import numpy as np

def dynamical_system(x, y, z, a, b, n_iter):

   x_values = np.empty(n_iter)
   y_values = np.empty(n_iter)
   z_values = np.empty(n_iter)

   x_values[0] = x
   y_values[0] = y
   z_values[0] = z

   for i in range(1, n_iter):
       x, y, z = x_values[i], y_values[i], z_values[i] = a * x * (1 - x) - 0.05 * (y + 0.35) * (1 - 2 * z), 0.1 * ((y + 0.35) * (1 + 2 * z) - 1) * (1 - 1.9 * x), 3.78 * z * (1 - z) + b * y

   return x_values, y_values, z_values

1. Parameters

a = 3.8 b = 0.2 x_init = 0.085 y_init = -0.121 z_init = 0.075 n_iter = 800000

x, y, z = dynamical_system(x_init, y_init, z_init, a, b, n_iter)

init_time = 1000

def normalize(x):

   a, b = np.quantile(x, 0.999), np.quantile(x, 0.001)
   return 2 * (x - (a+b)/2) / (a - b)

x = normalize(x)[init_time:] y = normalize(y)[init_time:] z = normalize(z)[init_time:] print(min(x), max(x), min(y), max(y), min(z), max(z))

%%capture import matplotlib.pyplot as plt import numpy as np import os

for i, theta in enumerate( np.linspace(0, 2*np.pi, 120)):

   fig, ax = plt.subplots(figsize=(20, 20))
   init_time = 1000
   ax.scatter(y*np.cos(theta) + z*np.sin(theta), x, color='k', s=0.005)
   ax.axis("off")
   ax.set_xlim(-1.06, 1.06)
   ax.set_ylim(-1.03, 1.03)
   dir_path = f"./towel"
   if not os.path.exists(dir_path):
       os.makedirs(dir_path)

   fig.savefig(f"{dir_path}/{i}.png")
   plt.close()

import imageio.v3 as iio import os from natsort import natsorted import moviepy.editor as mp

for dir_path in ["./towel"]:

   file_names = natsorted((fn for fn in os.listdir(dir_path) if fn.endswith('.png')))

   # Create a list of image files and set the frame rate
   images = []
   fps = 12

   # Iterate over the file names and append the images to the list
   for file_name in file_names:
       file_path = os.path.join(dir_path, file_name)
       images.append(iio.imread(file_path))

   filename = dir_path[2:]
   iio.imwrite(f"{filename}.gif", images, duration=1000/fps, rewind=True)
   clip = mp.ImageSequenceClip(images, fps=fps)
   clip.write_videofile(f"{filename}.mp4")