File:Hopf and homoclinic bifurcation.gif
Original file (1,600 × 1,600 pixels, file size: 47.63 MB, MIME type: image/gif, 240 frames, 19 s)
Note: Due to technical limitations, thumbnails of high resolution GIF images such as this one will not be animated.
This file is from a shared repository and may be used by other projects. The description on its file description page there is shown below.
Summary
DescriptionHopf and homoclinic bifurcation.gif |
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")``` |
Source |
This file is lacking source information.
Please edit this file's description and provide a source. |
Author |
This file is lacking author information.
|
|date=2023-04-26 |source=Own work |author=Cosmia Nebula }}
Licensing
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Items portrayed in this file
depicts
image/gif
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 19:00, 26 April 2023 | 1,600 × 1,600 (47.63 MB) | commons>Cosmia Nebula | Uploaded while editing "Bifurcation theory" on en.wikipedia.org |
File usage
There are no pages that use this file.