Is anyone here interested in chaos theory? Let's discuss it

Is anyone here interested in chaos theory? Let's discuss it.

Other urls found in this thread:

superfractalthing.co.nf/sft_maths.pdf
mathr.co.uk/mandelbrot/perturbation.pdf
youtube.com/watch?v=1UT_JE2X1jw
en.wikipedia.org/wiki/Baby_monster_group
dinkydauset.deviantart.com/art/Baby-Monster-498314232
youtube.com/watch?v=aAJkLh76QnM
github.com/gboeing/pynamical
youtube.com/watch?v=OWByH3FqrwM
mega.nz/#F!8llHnYZL!gWl-CwNddQFq5uVaPlltNg
a.pomf.cat/lebcbu.pdf
rutracker.org/forum/viewtopic.php?t=1098760
twitter.com/SFWRedditImages

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Mind = blown. Just realized that's supposed to be a butterfly. Chaos theory is prity tricky opie.

OP's attractor

Please contribute to discussion in a constructive manner.

...

Indeed it is. I am not very well versed in the subject but it is the area of mathematics which fascinates me the most. Correct me if I am wrong but basically this concept puts forward the idea that something small that happens (e.g the energy emitted from the flapping of a butterfly) can cause a massive reaction in nature (such as a tornado) somewhere else on Earth?

If that is the case wouldn't the energy from said butterfly be dissipated too much to have much of an impact on anything?

It's more that a very small change in the initial conditions of a system can lead to ildly different results later down the line, essentially the minor change in the starting conditions propagate and grow with time so that the system develops completely differently because of the small change.

The weather is a chaotic system like this, so a small change in the initial conditions, romantically put as a butterfly fluttering its wings, could lead to the development of the weather changing substantially, dramatically put as the formation of a tornado

You can also get chaos to arise in a damped/driven pendulum which is fun

>minor change in the starting conditions propagate and grow with time
That property is Lyapunov exponent.
also

Be honest, are you actually interested in the topic or do you just like the pretty pictures and the sound bites?

It's not suppose to a butterfly, if it looks like it, it is only coincidence. The Lorentz attractor is supposed to be ideal convection current, although it can model other things like variability of laser power.

I don't think there's anything wrong with liking chaos theory because of the images that are generated from chaotic systems. Mathematicians like to theorise about integer number sequences because it's interesting to them the same way people find fractals interesting. Also the processes of rendering fractals can involve some serious mathematical methods. For example this one is rendered using perturbation theory. Perturbation theory allows you to calculate one really big precision calculation, and approximate the rest of the image really well with low precision calculations.

superfractalthing.co.nf/sft_maths.pdf
mathr.co.uk/mandelbrot/perturbation.pdf

Did not want to start own thread but

On my BAII plus calculator, how the fuck can I clear the AMORT worksheet?

I've just gone to resetting the whole calculator because CLR WORK does nothing along with all others. That's what the manual suggests

I am legitimately interested in it. To me it is sort of like the fabric of our world. That sounds 2deep4u but it is like a blend of matj and philosophy. I am also studying atmospheric sciences so it is of considerable interest to me.

youtube.com/watch?v=1UT_JE2X1jw

It would be interesting to see what kind of shape the set of chaotic coefficients makes on it's 12-dimensional hilbert space. It would be analogous to the mandelbrot set on the complex plane. Too bad there's no good way of visualising space greater than 3 dimensions.

>baby monster

Is this en.wikipedia.org/wiki/Baby_monster_group ?

It's named after it by the artist, but it's not the actual baby monster group.

dinkydauset.deviantart.com/art/Baby-Monster-498314232

Also the full resolution image is glorious. 4000x3000 resolution. The zoom is also ridiculous at 2^6901.591. The center coordinates are also ridiculous precision, the decimal points are so large they can't fit in this post.

here's a cropped part of the full image

I'm taking a class reviolving around General system (field) theory. If anyone has some good online resources on the topic I'd greatly appreciate it

Absolutely beautiful

Math*

Okay, but why would they choose the one that looks like a butterfly if not for the butterfly effect. I've seen that so often on books or just in relation to chaos theory. Pic related.

Butterfly effect is related but it means that small changes like flapping of a butterfly's wings can have large effects given time. The graphic that is shown is the Lorentz attractor and Edward Lorentz was specifically studying it to model an ideal convection current. While he was studying this with computer models, he discovered the butterfly effect. The fact that the Lorentz attractor looks like a butterfly is purely coincidence.

So wait, the energy created from the flapping of a butterfly never dissipates or weakens? I guess the law of conservation of energy would be applied here?

Energy is conserved, but over time the causal effect becomes large.

My only background in it is Taylor's chapter on it, Piqued my interest.

Btw is there a software package out there to draw these strange attractors for arbitrary functions? Preferably written in Python.

Can someone explain me what is the chaos theory?

Interesting, thank you.

It's essentially the study of the behaviour of non-linear systems.

Important topics are system sensitivity to initial conditions and parameters, predictability and error propagation over time.

Important tools and results are
>the Feigenbaum number (a dimensionless empirical result which can be used to predict the onset of chaos of a non-linear system)
>bifurcation diagrams (used to visualize the periodicity of a system and the onset of chaos for different system parameters)
>Poincare sections (A plot of state space representations simulated over time)
and
>Strange attractors ( the OP pic is an example of one, many of these strange attractors are fractal like which is cool.)


The term "onset of chaos" refers to when a system does not reach a known steady state (including periodic behaviour) after a long period of time.

A good introductory paper to read is the study of the logistics map.

Hi. This is a very good short video for laypersons.

youtube.com/watch?v=aAJkLh76QnM

Found this
github.com/gboeing/pynamical
Looks promising.

>muh attractors
suave

Here's an attractor transitioning from chaos to non-chaos. It disapears a couple times while its doing this because its converging into a group of points for some reason.

I posted a mega link in the archived thread with source code of my program. I'ts not up to date, I've been adding features to it since I posted it. I'll probably update the link soon.

Reminds me of polynomial game

youtube.com/watch?v=OWByH3FqrwM

They literally used attractors for their effects. Also they have fractals in the background (I see a buddabrot in your pic).

Yep, there's a bunch of them. I even remember something like Lorentz arch or something

...

This looks really good user. Do you have a Github repo or something we can follow/collaborate with?

>models on which perturbation theory doesn't work
nty
t. physics student

Not yet, just a mega link with the source code:
mega.nz/#F!8llHnYZL!gWl-CwNddQFq5uVaPlltNg

I could create a github project though. I'd have to clean up my code and make it modular and not embarrassing.

im beeing a nice guy. also nice thread.

oh dear me i forgot the link. a.pomf.cat/lebcbu.pdf

I looked through the code and I still can't really figure out what's going on here.

What dynamical system's trajectory are these points tracking? I know you said it was

xn = c[0] + c[1] * x + c[2] * x*x + c[3] * x*y + c[4] * y + c[5] * y*y;
yn = c[6] + c[7] * x + c[8] * x*x + c[9] * x*y + c[10] * y + c[11] * y*y;
zn = ...

But how does this translate into ODEs? I've tried simulating it as

x'[t] = c[0] + c[1] * x[t] + ...
etc.

but that just seems to just blow up in a hurry. I also tried starting at {0.1, 0.1, 0.1} and just iterating the formula over and over again but that also blows up.

>I could create a github project though. I'd have to clean up my code and make it modular and not embarrassing.
That's fine people don't need to contribute immediately, also some you can delegate some donkey work to us if any when you are refactoring. Just put up an issue, you'll be surprised how many people chip in.

I wish Veeky Forums bros would collaborate more on projects in general, like /g/ often does.

rutracker.org/forum/viewtopic.php?t=1098760

1. The Chaos Revolution
2. The Clockwork Universe
3. From Clockwork to Chaos
4. Chaos Found and Lost Again
5. The Return of Chaos
6. Chaos as Disorder—The Butterfly Effect
7. Picturing Chaos as Order—Strange Attractors
8. Animating Chaos as Order—Iterated Maps
9. How Systems Turn Chaotic
10. Displaying How Systems Turn Chaotic
11. Universal Features of the Route to Chaos
12. Experimental Tests of the New Theory
13. Fractals—The Geometry of Chaos
14. The Properties of Fractals
15. A New Concept of Dimension
16. Fractals Around Us
17. Fractals Inside Us
18. Fractal Art
19. Embracing Chaos—From Tao to Space Travel
20. Cloaking Messages with Chaos
21. Chaos in Health and Disease
22. Quantum Chaos
23. Synchronization
24. The Future of Science

It's not a system of ODE's it's a mapping. One point directly maps to a new point. You could change it to ODE's though but you would have to calculate a dt to track smoothly over time and scale everything by that. Also there are keyboard commands that you have to use to find the attractors (like finding points in the mandelbrot set). It's hard to represent things over a 30 dimensional space so you can only explore manually/randomly instead of graphing the whole space at once like the mandelbrot set. The keyboard commands are:

Pg Up, Pg Dn : cycle through random seeds used to generate the coefficients
Home, End : zoom in and out
Arrow Keys : moves center position
Insert : reset position and zoom

I took a course on it my last year getting a physics degree. It was a lot less regimented than courses up to that point and we had a lot of interesting simulation projects. Everyone who took it that year loved the course.

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