j_bertolottiFeb 21, 2024
#PhysicsFactlet
The Lorenz system is a common example of chaotic dynamics and of a strange attractor.
Points with very similar initial conditions initially evolve very similarly to each other, until their trajectories diverge from each other, and start moving on a "butterfly"-shaped fractal.
#Physics #Chaos
Show threadEtienne JacobMar 6, 2024
@j_bertolotti if you used approximations with a numerical method to draw these curves, aren't the approximation errors very much impacting the curves after some time because it's chaotic? I've often wondered about that
Show threadj_bertolotti
@bleuje Yes, but you can track the error and stop when it becomes unacceptably large.
Mar 7, 2024 at 1:40pmWeb
Show threadRahul NarainMar 7, 2024

@j_bertolotti @bleuje But Etienne is correctly concerned that in chaotic dynamics, the error grows exponentially with time, so you would have to stop very quickly! A more relevant result is the shadowing lemma:

"Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one. Therefore, the fractal structure of chaotic trajectories seen in computer maps is real."

https://mathworld.wolfram.com/ShadowingTheorem.html
http://www.scholarpedia.org/article/Shadowing

Shadowing Theorem -- from Wolfram MathWorld

Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one. Therefore, the fractal structure of chaotic trajectories seen in computer maps is real.

Show threadj_bertolottiMar 7, 2024
@narain
Then I misunderstood the question 😉
@bleuje
Show threadEtienne JacobMar 9, 2024
@j_bertolotti @narain I appreciate both of your answers!