from maplib import Logistic, Sine
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
def bifurcation_diagram(map_type, params,
ntransients=100, ncycles=200, dotcolor="0.5",
fig=None,
nboundaries=2):
"""Plot a bifurcation diagram for an iterated map object.
Parameters
----------
map_type : functor
An iterated map constructor.
"""
nparam = len(params)
if nboundaries>0:
boundaries = np.zeros((nparam, nboundaries))
bound_rng = range(nboundaries)
xs = []
ys = []
for idx,param in enumerate(params):
m = map_type(param)
y = m.trajectory(m.iterate_from(0.5, ntransients), ncycles)
xs.extend(param*np.ones_like(y))
ys.extend(y)
if nboundaries:
# the boundaries are the iterates of the map's maximum, we assume
# here that it's located at 0.5
boundaries[idx] = m.trajectory(0.5, nboundaries+1)[1:]
# Make final figure
if fig is None:
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(xs, ys, '.', mfc=dotcolor, mec=dotcolor, ms=1, mew=0)
ax.set_xlim(params[0], params[-1])
if nboundaries>0:
bound_lines = ax.plot(params, boundaries)
def toggle(event):
if event.key!='t': return
for a in bound_lines:
v = a.get_visible()
a.set_visible(not v)
fig.canvas.draw()
fig.canvas.mpl_connect('key_press_event', toggle)
return fig
def cobweb(mu, walkers=10, steps=7):
f = plt.figure()
ax = f.add_subplot(111)
interval = np.linspace(0.0, 1.0, 100)
logmap = Logistic(mu)
logmap.plot(ax, interval, lw=2)
for x0 in np.random.rand(walkers):
logmap.plot_cobweb(ax, x0, steps, lw=2)
ax.set_title('Ex 2A: Random init. cond. for mu=%1.3f'%mu)
return f
def invariant_density(mu, x0,cycles=1000000,ret_all=False, bins=500):
transients = 1000
f = plt.figure()
ax = f.add_subplot(111)
logmap = Logistic(mu)
y0 = logmap.iterate_from(x0, transients)
y = logmap.trajectory(y0, cycles)
n, bins, patches = ax.hist(y, bins, normed=1)
ax.set_xlim(0,1)
if ret_all:
return f,logmap,n
else:
return f
# Exercise solutions
def ex2A():
cobweb(0.2)
cobweb(0.4)
cobweb(0.6)
def ex2B():
def rho(x):
return 1./(np.pi * np.sqrt( x*(1.-x)))
# Don't start from 0.5, which is a fixed point!
f = invariant_density(1.0, 0.567, bins=100)
ax = f.gca()
# avoid the edges: rho(x) is singular at 0 and 1!
x0 = np.linspace(0.001, 0.999, 1000)
l, = ax.plot(x0, rho(x0), 'r-', lw=3, alpha=0.7)
ax.set_title('Ex 2B: invariant density')
ax.legend((ax.patches[0], l), ('empirical', 'analytic'), loc='upper center')
ax.set_xlim(0,1)
ax.set_ylim(0,10)
def ex2CD(mu=0.9,x0=0.64):
fig, logmap, n = invariant_density(mu, x0, ret_all=True)
ax = fig.gca()
ax.set_xticks(np.linspace(0, 1, 10))
ax.grid(True)
# Now, iterate x=1/2 a few times and plot this 'orbit', which corresponds
# to peaks in the invariant density.
x0 = 0.5
pts = logmap.trajectory(x0,10)
pts_y = 0.5*np.linspace(1, max(n), len(pts))
ax.plot(pts,pts_y,'ro-')
ax.set_title('**Ex 2C/D: Analytics of cusps at mu=%0.2g' % mu)
def ex2E():
# Parameter grid to sample each map on
params = np.linspace(0.5, 1, 500)
fig = bifurcation_diagram(Logistic, params, nboundaries=8)
fig.gca().set_title('Ex 2E: Bifurcation diag. with boundaries (press t)')
fig = bifurcation_diagram(Logistic, params, dotcolor='blue')
fig = bifurcation_diagram(Sine, params, dotcolor='red', fig = fig)
fig.gca().set_title('Ex 2E: Bifurcation diag. logistic/sine maps')
if __name__=='__main__':
ex2A()
ex2B()
ex2CD()
ex2E()
plt.show()
