"""
Period Doubling exercise.
"""
from IterateLogistic import *
def fsin(x, B):
"""
Sine map f_sin(x) = B sin(pi x), which also folds the unit interval
(0,1) into itself."""
return B*scipy.sin(scipy.pi * x)
def PlotIterate(g, x0, N, args=()):
"""
Plots g, the diagonal y=x, and the boxes made of the segments
[[x0,x0], [x0, g(x0)], [g(x0), g(x0)], [g(x0), g(g(x0))], ...
"""
traj = IterateList(g, x0, N, args)
xs = []
for x in traj:
xs.extend([x,x])
ys = xs[1:]
xs = xs[0:-1]
funcplotx = scipy.arange(0., 1.001, 0.01)
funcploty = scipy.array([g(x, *args) for x in funcplotx])
matplot.plot(xs, ys, 'b-', linewidth=1, antialiased=True)
matplot.plot(funcplotx, funcploty, 'g-', linewidth=3, antialiased=True)
matplot.plot([0.0, 1.0], [0.0, 1.0], 'r-', linewidth=3, antialiased=True)
matplot.show()
# if save:
# output = file('PlotIterate.dat','w')
# for x,y in zip(funcplotx, funcploty):
# output.write("%g %g\n"%(x,y))
# output.write("\n")
# for x,y in zip(xs, ys):
# output.write("%g %g\n"%(x,y))
# output.write("\n")
# for x,y in [(0.0,0.0),(1.0,1.0)]:
# output.write("%g %g\n"%(x,y))
# output.close()
# Just for figure
# matplot.grid('off')
# matplot.xaxis('fit')
# FastBifurcationDiagram(0.1, 2000, 256, scipy.arange(0.6,1.00000001, 0.0001))
def FastBifurcationDiagram(x0, ntransient, ncycleMax, muarray, g=f):
mus = []
trajs = []
for mu in muarray:
if mu < 0.75:
ncycle = 2
elif mu < (1. + scipy.sqrt(6.))/4.0:
ncycle = 4
else:
ncycle = ncycleMax
mus.extend([mu]*ncycle)
trajs.extend(IterateList(g, Iterate(g, x0, ntransient, (mu,)),
ncycle, (mu,)))
matplot.plot(mus, trajs, 'k.')
matplot.show()
# Just for figure
# mus, deltas, alphas, muInfinity = FindScalingConvergence(f, 9)
# FastBifurcationDiagramWithBoxes(0.1,2000,128,scipy.arange(0.4,1.0,0.00005),\
# mus, muInfinity)
def FastBifurcationDiagramWithBoxes(x0, ntransient, ncycleMax, muarray,
mu, muInfinity, g=f, nBoxes=7,
sY1=0.07, sY2=0.16, xMax=0.5,
plotXMin = 0.4, plotXMax = 1.0,
plotYMin = 0.3, plotYMax = 0.7):
matplot.plot()
matplot.grid('off')
matplot.axis([plotXMin,plotXMax,plotYMin,plotYMax])
xMin = g(g(0.5,muInfinity),muInfinity)
for n in range(nBoxes):
nIterated = 2**n
xMax = Iterate(g, xMin, 2*nIterated, (muInfinity,))
boxHoriz = [mu[n], mu[n], muInfinity, muInfinity, mu[n]]
boxVert = [xMin, xMax, xMax, xMin, xMin]
matplot.plot(boxHoriz, boxVert, 'r-', linewidth=3, antialiased=True)
xMin = xMax
mus = []
trajs = []
for mu in muarray:
if (mu>=plotXMin) and (mu <= plotXMax):
if mu < 0.75:
ncycle = 2
elif mu < (1. + scipy.sqrt(6.))/4.0:
ncycle = 4
else:
ncycle = ncycleMax
traj = IterateList(g, Iterate(g, x0, ntransient, (mu,)), ncycle, (mu,))
trajPruned = [y for y in traj if (y>=plotYMin) and (y <= plotYMax)]
trajs.extend(trajPruned)
mus.extend([mu]*len(trajPruned))
matplot.plot(mus, trajs, 'k.')
matplot.show()
def makeFBDWB():
mus, deltas, alphas, muInfinity = FindScalingConvergence(f,9)
FastBifurcationDiagramWithBoxes(0.1,2000,128,scipy.arange(0.4,1.0,0.001),
mus, muInfinity)
# Only for solution
def FixedPointIteratedMap(g, x0, nIterated, args=()):
"""
Returns fixed point g^[nIterated](x*) = x*
Defines a temporary function, fn(x)
which iterates g nIterated times (Iterate except with only one argument)
"""
fn = lambda x: Iterate(g, x, nIterated, args)
xf = scipy.optimize.fixed_point(fn, x0)
return xf
def FindSuperstable(g, nIterated, etaMin, etaMax, xMax=0.5):
"""
Finds superstable orbit g^[nIterated](xMax, eta) = xMax
in range (etaMin, etaMax).
Must be started with g-xMax of different sign at etaMin, etaMax.
Uses scipy.optimize.brentq, defining a temporary function, fn(eta)
which is zero for the superstable value of eta.
fn iterates g nIterated times and subtracts xMax
(basically Iterate - xMax, except with only one argument eta)
"""
fn = lambda eta: Iterate(g, xMax, nIterated, (eta,))-xMax
eta_root = scipy.optimize.brentq(fn, etaMin, etaMax)
return eta_root
def GetSuperstablePoints(g, nMax, eta0, eta1, xMax=0.5):
"""
Given the parameters for the first two superstable parameters eta_ss[0]
and eta_ss[1], finds the next nMax-1 of them up to eta_ss[nMax].
Returns dictionary eta_ss
Usage:
Find the value of the parameter eta_ss[0] = eta0 for which the fixed
point is xMax and g is superstable (g(xMax) = xMax), and the
value eta_ss[1] = eta1 for which g(g(xMax)) = xMax, either
analytically or using FindSuperstable by hand.
mus = GetSuperstablePoints(f, 9, eta0, eta1)
Searches for eta_ss[n] in the range (etaMin, etaMax),
with etaMin = eta_ss[n-1] + epsilon and
etaMax = eta_ss[n-1] + (eta_ss[n-1]-eta_ss[n-2])/A
where A=3 works fine for the maps I've tried.
(Asymptotically, A should be smaller than but comparable to delta.)
"""
eps = 1.0e-06
A = 3.
eta_ss = {0:eta0, 1:eta1}
for n in scipy.arange(2, nMax+1):
etaMin = eta_ss[n-1] + eps
etaMax = eta_ss[n-1] + (eta_ss[n-1]-eta_ss[n-2])/A
nIterated = 2**n
eta_ss[n] = FindSuperstable(g, nIterated, etaMin, etaMax, xMax)
return eta_ss
def DeltaAlpha(g, eta_ss, xMax=0.5):
"""
Given superstable 2^n cycle values eta_ss[n], calculates
delta[n] = (eta_{n-1}-eta_{n-2})/(eta_{n}-eta_{n-1}),
sep[n] = distance from xMax to attractor point halfway around circle,
and alpha = sep[n-1]/sep[n]
Also extrapolates eta to etaInfinity using definition of delta and
most reliable value for delta:
delta = lim{n->infinity} (eta_{n-1}-eta_{n-2})/(eta_n - eta_{n-1})
Returns delta and alpha dictionaries for n>=2, and etaInfinity
"""
delta = {}
alpha = {}
sep = {}
sep[1] = g(xMax, eta_ss[1]) - xMax
nMax = max(eta_ss.keys())
for n in range(2, nMax+1):
delta[n] = (eta_ss[n-1]-eta_ss[n-2])/(eta_ss[n]-eta_ss[n-1])
nIterated = 2**n
sep[n] = Iterate(g, xMax, nIterated/2, (eta_ss[n],))-xMax
alpha[n] = sep[n-1]/sep[n]
etaInfinity = eta_ss[nMax] + (eta_ss[nMax]-eta_ss[nMax-1])/(delta[nMax]-1.0)
return delta, alpha, etaInfinity
def FindScalingConvergence(g, nMax, xMax=0.5):
"""
Finds eta0, eta1:
Assumes superstable fixed point eta0 is between 0.3 and 1,
superstable period two cycle eta1 is between eta0+epsilon and 1
Calls GetSuperstablePoints
Calls DeltaAlpha
Returns etas, deltas, alphas, etaInfinity
"""
eps = 1.0e-6
eta0 = FindSuperstable(g, 1, 0.3, 1.0, xMax)
eta1 = FindSuperstable(g, 2, eta0 + eps, 1.0, xMax)
etas = GetSuperstablePoints(g, nMax, eta0, eta1, xMax)
deltas, alphas, etaInfinity = DeltaAlpha(g, etas, xMax)
return etas, deltas, alphas, etaInfinity
def PlotFIterated(g, n, args):
"""
Plots g^[2^n](x,*args) versus x, along with the curve y=x.
Also can plot box with corners (1-x*, 1-x*), (x*, x*)
"""
nIter = 2**n
# Curve
xs = scipy.arange(0.,1.000001,0.00002)
ys = [Iterate(g, x, nIter, args) for x in xs]
# Diagonal
matplot.plot(xs, ys, 'b-', linewidth=3)
matplot.plot([0.0,1.0],[0.0,1.0],'k-', linewidth=3)
# Box
fnMax = lambda x: g(x,*args) -x
xStarMax = scipy.optimize.brentq(fnMax, 0.5,1.0)
fnMin = lambda x: g(x,*args) -xStarMax
xStarMin = scipy.optimize.brentq(fnMin, 0.0,0.5)
boxX = [xStarMin, xStarMin, xStarMax, xStarMax, xStarMin]
boxY = [xStarMin, xStarMax, xStarMax, xStarMin, xStarMin]
matplot.plot(boxX, boxY,'r-', linewidth=2)
matplot.show()
# Just for figure
# if save:
# output = file('PlotFIterate.dat','w')
# for x,y in zip(xs, ys):
# output.write("%g %g\n"%(x,y))
# output.write("\n")
# for x,y in zip(boxX,boxY):
# output.write("%g %g\n"%(x,y))
# output.write("\n")
# for x,y in [(0.0,0.0),(1.0,1.0)]:
# output.write("%g %g\n"%(x,y))
# output.close()
# Just for figure
# mus, deltas, alphas, muInfinity = FindScalingConvergence(f, 9)
# PlotFIteratedAllBoxes(f, 8, (muInfinity,), alphas[9])
def PlotFIteratedAllBoxes(g, nMax, args, alpha, save=False):
# Curve
xs = scipy.arange(0.,1.000001,0.0002)
ys = [Iterate(g, x, 2.0**nMax, args) for x in xs]
# Diagonal
matplot.plot(xs, ys, 'b-', linewidth=3)
matplot.plot([0.0,1.0],[0.0,1.0],'k-', linewidth=3)
# Box
boxX = {}
boxY = {}
scale = 1.0
maxBoxLog2n = nMax
for n in range(maxBoxLog2n):
fnMax = lambda x: Iterate(g, x, 2**n, args) -x
xStarMax = scipy.optimize.brentq(fnMax, 0.5,0.5+0.5/scale)
fnMin = lambda x: Iterate(g, x, 2**n, args) - xStarMax
xStarMin = scipy.optimize.brentq(fnMin, 0.5-0.5/scale,0.5)
boxX[n] = [xStarMin, xStarMin, xStarMax, xStarMax, xStarMin]
boxY[n] = [xStarMin, xStarMax, xStarMax, xStarMin, xStarMin]
matplot.plot(boxX[n], boxY[n],'r-', linewidth=2)
scale *= alpha
matplot.hold('off')
matplot.show()
if save:
output = file('PlotFIterate.dat','w')
for x,y in zip(xs, ys):
output.write("%g %g\n"%(x,y))
output.write("\n")
for n in range(maxBoxLog2n):
for x,y in zip(boxX[n],boxY[n]):
output.write("%g %g\n"%(x,y))
output.write("\n")
for x,y in [(0.0,0.0),(1.0,1.0)]:
output.write("%g %g\n"%(x,y))
output.close()
def XStar(g, args=(), xMax = 0.5):
"""
Finds fixed point of one-humped map g, which is assumed to be
between xMax and 1.0.
"""
gXStar = lambda x: g(x, *args) - x
return scipy.optimize.brentq(gXStar, xMax, 1.0)
def Alpha(g, args=(), xMax = 0.5):
"""
Finds the (negative) scale factor alpha which inverts and rescales
the small inverted region of g(g(x)) running from (1-x*) to x*.
"""
gWindowMax = XStar(g, args, xMax)
gWindowMinFunc = lambda x: g(x, *args) - gWindowMax
gWindowMin = scipy.optimize.brentq(gWindowMinFunc, 0.0, xMax)
return 1.0/(gWindowMin-gWindowMax)
class T:
"""
Creates a new function T[g] from g, implementing Feigenbaum's
renormalization-group transformation of function space into itself.
We define it as a class so that we can initialize alpha and xStar,
which otherwise would need to be recalculated each time T[g] was
evaluated at a point x.
Usage:
Tg = T(g, args)
Tg(x) evaluates the function at x
"""
def __init__(self, g, args=(), xMax=0.5):
"""
Stores g and args.
Calculates and stores xStar and alpha.
"""
self.args = args
self.xStar = XStar(g, args, xMax)
self.alpha = Alpha(g, args, xMax)
self.g = g
def __call__(self, x):
"""
Defines xShrunk to be x/alpha + x*
Evaluates g2 = g(g(xShrunk))
Returns expanded alpha*(g2-xStar)
"""
xShrunk = x/self.alpha + self.xStar
g2 = self.g(self.g(xShrunk, *(self.args)), *(self.args))
return self.alpha * (g2 - self.xStar)
def PlotTgVsG(g, args, xMax=0.5):
"""Plots Tg(x) and g(x) on the same plot, for x from (0,1)"""
xs = scipy.arange(0., 1., 0.01)
gs = []
Tg = T(g, args)
Tgs = []
for x in xs:
gs.append(g(x, *args))
Tgs.append(Tg(x))
matplot.plot(xs, gs, xs, Tgs)
matplot.show()
# mus, deltas, alphas, muInfinity = FindScalingConvergence(f, 9)
# PlotTIterates(f, (muInfinity,))
# mus, deltas, alphas, muInfinity = FindScalingConvergence(fsin, 9)
# PlotTIterates(fsin, (muInfinity,))
def PlotTIterates(g, args, nMax = 2, xMax=0.5):
"""
Plots g(x), T[g](x), T[T[g]](x) ...
on the same plot, for x from (0,1)
"""
xs = scipy.arange(0., 1., 0.01)
Tg = {}
Tg[0] = lambda x: g(x, *args)
for n in range(1,nMax+1):
Tg[n] = T(Tg[n-1], xMax=xMax)
TgValues = {}
matplot.plot([0.0,1.0],[0.0,1.0])
for n in range(nMax+1):
TgValues[n] = []
for x in xs:
TgValues[n].append(Tg[n](x))
matplot.plot(xs, TgValues[n])
matplot.show()
def PlotT2FVsT2Fsin():
"""
Plots T[T[f]](x), T[T[fsin]](x) ...
on the same plot, for x from (0,1)
"""
xs = scipy.arange(0., 1., 0.01)
mus, deltas, alphas, muInfinity = FindScalingConvergence(f, 9)
mus, deltas, alphas, BInfinity = FindScalingConvergence(fsin, 9)
T2F = T(T(f, args=(muInfinity,)))
T2Fsin = T(T(fsin, args=(BInfinity,)))
matplot.plot([0.0,1.0],[0.0,1.0])
T2FValues = []
T2FsinValues = []
for x in xs:
T2FValues.append(T2F(x))
T2FsinValues.append(T2Fsin(x))
matplot.plot(xs, T2FValues, "r-", xs, T2FsinValues, "g-")
matplot.show()
def demo():
"""Demonstrates solution for exercise: example of usage"""
print "Period Doubling Demo"
print "Close plots to continue"
mus, deltas, alphas, muInfinity = FindScalingConvergence(f,9)
print " Fixed Point, mu=0.7"
print " (Close plot to continue)"
PlotIterate(f,0.1,20,(0.7,))
print " Period Doubling: mu=0.8"
PlotIterate(f,0.1,100,(0.8,))
print " Chaos: mu=0.9"
x0 = Iterate(f,0.5,100,(0.9,))
matplot.figure()
PlotIterate(f,x0,100,(0.9,))
print " Onset of Chaos: mu=muInfinity"
x0 = Iterate(f,0.5,100,(muInfinity,))
PlotIterate(f,x0,100,(muInfinity,))
print " Scaling in Bifurcation Diagram"
# Optional
matplot.figure()
FastBifurcationDiagramWithBoxes(0.1,2000,128,scipy.arange(0.4,1.0,0.002),
mus,muInfinity)
print " Convergence of alpha, delta, mu to scaling values"
matplot.figure()
matplot.plot(mus.keys(), mus.values(), "ro", label="mu")
matplot.plot(deltas.keys(), deltas.values(), "go", label = "delta")
matplot.plot(alphas.keys(), alphas.values(), "bo", label = "alpha")
matplot.legend()
matplot.show()
print " Renormalization-group Transformation"
matplot.figure()
PlotFIterated(f, 1, (muInfinity,))
print " Renormalization-group Transformation Twice"
# Optional
matplot.figure()
PlotFIteratedAllBoxes(f, 2, (muInfinity,), alphas[9])
print " Repeated renormalization-group transformations"
matplot.figure()
PlotTIterates(f, (muInfinity,))
print " Universality: T^2 of sine map and logistic map agree"
print " (Zoom in to see difference)"
matplot.figure()
PlotT2FVsT2Fsin()
if __name__=="__main__":
demo()
