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
Pynamical can simulate, visualize, and explore any discrete dynamical system. This notebook demonstrates how to do this with two additional models that come included in the Pynamical module (the Cubic Map and the Singer Map) as well as with one additional model defined in this notebook (the Mandelbrot Map). Any other models can be created and easily plugged-in, as demonstrated below.
In [1]:
from pynamical import simulate, cubic_map, singer_map, bifurcation_plot, phase_diagram, phase_diagram_3d
from numba import jit
%matplotlib inline
First we visualize the behavior of the Cubic Map. To do this, our code is nearly identical to the previous examples using the Logistic Map, with the exception that we are now passing model=cubic_map into our simulate function.
In [2]:
pops = simulate(model=cubic_map, num_gens=100, rate_min=1, rate_max=4, num_rates=1000, num_discard=100)
bifurcation_plot(pops, title='Cubic Map Bifurcation Diagram', xmin=1, xmax=4, save=False)
In [3]:
pops = simulate(model=cubic_map, num_gens=1000, rate_min=3.99, num_rates=1, num_discard=100)
phase_diagram(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, save=False, title='Cubic Map 2D Phase Diagram')
In [4]:
pops = simulate(model=cubic_map, num_gens=1000, rate_min=3.99, num_rates=1, num_discard=100)
phase_diagram_3d(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, zmin=-1, zmax=1, save=False, title='Cubic Map 3D Phase Diagram')
In [5]:
pops = simulate(model=cubic_map, num_gens=3000, rate_min=3.5, num_rates=30, num_discard=100)
phase_diagram_3d(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, zmin=-1, zmax=1, save=False, alpha=0.2, color='viridis',
azim=330, title='Cubic Map 3D Phase Diagram, r = 3.5 to 4.0')
In [6]:
pops = simulate(model=singer_map, num_gens=100, rate_min=0.9, rate_max=1.08, num_rates=1000, num_discard=100)
bifurcation_plot(pops, title='Singer Map Bifurcation Diagram', xmin=0.9, xmax=1.08, save=False)
In [7]:
pops = simulate(model=singer_map, num_gens=1000, rate_min=1.07, num_rates=1, num_discard=100)
phase_diagram(pops, title='Singer Map 2D Phase Diagram, r = 1.07', save=False)
In [8]:
pops = simulate(model=singer_map, num_gens=1000, rate_min=1.07, num_rates=1, num_discard=100)
phase_diagram_3d(pops, title='Singer Map 3D Phase Diagram, r = 1.07', save=False)
In [9]:
pops = simulate(model=singer_map, num_gens=3000, rate_min=1.04, rate_max=1.07, num_rates=30, num_discard=100)
phase_diagram_3d(pops, save=False, alpha=0.2, color='viridis', title='Singer Map 3D Phase Diagram, r = 1.04 to 1.07')
In [10]:
@jit(nopython=True)
def mandelbrot_map(pop, rate):
return pop ** 2 + rate
In [11]:
pops = simulate(model=mandelbrot_map, num_gens=100, rate_min=-2, rate_max=0, num_rates=1000, num_discard=100)
bifurcation_plot(pops, title='Mandelbrot Map Bifurcation Diagram', xmin=-2, xmax=0, ymin=-2, ymax=2, save=False)
In [12]:
pops = simulate(model=mandelbrot_map, num_gens=1000, rate_min=-1.99, num_rates=1, num_discard=100)
phase_diagram(pops, title='Mandelbrot Map 2D Phase Diagram', xmin=-2, xmax=2, ymin=-2, ymax=2, save=False)
In [13]:
pops = simulate(model=mandelbrot_map, num_gens=1000, rate_min=-1.99, num_rates=1, num_discard=100)
phase_diagram_3d(pops, title='Mandelbrot Map 3D Phase Diagram', xmin=-2, xmax=2, ymin=-2, ymax=2, zmin=-2, zmax=2, save=False)
In [14]:
pops = simulate(model=mandelbrot_map, num_gens=3000, rate_min=-1.99, rate_max=-1.5, num_rates=30, num_discard=100)
phase_diagram_3d(pops, title='Mandelbrot Map 3D Phase Diagram', alpha=0.2, color='viridis', color_reverse=True,
xmin=-2, xmax=2, ymin=-2, ymax=2, zmin=-2, zmax=2, save=False)