![``` 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](http://upload.wikimedia.org/wikipedia/commons/8/8f/Hopf_and_homoclinic_bifurcation.gif)
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Summary
| Description |
```
})
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")
``` |
| Source | |
| Author |
|date=2023-04-26 |source=Own work |author=Cosmia Nebula }}
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 19:00, 26 April 2023 | | 1,600 × 1,600 (47.63 MB) | Cosmia Nebula | Uploaded while editing "Bifurcation theory" on en.wikipedia.org |
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