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.
You use the butterfly as a literal example but it's more of a metaphor. When I think of Chaos Theory, I think of the Cascadian Subduction Zone with 2 plates colliding during an attempted subduction. As for being able to predict the weather accurately, for say 2 weeks, it is virtually impossible because of the dirth of unknown variables. And these variables are Not static. It's exactly why climatologists are finding their predictions are Way behind the reality - Unknown Variables. Humans Love Predictability, Safety, Constants because by labeling things and putting things in boxes you automatically cage that thing by naming and confining it. We remove its ability to flow freely, to whatever it is whenever it is, if that makes sense. Sadly, the majority of humanity are completely Out of Sync with nature and it's killing us and our planet. I think the Earth is responding to Our ignorance, abuse and carelessness - not just with her, but with each other also. We are throwing nature out of balance with our divisiveness and negativity. Just my 2¢
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.
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".
@@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.
