File:Duffing oscillator.webm

def potential_well(x, a, b):
    return - a * x ** 2 + b * x ** 4

def draw_potential_well(potential_well, F, x0, xmin, xmax, ax, safety_factor=0.03):
    x = np.linspace(xmin, xmax, 200)
    y = potential_well(x)
    ymin, ymax = min(y), max(y)
    ymin -= (ymax - ymin) * safety_factor
    ymax += (ymax - ymin) * safety_factor
    
    ax.plot(x, y, color='gray')

    arrow_props = {'width': (ymax-ymin) * 5e-3, 'head_width': (ymax-ymin) * 2e-2, 
                   'head_length': (xmax-xmin) * 2e-2, 'length_includes_head': True,
                   'facecolor': '#4a5a90', 'edgecolor': 'none'}

    ax.arrow(x0, potential_well(x0), F, 0, **arrow_props)
    
    ax.scatter(x0, potential_well(x0), color='#938fba', s=100)

    ax.set_xlim(xmin, xmax)
    ax.set_ylim(ymin, ymax)
    return ax

# Code modified from https://github.com/vkulkar/Duffing by Vikram Kulkarni

import numpy as np
import matplotlib.pyplot as plt

# parameters (mass = 1)
a, b = 0.5, 1/16 # potential coefficients
gamma = 0.1 # damping coefficient
F_0 = 2.5 # driving force
omega = 2.0 # driving angular frequency
period = 2*np.pi/omega
cycles, steps_per_cycle = 100000, 65
h = period/steps_per_cycle # time step

# length of the simulation
T = period * cycles
t = np.arange(0,T,h)

def x_2(x,v):    return -gamma*v + 2.0*a*x - 4.0*b*x*x*x
def x_3(x2,x,v):     return -gamma*x2 + 2.0*a*v -12.0*b*x*x*v
def x_4(x3,x2,x,v):    return -gamma*x3 + 2.0*a*x2 -12.0*b*x*x*x2 - 24.0*b*v*v*x
def x_5(x4,x3,x2,x,v):    return -gamma*x4 + 2*a*x3 -12.0*b*(x*x*x3 + 2.0*x2*x*v) -24.0*b*(v*v*v+2*x*v*x2)

# Trigonometric terms in derivatives
x2F =  F_0*np.cos(omega*t)
x3F = -F_0*omega*np.sin(omega*t)
x4F = -F_0*omega*omega*np.cos(omega*t)
x5F =  F_0*omega*omega*omega*np.sin(omega*t)

# Taylor series coefficients
coef1 = 1/2  *h**2
coef2 = 1/6  *h**3
coef3 = 1/24 *h**4
coef4 = 1/120*h**5

# initial conditions
x, v = 0.5, 0.0

position = np.zeros(len(t))
velocity = np.zeros(len(t))
position[0] = x

for i in range(1,len(t)):
    d2 = x_2(x,v) + x2F[i]
    d3 = x_3(d2,x,v) + x3F[i]
    d4 = x_4(d3,d2,x,v) + x4F[i]
    d5 = x_5(d4,d3,d2,x,v) + x5F[i]
    # Taylor series expansion for x,v. Order h^5
    x += v*h + coef1*d2 + coef2*d3 + coef3*d4 + coef4*d5
    v += d2*h + coef1*d3 + coef2*d4 + coef3*d5
    position[i] = x
    velocity[i] = v

def get_lims(tensor, safety_factor = 0.03):
    if tensor.shape[1] != 2:
        tensor = tensor.T
    xmin, xmax = min(tensor[:,0]), max(tensor[:,0])
    ymin, ymax = min(tensor[:,1]), max(tensor[:,1])
    xmin -= (xmax - xmin) * safety_factor
    xmax += (xmax - xmin) * safety_factor
    ymin -= (ymax - ymin) * safety_factor
    ymax += (ymax - ymin) * safety_factor
    return xmin, xmax, ymin, ymax

def plotting(trail, tmin, tmax, t, position, velocity):
    xmin, xmax, ymin, ymax = get_lims(np.array([position, velocity]))

    fig, axs = plt.subplot_mosaic("AAAAAB;AAAAAC;AAAAAC;AAAAAD", figsize=(20, 16))

    # Poincare plot
    ax=axs['A']
    poincare_plot = np.array([position, velocity]).T[(tmax%steps_per_cycle)::steps_per_cycle,:]
    ax.scatter(poincare_plot[:,0],poincare_plot[:,1], color='#938fba', s=2)
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_xlim(xmin, xmax)
    ax.set_ylim(ymin, ymax)
    ax.set_xlabel('$x$',{'fontsize':16})
    ax.set_ylabel('$\dot x$',{'fontsize':16})
    # ax.set_title(r'Poincare Plot (Phase space at time = $\frac{2\pi N}{\omega}$, N = 1,2,3...)',{'fontsize':16})
    ax.tick_params(axis='both',labelsize=16)
    
    # Vector field plot
    x_axis = np.linspace(xmin, xmax, 20)
    y_axis = np.linspace(ymin, ymax, 20)
    x_values, y_values = np.meshgrid(x_axis, y_axis)
    
    dx = 1.0*y_values
    dy = x_2(x_values, y_values) + x2F[tmax]
    arrow_lengths = np.sqrt(dx**2 + dy**2)
    alpha_values = 1 - (arrow_lengths / np.max(arrow_lengths))**0.4
    ax.quiver(x_values, y_values, dx, dy, color='blue', linewidth=0.5, alpha=alpha_values)

    # Potential well plot
    ax = axs['B']
    draw_potential_well(potential_well=(lambda x: potential_well(x, a, b)), 
                        F=x2F[tmax], x0=position[tmax], xmin=xmin, xmax=xmax, ax=ax, safety_factor=0.03)
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_xlabel('$x$',{'fontsize':16})
    ax.set_ylabel('$V(x)$',{'fontsize':16})

    # Trajectory plot
    ax = axs['C']
    ax.plot(position[max(0, tmin):tmax], t[max(0, tmin):tmax], color='#4a5a90', linewidth=1)
    ax.scatter(position[tmax], t[tmax], color='#938fba')
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_xlim(xmin, xmax)
    ax.set_ylim((tmin-2)*h, (tmax+2)*h)
    # ax.set_title('Trajectory of the oscillator',{'fontsize':16})
    ax.set_xlabel('$x$',{'fontsize':16})
    ax.set_ylabel('$t$',{'fontsize':16})
    ax.tick_params(axis='both',labelsize=16)

    # Phase space plot
    ax=axs['D']
    for j in range(max(0, tmin), tmax):
        alpha = (j - (tmax - trail)) / trail
        ax.plot(position[j-1:j+1], velocity[j-1:j+1], '.-', markersize=2, color='#4a5a90', alpha=alpha)
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_xlim(xmin, xmax)
    ax.set_ylim(ymin, ymax)
    # ax.set_title('Phase space',{'fontsize':16})
    ax.set_xlim([-4.5,4.5])
    ax.set_xlabel('$x$',{'fontsize':16})
    ax.set_ylabel('$\dot x$',{'fontsize':16})
    ax.tick_params(axis='both',labelsize=16)

    fig.suptitle(fr'Duffing, $(\alpha, \beta, \gamma, F_0, \omega) = ({a}, {b}, {gamma}, {F_0}, {omega})$', fontsize=20,y=0.92)
    return fig, ax

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

def export_to_video(dir_paths, fps=12):
    for dir_path in dir_paths:
        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 = []

        # 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")
    return

from tqdm import trange
import os
for i, tmax in enumerate(trange(0, 10 * steps_per_cycle)):
    trail = 3 * steps_per_cycle
    tmin = tmax-trail
    fig, ax = plotting(trail=trail, tmin=tmin, tmax=tmax, t=t, position=position, velocity=velocity)
    
    dir_path = "./duffing"
    if not os.path.exists(dir_path):
        os.makedirs(dir_path)

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

export_to_video(["./duffing"], fps=12)
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