def partD(numpts=6400):
kwargs = {'F0':2e0, 'omega':2.4e0, 'gam':0.1e0, 'm':1e0, 'a':.5e0, 'b':.25e0}
T = 2 * pi / kwargs['omega']
t = arange(0, numpts*T, T)
sol = duffing(t, x0=[0.5e0, 1e0], **kwargs)
x, p = (sol[:, 0], kwargs['m'] * sol[:, 1])
xmin, xmax = (min(x), max(x))
pmin, pmax = (min(p), max(p))
Lx = xmax - xmin
Lp = pmax - pmin
L = max(Lx, Lp)
def occupied(divisions):
"""
Create array showing what squares are occupied.
"""
N = float(divisions)
inside = [[any((x >= xmin + i * L / N) * (x < xmin + (i + 1) * L / N) *
(p >= pmin + j * L / N) * (p < pmin + (j + 1) * L / N))
for j in range(divisions)]
for i in range(divisions)]
return array(inside, dtype='bool')
fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')
attractor, = ax.plot(x, p, color='black', marker=',', linestyle='None')
divisions = 2**4
occ = occupied(divisions)
# print(occ)
print('sum(occ)={0}'.format(sum(sum(occ))))
l = L / float(divisions)
indices = zip(*where(occ))
for (i, j) in indices:
x_i = xmin + i * l
p_i = pmin + j * l
ax.add_patch(Rectangle((x_i, p_i), l, l, alpha=.1))
plt.show()
def count(divisions):
"""
Find the number of occupied squares.
This is a simpler and memory friendly version of `occupied` that only
computes the total number of occupied squares.
"""
N = float(divisions)
occupied = sum(any((x >= xmin + i * L / N) * (x < xmin + (i + 1) * L / N) *
(p >= pmin + j * L / N) * (p < pmin + (j + 1) * L / N))
for i in range(divisions) for j in range(divisions))
return occupied
divisions = range(2, 8)
logb = log(L / array(divisions, dtype='float'))
num_occupied = map(count, divisions)
print(num_occupied)
lognum = log(num_occupied)
z = polyfit(logb, lognum, 1)
print(z)
plt.plot(logb, lognum, 'r.')
plt.plot(logb, poly1d(z)(logb), 'b-', label=r'$\log\left(N\right)={0:.3g}-{1:.3g}\cdot\log\left(b\right)$'.format(z[1], -z[0]))
plt.legend()
plt.show()
def partA(numpts=2000):
t = linspace(0, 100, numpts)
positions = (0.5e0, 0.5001e0)
v0 = 0e0
solutions = [duffing(t, [x0, v0], m=1e0, a=.5e0, b=0.25e0, F0=2.0e0, omega=2.4e0, gam=0.1e0) for x0 in positions]
for pos, sol in zip(positions, solutions):
plt.plot(t, sol[:,0], label='$x_0={0}$'.format(pos))
plt.legend()
plt.show()
def partB(numpts=2000):
kwargs = {'F0':2e0, 'omega':2.4e0, 'gam':0.1e0, 'm':1e0, 'a':.5e0, 'b':.25e0}
T = 2 * pi / kwargs['omega']
t = arange(0, numpts*T, T)
# t = linspace(0, 10*T, numpts)
sol = duffing(t, x0=[0.5e0, 1e0], **kwargs)
x, v = (sol[:, 0], sol[:, 1])
plt.plot(x, kwargs['m']*v, color='black', marker=',', linestyle='None')
plt.xlabel('$x$')
plt.ylabel('$p$')
plt.title('Strange Attractor for Duffing Oscillator')
plt.show()
def partA(numpts=2000):
t = linspace(0, 100, numpts)
positions = (0.5e0, 0.5001e0)
v0 = 0e0
solutions = [
duffing(t, [x0, v0],
m=1e0,
a=.5e0,
b=0.25e0,
F0=2.0e0,
omega=2.4e0,
gam=0.1e0) for x0 in positions
]
for pos, sol in zip(positions, solutions):
plt.plot(t, sol[:, 0], label='$x_0={0}$'.format(pos))
plt.legend()
plt.show()
def partB(numpts=2000):
kwargs = {
'F0': 2e0,
'omega': 2.4e0,
'gam': 0.1e0,
'm': 1e0,
'a': .5e0,
'b': .25e0
}
T = 2 * pi / kwargs['omega']
t = arange(0, numpts * T, T)
# t = linspace(0, 10*T, numpts)
sol = duffing(t, x0=[0.5e0, 1e0], **kwargs)
x, v = (sol[:, 0], sol[:, 1])
plt.plot(x, kwargs['m'] * v, color='black', marker=',', linestyle='None')
plt.xlabel('$x$')
plt.ylabel('$p$')
plt.title('Strange Attractor for Duffing Oscillator')
plt.show()
def partD(numpts=6400):
kwargs = {
'F0': 2e0,
'omega': 2.4e0,
'gam': 0.1e0,
'm': 1e0,
'a': .5e0,
'b': .25e0
}
T = 2 * pi / kwargs['omega']
t = arange(0, numpts * T, T)
sol = duffing(t, x0=[0.5e0, 1e0], **kwargs)
x, p = (sol[:, 0], kwargs['m'] * sol[:, 1])
xmin, xmax = (min(x), max(x))
pmin, pmax = (min(p), max(p))
Lx = xmax - xmin
Lp = pmax - pmin
L = max(Lx, Lp)
def occupied(divisions):
"""
Create array showing what squares are occupied.
"""
N = float(divisions)
inside = [[
any((x >= xmin + i * L / N) * (x < xmin + (i + 1) * L / N) *
(p >= pmin + j * L / N) * (p < pmin + (j + 1) * L / N))
for j in range(divisions)
] for i in range(divisions)]
return array(inside, dtype='bool')
fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')
attractor, = ax.plot(x, p, color='black', marker=',', linestyle='None')
divisions = 2**4
occ = occupied(divisions)
# print(occ)
print('sum(occ)={0}'.format(sum(sum(occ))))
l = L / float(divisions)
indices = zip(*where(occ))
for (i, j) in indices:
x_i = xmin + i * l
p_i = pmin + j * l
ax.add_patch(Rectangle((x_i, p_i), l, l, alpha=.1))
plt.show()
def count(divisions):
"""
Find the number of occupied squares.
This is a simpler and memory friendly version of `occupied` that only
computes the total number of occupied squares.
"""
N = float(divisions)
occupied = sum(
any((x >= xmin + i * L / N) * (x < xmin + (i + 1) * L / N) *
(p >= pmin + j * L / N) * (p < pmin + (j + 1) * L / N))
for i in range(divisions) for j in range(divisions))
return occupied
divisions = range(2, 8)
logb = log(L / array(divisions, dtype='float'))
num_occupied = map(count, divisions)
print(num_occupied)
lognum = log(num_occupied)
z = polyfit(logb, lognum, 1)
print(z)
plt.plot(logb, lognum, 'r.')
plt.plot(
logb,
poly1d(z)(logb),
'b-',
label=r'$\log\left(N\right)={0:.3g}-{1:.3g}\cdot\log\left(b\right)$'.
format(z[1], -z[0]))
plt.legend()
plt.show()
