stretch factor is normally called the lyapunov exponent and treat them interchangeably. But actually the way I've written it, it should be that stretch factor = exp(lyapunov exponent). Nothing conceptually really changes but this is why in my demo of the sphere deforming the stretch factor becomes negative -- its because I'm treating it equivalently to the lyapunov exponent which can be negative.

Heart analysis: While people do indeed use chaos analysis on heart beat recordings, heart rate variability derived from EKG recordings is more commonly studied, where as I am just using the EKG signal directly. A healthy heart should have small variability, but if it is too small then your heart cannot respond to changing demand. Thus it could be that heart beats are actually optimally slightly chaotic or operate at an edge of chaos transition. There was a good paper on this that I can't find at the time of writing. I'll update it if I find it, but in the mean time, here is a somewhat recent example paper ("Chaos-Based Analysis of Heart Rate Variability Time Series in Obstructive Sleep Apnea Subjects", 2020) and "Chaos Theory, Heart Rate Variability, and Arrhythmic Mortality" is a nice little review, but it is from 2000. It is still well written.