File:Hopf and homoclinic bifurcation.gif

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English: ```

from tqdm import tqdm import numpy as np import matplotlib.pyplot as plt from scipy.integrate import solve_ivp import os

for i, mu in enumerate(tqdm(np.linspace(-1, 1, 240))): 

 def system(t, y):
     v, w = y
     dv = mu * v + w - v**2
     dw = -v + mu * w + 2 * v**2
     return [dv, dw]
 def system_reversed(t, y):
     v, w = y
     dv = mu * v + w - v**2
     dw = -v + mu * w + 2 * v**2
     return [-dv, -dw]

 x_root = (mu**2+1)/(2+mu)
 y_root = -mu * x_root + x_root ** 2
 vmin, vmax, wmin, wmax = -1,1,-1,1
 # vmin,vmax,wmin,wmax= x_root-0.0005,x_root+0.0005, y_root-0.0005, y_root+0.0005

 t_span = [0, 10]
 trajectory_resolution = 10

 epsilon = 0.01
 
 initial_conditions = [(x, y)  for x in np.linspace(vmin, vmax, trajectory_resolution) for y in np.linspace(wmin, wmax, trajectory_resolution)]
 initial_conditions_2 = [(x_root + dx, y_root + dy) for dx in np.linspace(-epsilon, epsilon, 10) for dy in np.linspace(-epsilon, epsilon, 10)]
 sols = {}
 sols_2 = {}
 sols_reversed = {}
 sols_reversed_2 = {}
 for ic in initial_conditions:
     sols[ic] = solve_ivp(system, t_span, ic, dense_output=True, max_step=0.05)
     sols_reversed[ic] = solve_ivp(system_reversed, t_span, ic, dense_output=True, max_step=0.05)
 for ic in initial_conditions_2:
     sols_2[ic] = solve_ivp(system, t_span, ic, dense_output=True, max_step=0.05)
     sols_reversed_2[ic] = solve_ivp(system_reversed, t_span, ic, dense_output=True, max_step=0.05)

 vs = np.linspace(vmin, vmax, 200)
 v_axis = np.linspace(vmin, vmax, 20)
 w_axis = np.linspace(wmin, wmax, 20)

 v_values, w_values = np.meshgrid(v_axis, w_axis)

 dv, dw = system(0, [v_values, w_values])

 fig, ax = plt.subplots(figsize=(16,16))
 # ax.scatter(x_root, y_root)
 # integral curves
 for ic in initial_conditions:
   sol = sols[ic]
   ax.plot(sol.y[0], sol.y[1],alpha=0.4, linewidth=0.5, color='k')
   sol = sols_reversed[ic]
   ax.plot(sol.y[0], sol.y[1], alpha=0.4, linewidth=0.5, color='k')
 for ic in initial_conditions_2:
   sol = sols_2[ic]
   ax.plot(sol.y[0], sol.y[1],alpha=0.8, linewidth=0.5, color='r')
   sol = sols_reversed_2[ic]
   ax.plot(sol.y[0], sol.y[1], alpha=0.8, linewidth=0.5, color='b')

 # vector fields
 arrow_lengths = np.sqrt(dv**2 + dw**2)
 alpha_values = 1 - (arrow_lengths / np.max(arrow_lengths))**0.4
 ax.quiver(v_values, w_values, dv, dw, color='blue', linewidth=0.5, scale=25, alpha=alpha_values)

 # nullclines
 ax.plot(vs, vs**2-mu*vs,  color="green", alpha=0.2, label="x nullcline")
 if np.abs(mu) < 0.001:
   ax.axvline(0, wmin, wmax, color="red", alpha=0.2, label="y nullcline")
   ax.axvline(1/2, wmin, wmax, color="red", alpha=0.2, label="y nullcline")
 else:  
   ax.plot(vs, (vs-2*vs**2)/mu, color="red", alpha=0.2, label="y nullcline")
ax.set_title(f'Hopf Bifurcation Model\n$\mu={mu:.3f
}) 
 # ax.legend()
 ax.set_xlim(vmin, vmax)
 ax.set_ylim(wmin, wmax)
 ax.set_xticks([])
 ax.set_yticks([])
 dir_path = f"./hopf"
 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 from natsort import natsorted import moviepy.editor as mp

for dir_path in ["./hopf"]: 

   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 = 24

   # 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")
```
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|date=2023-04-26 |source=Own work |author=Cosmia Nebula }}

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current19:00, 26 April 2023Thumbnail for version as of 19:00, 26 April 20231,600 × 1,600 (47.63 MB)Cosmia Nebula (talk | contribs)Uploaded while editing "Bifurcation theory" on en.wikipedia.org

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