from maplib import Logistic,Sine
from matplotlib.numerix import arange, pi, sqrt, sort, zeros,ones, Float
from matplotlib.numerix.mlab import rand
from pylab import figure, draw, show, ion, ioff,frange,gcf,rcParams,rc
def bifurcation_diagram(map_type,param0=0,param1=1,nparam=300,
ntransients=100,ncycles=200, dotcolor="0.5",
fig=None,
nboundaries=0):
if nboundaries:
boundaries = zeros((nparam,nboundaries),Float)
params = frange(param0,param1,npts=nparam)
bound_rng = range(nboundaries)
xs = []
ys = []
for idx,param in enumerate(params):
m = map_type(param)
y = m.iterate(m.iterate(0.5,ntransients,lastonly=True),
ncycles, lastonly=False)
xs.extend(param*ones(len(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.iterate(0.5,nboundaries,lastonly=False)[1:]
if fig is None:
fig = figure()
ax = fig.add_subplot(111)
# save state (John, is there a cleaner way to do this?)
ax.plot(xs, ys, '.', mfc=dotcolor, mec=dotcolor, ms=1, mew=0)
bound_lines = []
for i in bound_rng:
bound_lines.extend(ax.plot(params,boundaries[:,i]))
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 = figure()
ax = f.add_subplot(111)
interval = frange(0.0, 1.0, npts=100)
logmap = Logistic(mu)
logmap.plot(ax, interval, lw=2)
for x0 in 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):
transients = 1000
bins = 500
f = figure()
ax = f.add_subplot(111)
logmap = Logistic(mu)
y = logmap.iterate(x0, transients, lastonly=True)
y = logmap.iterate(y, cycles, lastonly=False)
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./(pi * sqrt( x*(1.-x)))
# Don't start from 0.5, which is a fixed point!
f = invariant_density(1.0,0.567)
ax = f.gca()
# avoid the edges: rho(x) is singular at 0 and 1!
x0 = frange(0.001, 0.999, npts=1000)
l, = ax.plot(x0, rho(x0), 'r-', lw=3, alpha=0.5)
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):
f,logmap,n = invariant_density(mu,x0,ret_all=True)
ax = f.gca()
ax.set_xticks(frange(0,1,npts=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.iterate(x0,10)
pts_y = 0.5*frange(1,max(n),npts=len(pts))
ax.plot(pts,pts_y,'ro-')
ax.set_title('Ex 2C/D: Analytics of cusps at mu=%0.2g' % mu)
def ex2E():
par0 = 0.8
par1 = 1.0
fig = bifurcation_diagram(Logistic,par0,par1,nparam=1000,
nboundaries=8)
fig.gca().set_title('Ex 2E: Bifurcation diag. with boundaries (press t)')
fig = bifurcation_diagram(Logistic,par0,par1,dotcolor='blue')
fig = bifurcation_diagram(Sine,par0,par1,dotcolor='red',fig = fig)
fig.gca().set_title('Ex 2E: Bifurcation diag. logistic/sine maps')
if __name__=='__main__':
ex2A()
ex2B()
ex2CD()
ex2E()
show()
