The Butterfly Effect in Cat Purrs

If you'd like to support these videos:

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.
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):
Road Trip - Chris Doerksen (see previous link)
Broken Drum Machine - Godmode (youtube audio library)
It's nice to be home - Not David (not posted yet)
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:

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.

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.

[not_David]

In lieu of pictures, please tell me about your cat in great enough detail that I can visualize them in my head thank you (or dog or other pet, or if you don't have one (like me) I accept house plants) edit: I've been keeping up with all of these and love them all, if I happen to not respond to your comment please know I've read it and probably went 'awwweee' to it. Thank you :)

EDIT: I just realized I never stayed the delay I used for Helga's caterfly - it was about 35 time steps. Audacity says the recording is sampled at 44.1 kHz, so 1 time step is about 22 microseconds. However, A viewer made their own caterfly (!!) and it does look quite a bit different to the one shown in the video. For example, their caterfly does not have two clear lobes and the delay they used was very different as well (still looks awesome though imo). It very could be that it depends on the recording method and probably even the cat.

[samuelthecamel]

If cat purring is chaotic, does that mean that I can use it as a pseudo-random number generator?

[kackers]

the visuals of the shadow manifold vs. the original butterfly being coloured black/red and blue/beige is an excellent visual gag

[KoneSkirata]

7:08 „Shadow looks like a much edgier version of the original“
Spit out my drink on that joke. Brilliant and unexpected.

[YandiBanyu]

"And that's where today sponsor, nah I'm just kidding". That got me there laughing hard

[zigzagcheef]

Fun fact, cougars and cheetahs can not only purr, but they can meow too!

[DaGrEeNtEaCoNnOiSsEr]

"Biologist" here. It's in quotation marks since I'm a wet lab biologist that's moved into computational biology. It's a struggle to make this transition at times but videos like this one really makes learning and transitioning easier. The video is well done and breaks down concepts super clearly. Never thought I'd learn so much from cat purrs on chaos theory. Great work and I'm excited to learn more from ur vids. :)

P.S. I don't know much about cats so can't help with explaining why cat purrs would be chaotic even though I am a biologist. 😂

[an_asp]

The biggest thing I got out of this video is that I really need to spend more time on data visualization. It's amazing how much more visible important insights like the dynamics of these systems can be when you put the data into a more human-friendly format. I'm always blown away by the effort you put into the visuals on these videos!

[jeffreychandler]

I think what "cat purring is chaotic" actually means is more as a fun proof of concept. Small starting point changes in the position, airflow, etc of the mouth, vocal tracts, esophagus, etc results in disjunct responses. I found a study trying to see if voices are chaotic ("Chaos in Voice, From Modeling to Measurement" Jiang et al 2006) and they in fact seem to make a voice logistic map xD, which is pretty cool.

[mince]

this video slaps. i found the handwavey explanation pretty neat. it would be cool to see that explanation further developed.

[LoyDizak]

Nice to see bird the physics explainer flying around

[gauthamnair]

I love the humor, visuals, and the actual explanations. I wouldn’t know anything about physics, but this was very educational.

[MicroBlogganism]

Biologist here. The jargon-free summery of the quote at 5:23 is that purring is done by having muscles in the voice box contract and relax 20-30 times a second. Each time they contract it blocks the airway, causing pressure to build up, which is then released when the muscles are relaxed again. When the cat breathes in while purring, the contraction of the diaphragm, the muscle that does the breathing, is not a smooth movement like during a normal breath, but choppy, which also makes the breathing choppy. The choppy breathing and the opening and closing of the voice box is timed so that it doesn't impact the breathing too much, because the intake of air happens mostly when their airway is open and air can pass through it the easiest.

[Lelle_Berg]

This is honestly one of the most well explained and entertaining (16:55) videos I have seen of such a complicated topic. I would never have read about this otherwise, but I'm very happy to have seen the video.
Looking forward to seeing this channel grow :)

[pathagas]

i’ve been interested in synthesizing cat purr sounds through physical modeling synthesis, and this video was incredibly helpful! it looks like this means there are four different systems i’ll have to model and have feeding back into one another. i’m guessing the chaotic-ness comes from the feedback loops happening in the system, which is very common in physical modeling synthesis.

[minotower_]

literally the best video on the internet consisting of an extremely underrated field of math, silly ideas, amazing cute cats and amazing really beautiful animations and all of that while being accessible to almost everyone!! i love your stuff a lot, hope you continue doing it

[yousseftamer]

Honestly, I'm not even surprised that cats are chaos incarnate. You can NEVER predict what a cat will do, and I say that out of experience.

[kaelhop]

@not_David You got a hobby, YouTube. We the viewers thank you for that. You are doing great.

[coffeeandproofs]

hey -- math phd (& cat owner!) here, this video is AMAZING. regarding the curiosity on the chaos embedded in cats purring, it also got me wondering. i'd be happy to talk more on this -- could it be the percision of measurements when purring was recorded? i go back and forth on this as the heart data you visualized was clearly periodic and non-chaotic, so i suppose i now wonder how they measured that data...

the reason i wonder on about the chaos of purring is because of two ideas that came to mind: (1) there's a neat paper, and quanta had a nice article explaining it, on "non-deterministic approaches to physics" in relation to this neat concept called "intuitionism" developed by brouwer, a mathematician from ages ago -- and how our inability to measure to "infinite-precision" requires "fuzz" in our measurements, which would mean, as you fantastically noted, everything in our real life is chaotic!

reason (2) i wondered about the chaos being likely, related to (1) slightly, concerns what's called the "dense line on the torus" -- lets say Earth is a donut, and you're standing somewhere on it. the direction towards the "hole" from you, lets call that the y-axis direction, and the direction "around" the hole, is the x-axis direction. you are the origin. (this is called an "affine chart"/"local chart"/etc etc).

now, if you walk off in the x-direction, you just loop all around and are back (same with y-direction). but if you point at some random direction, say it makes a *rational* angle θ with your finger and the x-axis, then walk that direction, that will be *periodic*, you'll eventually get back to your original spot. (rational means it's some fraction, like angle is 326/728, etc.)

if it so happened you pointed at *irrational* angle with the x-axis, you get the dense line on the torus. in other words, you will never get back to where you started, always and forever walking somewhere you haven't been.

i am curious and wonder if it's this, embedded in some shape or form here, that causes the chaos.

for instance, what if you change the "time-shift" interval you mentioned doing? maybe instead of 1 second, it's 0.9 seconds? how does that change the structure? does there exist a correct shift such that it *isn't* chaotic?

what if you had (unobtainably...) perfect recording equipment, with no noise?

but i suppose maybe most likely, like us trying to sing a consistent *note* and hold it still -- that biological beings just "can't be consistent"? though, this loops back to the strangeness of that heart diagram... hmmmm...

ok brb i need to go record my cat purring

[zegevlier]

I think a company 3d printing your cat's purr print would be neat