Citation info: Boeing, G. 2016. "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction." Systems, 4 (4), 37. doi:10.3390/systems4040037.
Pynamical documentation: http://pynamical.readthedocs.org
This notebook demonstrates several ways to create phase diagrams to visualize system attractors and differentiate random noise from chaos. Phase diagrams embed 1-dimensional data (like time series) from a dynamical system into 2- or 3-dimensional phase space by plotting the value at time t vs the value at time t+1 (vs the value at time t+2, if it's a 3-D plot).
In [1]:
import pynamical
from pynamical import simulate, save_fig, phase_diagram, phase_diagram_3d
import pandas as pd, numpy as np, matplotlib.pyplot as plt, IPython.display as IPdisplay
%matplotlib inline
In [2]:
title_font = pynamical.get_title_font()
label_font = pynamical.get_label_font()
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# draw a phase diagram for 100 generations for the growth rate parameter 2.9
# it shows points converging on 0.655 because the logistic map has a fixed-point attractor at 0.655 when r=2.9
pops = simulate(num_gens=100, rate_min=2.9, num_rates=1, num_discard=100)
phase_diagram(pops, title='Logistic Map Attractor, r=2.9', size=20)
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# draw a phase diagram for 100 generations for the growth rate parameter 3.5
# it shows 4 points because the logistic map has a period of 4 when r=3.5
pops = simulate(num_gens=100, rate_min=3.5, num_rates=1, num_discard=100)
phase_diagram(pops, title='Logistic Map Attractor, r=3.5', size=20)
In [5]:
# draw a phase diagram for 100 generations for the growth rate parameter 3.56
# it shows 8 points because the logistic map has a period of 8 when r=3.56
pops = simulate(num_gens=100, rate_min=3.56, num_rates=1, num_discard=100)
phase_diagram(pops, title='Logistic Map Attractor, r=3.56', size=20)
In [6]:
# draw a phase diagram for 100 generations for the growth rate parameter 3.57
# it shows n points because the logistic map has a period of n when r=3.57
pops = simulate(num_gens=100, rate_min=3.57, num_rates=1, num_discard=100)
phase_diagram(pops, title='Logistic Map Attractor, r=3.57', size=20)
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# draw a phase diagram for 2,000 generations for the growth rate parameter 3.9
# the plot reveals the strange attractor - the logistic map is chaotic when r=3.9
pops = simulate(num_gens=2000, rate_min=3.9, num_rates=1)
phase_diagram(pops, xmin=0.25, xmax=0.75, ymin=0.8, ymax=1.01, size=20, title='Logistic Map Attractor, r=3.9')
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# draw a phase diagram for 2,000 generations across 50 growth rate steps from 3.6 to 4.0
# each chaotic growth rate has its own parabola
pops = simulate(num_gens=2000, rate_min=3.6, rate_max=4.0, num_rates=50)
phase_diagram(pops, xmin=0.25, xmax=0.75, ymin=0.8, ymax=1.01, size=7,
title='Logistic Map Attractor, r=3.6 to r=4.0', color='viridis')
In [9]:
# sometimes it is hard to tell if a time series is chaotic or random
# generate two time series of 1,000 steps, one chaotic and one random
# generate 30,000 time steps for the chaotic series but only keep the final 1,000 (when system is fully evolved)
total_gens = 30000
gens = 1000
np.random.seed(1)
chaos_pops = simulate(num_gens=total_gens, rate_min=3.99, num_rates=1)
chaos_pops = chaos_pops.iloc[total_gens-gens:].reset_index().drop(labels='index', axis=1)
random_pops = pd.DataFrame(np.random.random(gens), columns=['value'])
time_series = pd.concat([chaos_pops, random_pops], axis=1)
time_series.columns = ['chaos', 'random']
time_series.head()
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In [10]:
# plot the chaotic and random time series to show how they are sometimes tough to differentiate
ax = time_series.plot(kind='line', figsize=[10, 6], linewidth=3, alpha=0.6, style=['#003399','#cc0000'])
ax.grid(True)
ax.set_xlim(40, 90)
ax.set_ylim(0, 1)
ax.set_title('Time Series, Deterministic Chaos vs Random Data', fontproperties=title_font)
ax.set_xlabel('Generation', fontproperties=label_font)
ax.set_ylabel('Population', fontproperties=label_font)
ax.legend(loc=3)
save_fig('chaos-vs-random-line')
plt.show()
In [11]:
# plot same data as 2D phase diagram instead
pops = pd.concat([chaos_pops, random_pops], axis=1)
pops.columns = ['chaos', 'random']
phase_diagram(pops, size=20, color=['#003399','#cc0000'], ymax=1.005, legend=True,
filename='logistic-attractor-chaos-random')
In [12]:
# plot same data as 3D phase diagram instead
phase_diagram_3d(pops, color=['#003399','#cc0000'], filename='logistic-attractor-chaos-random-3d',
legend=True, legend_bbox_to_anchor=(0.94, 0.9))
In [13]:
# run logistic model for 4,000 generations across 50 growth rate steps from 3.6 to 4.0
pops = simulate(num_gens=4000, rate_min=3.6, rate_max=4.0, num_rates=50)
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# phase diagram: each chaotic growth rate has its own strange attractor curling through state space
phase_diagram_3d(pops, title='Logistic Map Attractor, r=3.6 to r=4.0', alpha=0.1, color='viridis', color_reverse=False,
azim=230, filename='3d-logistic-map-attractor-1')
In [15]:
# phase diagram: each chaotic growth rate has its own strange attractor curling through state space
phase_diagram_3d(pops, title='Logistic Map Attractor, r=3.6 to r=4.0', alpha=0.1, color='viridis', color_reverse=False,
elev=7, azim=320, filename='3d-logistic-map-attractor-2')
In [16]:
# here's an example of the animated phase diagrams that I create in pynamical-demo-3d-animation.ipynb
IPdisplay.Image(url='images/phase-animate/05-logistic-3d-phase-diagram-chaotic-regime.gif')
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