Fractal Dimension ¶

In the study of dynamical systems there are many quantities that identify as "entropy". Notice that these quantities are not the more commonly known thermodynamic ones, used in Statistical Physics. Rather, they are more like the to the entropies of information theory, which represents information contained within a dataset, or information about the dimensional scaling of a dataset. Based on the definition of the generalized entropy, one can calculate an appropriate dimension.

Work in progress

      %matplotlib inline

import utils
import box_count
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm

samples=50
t, x_t = utils.solve_lorenz(N=samples, t_max=1000)

     

      # Pick a colormap: viridis, plasma, gray, binary, seismic, gnuplot
viridis = cm.get_cmap('viridis', 50)
colors = viridis(np.linspace(0, 1, 50))

plt.figure(figsize=(12, 12))

plt.style.use('bmh')
plt.xlabel('$x$')
plt.ylabel('$z$')

for i in range(1):
    x, y, z = x_t[i,:,:].T
    plt.plot(x, z, "-")

     

      plt.style.use('bmh')
fig, ax = plt.subplots(figsize=[10, 10])

# zoom-in lower-left corner of inset axes, and its width and height.
# axins = ax.inset_axes([0.675, 0.05, 0.25, 0.25])
axins = ax.inset_axes([0.1, 0.1, 0.3, 0.3])

# sub-region of the original image
x1, x2, y1, y2 = -5, -4.4, 28.4, 29.5

axins.set_xlim(x1, x2)
axins.set_ylim(y1, y2)
axins.set_xticklabels('')
axins.set_yticklabels('')
axins.set_facecolor('white')

rectpatch, connects=ax.indicate_inset_zoom(axins)
connects[0].set_visible(False)
connects[1].set_visible(True)
connects[2].set_visible(False)
connects[3].set_visible(True)


for i in range(samples):
    x, y, z = x_t[i,:,:].T
    xi, zi = utils.find_zero_crossings(x,y,z)

    ax.plot(xi, zi, '.')
    axins.plot(xi, zi, '.')

plt.xlabel('$x$')
plt.ylabel('$z$')

     

Out[69]:

Text(0, 0.5, '$z$')

      fd = []
for i in range(1, samples+1):
    d = box_count.dimension(np.concatenate(x_t[:i,:,:], axis=0))
    fd.append(d)

# print(fd)

# plt.style.use('bmh')
# fig, ax = plt.subplots(figsize=(12, 6))
# ax.set_xlabel("Samples")
# ax.set_ylabel("Dimension")

# ax.scatter(range(1, samples+1), fd, c="red")

     

      import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

def fit_function(xd, a, b, n):
    return a * xd ** n  / (xd ** n + b)

x = np.arange(1., samples + 1.)

popt, pcov = curve_fit(fit_function, x, fd)

plt.style.use('bmh')
fig, ax = plt.subplots(figsize=(12, 6))

plt.scatter(x, fd)
plt.plot(x, fit_function(x, *popt), 'r-')

plt.show()

# Standard deviation errors on the parameters
sd = np.sqrt(np.diag(pcov))

# Optimal values
print(f"Fitted parameters: a = {popt[0]:.3f} ± {sd[0]:.2f}, b = {popt[1]:.3f}, n = {popt[2]:.3f}")

print(f" {sd}")

     

Fitted parameters: a = 2.101 ± 0.09, b = 0.134, n = 0.175
 [0.09333063 0.04731838 0.0860685 ]

Fractal Dimension ¶

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