def pr1(file_prefix):
df = threebody(1.0, 0.5, 0.5, 0.5)
x0 = numpy.array([0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3000, 0, 0, 0, 0],
dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 7600)
xs = xs.transpose()
plot.render(xs[:, 0],
xs[:, 1],
'b',
xs[:, 6],
xs[:, 7],
'r',
xs[:, 12],
xs[:, 13],
'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
xbound=(-10.0, 10.0),
ybound=(0.0, 20.0),
aspect='equal',
title='3-body Orbital Trajectory',
legend=('Star A', 'Star B'),
file_prefix=suffixed(file_prefix, '_1a'))
plot.render(xs[:, 6] - xs[:, 0],
xs[:, 7] - xs[:, 1],
'r',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
aspect='equal',
title='3-body Trajectory, Star A at origin',
legend=('Star B', ),
file_prefix=suffixed(file_prefix, '_1b'))
def pr1(file_prefix):
df = threebody(1.0, 0.5, 0.5, 0.5)
x0 = numpy.array([
0, 0, 0,
0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 3000, 0,
0, 0, 0
], dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 7600)
xs = xs.transpose()
plot.render(xs[:,0], xs[:,1], 'b', xs[:,6], xs[:,7], 'r', xs[:,12], xs[:,13], 'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
xbound=(-10.0, 10.0),
ybound=(0.0, 20.0),
aspect='equal',
title='3-body Orbital Trajectory',
legend=('Star A', 'Star B'),
file_prefix=suffixed(file_prefix, '_1a')
)
plot.render(xs[:,6] - xs[:,0], xs[:,7] - xs[:,1], 'r',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
aspect='equal',
title='3-body Trajectory, Star A at origin',
legend=('Star B',),
file_prefix=suffixed(file_prefix, '_1b')
)
def pr2(file_prefix):
df = threebody(1.0, 0.5, 0.5, 0.5)
x0 = numpy.array(
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 20, 0, 0, -0.15, 0],
dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 50000)
xs = xs.transpose()
plot.render(xs[:5500, 0],
xs[:5500, 1],
'b',
xs[:5500, 6],
xs[:5500, 7],
'r',
xs[:5500, 12],
xs[:5500, 13],
'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title='3-body Orbital Interaction',
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix, '_2a'))
plot.render(xs[:, 0],
xs[:, 1],
'b',
xs[:, 12],
xs[:, 13],
'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title='3-body Orbital Interaction',
legend=('Star A', 'Star C'),
file_prefix=suffixed(file_prefix, '_2b'))
def pr2(file_prefix):
df = threebody(1.0, 0.5, 0.5, 0.5)
x0 = numpy.array([
0, 0, 0,
0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 20, 0,
0, -0.15, 0
], dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 50000)
xs = xs.transpose()
plot.render(xs[:5500,0], xs[:5500,1], 'b', xs[:5500,6], xs[:5500,7], 'r', xs[:5500,12], xs[:5500,13], 'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title='3-body Orbital Interaction',
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix, '_2a')
)
plot.render(xs[:,0], xs[:,1], 'b', xs[:,12], xs[:,13], 'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title='3-body Orbital Interaction',
legend=('Star A', 'Star C'),
file_prefix=suffixed(file_prefix, '_2b')
)
def pr2b(file_prefix, delay, dim):
embed_run(
dset='data3',
delay=delay,
vdim=dim,
ydim=5,
file_prefix=suffixed(file_prefix, '_2b')
)
def pr2b(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, xxs = rungekutta.rk4(lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64), 0.0001, 1000000)
xs = xxs.transpose()
x0 = numpy.array([-30, -40, 4], dtype=numpy.float64)
d_cap = find_loglog_slope(xs[100000:400000, :], x0, file_prefix=suffixed(file_prefix, "_2b"))
print "Lorenz\tshort\td_cap = {0:.6f}".format(d_cap)
return xs
def pr2a(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, xxs = rungekutta.rk4(lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64), 0.0001, 300000)
xs = xxs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.mutual("-D", 16000, input=data)
tau = first_min(ms)
ws = tispy.delay("-d", int(tau), "-m", 7, input=data)
x0 = numpy.array([-35, -35], dtype=numpy.float64)
zs = numpy.array(zip(ws[100000:, 0], ws[100000:, 5]), dtype=numpy.float64)
d_cap = find_loglog_slope(zs, x0, file_prefix=suffixed(file_prefix, "_2a"))
print "Lorenz\tembed\td_cap = {0:.6f}".format(d_cap)
def pr2b(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, xxs = rungekutta.rk4(
lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64),
0.0001, 1000000)
xs = xxs.transpose()
x0 = numpy.array([-30, -40, 4], dtype=numpy.float64)
d_cap = find_loglog_slope(xs[100000:400000, :],
x0,
file_prefix=suffixed(file_prefix, '_2b'))
print 'Lorenz\tshort\td_cap = {0:.6f}'.format(d_cap)
return xs
def pr1(file_prefix=None):
data = numpy.loadtxt('data/ps8/data1', dtype=numpy.float64)
points = construct_angular_velocity(data[::30, 1], data[::30, 0])
thetas = numpy.array([pendulum.mod2pi(theta) for theta in points[:, 1]],
dtype=numpy.float64)
plot.render(thetas,
points[:, 2],
'b',
xlabel=r'$\theta$',
ylabel=r'$\omega$',
title='State-space Trajectory, Data Set 1',
file_prefix=suffixed(file_prefix, '_1'))
def pr2a(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, xxs = rungekutta.rk4(
lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64),
0.0001, 300000)
xs = xxs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.mutual('-D', 16000, input=data)
tau = first_min(ms)
ws = tispy.delay('-d', int(tau), '-m', 7, input=data)
x0 = numpy.array([-35, -35], dtype=numpy.float64)
zs = numpy.array(zip(ws[100000:, 0], ws[100000:, 5]), dtype=numpy.float64)
d_cap = find_loglog_slope(zs, x0, file_prefix=suffixed(file_prefix, '_2a'))
print 'Lorenz\tembed\td_cap = {0:.6f}'.format(d_cap)
def pr1(file_prefix=None):
data = numpy.loadtxt('data/ps8/data1', dtype=numpy.float64)
points = construct_angular_velocity(data[::30,1], data[::30,0])
thetas = numpy.array([
pendulum.mod2pi(theta) for theta in points[:,1]
], dtype=numpy.float64)
plot.render(
thetas,
points[:,2],
'b',
xlabel=r'$\theta$',
ylabel=r'$\omega$',
title='State-space Trajectory, Data Set 1',
file_prefix=suffixed(file_prefix, '_1')
)
def pr4(file_prefix=None):
data = numpy.loadtxt("data/ps8/data2.first250sec", dtype=numpy.float64)
vs = tispy.delay("-d", "155", "-m", "8", input=data)
plot.embedded(
vs,
0,
2,
"k.",
markersize=0.6,
ybound=(0, 2 * numpy.pi),
yticks=(0, numpy.pi / 2, numpy.pi, 3 * numpy.pi / 2, 2 * numpy.pi),
yticklabels=("0", r"$\frac{\pi}{2}$", r"$\pi$", r"$\frac{3\pi}{2}$", r"$2\pi$"),
title=r"Delay Coordinate Embedding: {0}, $\tau={1}$ s".format("data2", _get_actual_delay("data2", 1, 155)),
xlabel=r"$\theta(t + {0})$".format(_get_actual_delay("data2", 0, 155)),
ylabel=r"$\theta(t + {0})$".format(_get_actual_delay("data2", 2, 155)),
aspect="equal",
file_prefix=suffixed(file_prefix, "_3"),
)
def pr3(file_prefix=None):
data = numpy.loadtxt("data/ps8/data2.first250sec", dtype=numpy.float64)
fnns = tispy.false_nearest("-M", "1,10", "-d", "155", input=data)
dim = select_embedding_dimension(fnns)
def _render(axes):
axes.plot(fnns[:, 0], fnns[:, 1])
axes.plot([dim, dim], axes.get_ybound(), "k")
axes.plot(axes.get_xbound(), [0.1, 0.1], "k--")
axes.legend(("False nearest neighbors", r"Selected $m$", r"$\leq$10% FNNs required"))
plot.render(
xlabel=r"$m$",
ylabel="Fraction of false nearest neighbors",
title=r"False nearest neighbors vs. embedding dimension ($m$={0})".format(select_embedding_dimension(fnns)),
ax_callback=_render,
file_prefix=suffixed(file_prefix, "_2"),
)
def pr5(dim=8, file_prefix=None):
data = numpy.loadtxt('data/ps8/data2', dtype=numpy.float64)
ms = tispy.lyap_k('-d', 155, '-m', dim, '-M', dim, input=data)
pargs = cut(ms[:, 0], ms[:, 1])
# Find the average slope of the first two points.
total = 0.0
count = len(pargs) / 2
for i in xrange(0, len(pargs), 2):
[x1, x2] = pargs[i][:2]
[y1, y2] = pargs[i + 1][:2]
total += (y2 - y1) / (0.31 * (x2 - x1))
print 'Found Lyapunov exponent: {0:.6f}'.format(total / count)
plot.render(*pargs,
xlabel='Iterations',
ylabel='Logarithm of the Stretching Factor',
title='Calculating the Lyapunov Exponent',
file_prefix=suffixed(file_prefix, '_4'))
def pr4(file_prefix):
"""This is not the problem you are looking for."""
x0 = numpy.array([
0, 0, 0,
0, 0, 0,
1, 0, 0,
0, 1, 0,
0, 21.5, 0,
0, -0.15, 0
], dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 8000)
xs = xs.transpose()
plot.render(xs[:,0], xs[:,1], 'b', xs[:,6], xs[:,7], 'r', xs[:,12], xs[:,13], 'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title=r'3-body Orbital Interaction ($x_{14}=21.5$)',
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix, '_4')
)
def pr3(file_prefix=None):
data = numpy.loadtxt('data/ps8/data2.first250sec', dtype=numpy.float64)
fnns = tispy.false_nearest('-M', '1,10', '-d', '155', input=data)
dim = select_embedding_dimension(fnns)
def _render(axes):
axes.plot(fnns[:, 0], fnns[:, 1])
axes.plot([dim, dim], axes.get_ybound(), 'k')
axes.plot(axes.get_xbound(), [0.1, 0.1], 'k--')
axes.legend(('False nearest neighbors', r'Selected $m$',
r'$\leq$10% FNNs required'))
plot.render(
xlabel=r'$m$',
ylabel='Fraction of false nearest neighbors',
title=r'False nearest neighbors vs. embedding dimension ($m$={0})'.
format(select_embedding_dimension(fnns)),
ax_callback=_render,
file_prefix=suffixed(file_prefix, '_2'))
def pr4(file_prefix=None):
data = numpy.loadtxt('data/ps8/data2.first250sec', dtype=numpy.float64)
vs = tispy.delay('-d', '155', '-m', '8', input=data)
plot.embedded(
vs,
0,
2,
'k.',
markersize=0.6,
ybound=(0, 2 * numpy.pi),
yticks=(0, numpy.pi / 2, numpy.pi, 3 * numpy.pi / 2, 2 * numpy.pi),
yticklabels=('0', r'$\frac{\pi}{2}$', r'$\pi$', r'$\frac{3\pi}{2}$',
r'$2\pi$'),
title=r'Delay Coordinate Embedding: {0}, $\tau={1}$ s'.format(
'data2', _get_actual_delay('data2', 1, 155)),
xlabel=r'$\theta(t + {0})$'.format(_get_actual_delay('data2', 0, 155)),
ylabel=r'$\theta(t + {0})$'.format(_get_actual_delay('data2', 2, 155)),
aspect='equal',
file_prefix=suffixed(file_prefix, '_3'))
def pr3(file_prefix):
ic = lambda y: numpy.array([
0, 0, 0,
0, 0, 0,
1, 0, 0,
0, 1, 0,
0, y, 0,
0, -0.15, 0
], dtype=numpy.float64)
df = threebody(1.0, 0.5, 0.5, 0.5)
for i in xrange(8):
ts, xs = rungekutta.rk4(df, 0, ic(0.3*i + 20.0), 0.005, 8000)
xs = xs.transpose()
plot.render(xs[:,0], xs[:,1], 'b', xs[:,6], xs[:,7], 'r', xs[:,12], xs[:,13], 'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title=r'3-body Orbital Interaction ($x_{14}=%.1f$)' % (0.3*i+20.0),
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix, '_3{0}'.format(chr(97+i)))
)
def pr4(file_prefix):
"""This is not the problem you are looking for."""
x0 = numpy.array(
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 21.5, 0, 0, -0.15, 0],
dtype=numpy.float64)
ts, xs = rungekutta.rk4(df, 0, x0, 0.005, 8000)
xs = xs.transpose()
plot.render(xs[:, 0],
xs[:, 1],
'b',
xs[:, 6],
xs[:, 7],
'r',
xs[:, 12],
xs[:, 13],
'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title=r'3-body Orbital Interaction ($x_{14}=21.5$)',
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix, '_4'))
def pr2(delay=None, file_prefix=None):
data = numpy.loadtxt("data/ps8/data2.first250sec", dtype=numpy.float64)
if delay is not None:
ms = tispy.mutual("-D", str(delay), input=data)
else:
ms = tispy.mutual(input=data)
tau = first_min(ms)
def _render(axes):
axes.plot(ms[:, 0], ms[:, 1])
(lbound, ubound) = axes.get_ybound()
axes.plot([tau, tau], [lbound, ubound], "k")
axes.legend(("Mutual information", r"Optimal $\tau$"))
plot.render(
xlabel=r"$\tau$",
ylabel="Mutual information",
title=r"Mutual information vs. $\tau$ (min at $\tau$={0})".format(first_min(ms)),
ax_callback=_render,
file_prefix=suffixed(file_prefix, "_1"),
)
def pr2(delay=None, file_prefix=None):
data = numpy.loadtxt('data/ps8/data2.first250sec', dtype=numpy.float64)
if delay is not None:
ms = tispy.mutual('-D', str(delay), input=data)
else:
ms = tispy.mutual(input=data)
tau = first_min(ms)
def _render(axes):
axes.plot(ms[:, 0], ms[:, 1])
(lbound, ubound) = axes.get_ybound()
axes.plot([tau, tau], [lbound, ubound], 'k')
axes.legend(('Mutual information', r'Optimal $\tau$'))
plot.render(
xlabel=r'$\tau$',
ylabel='Mutual information',
title=r'Mutual information vs. $\tau$ (min at $\tau$={0})'.format(
first_min(ms)),
ax_callback=_render,
file_prefix=suffixed(file_prefix, '_1'))
def pr5(dim=8, file_prefix=None):
data = numpy.loadtxt("data/ps8/data2", dtype=numpy.float64)
ms = tispy.lyap_k("-d", 155, "-m", dim, "-M", dim, input=data)
pargs = cut(ms[:, 0], ms[:, 1])
# Find the average slope of the first two points.
total = 0.0
count = len(pargs) / 2
for i in xrange(0, len(pargs), 2):
[x1, x2] = pargs[i][:2]
[y1, y2] = pargs[i + 1][:2]
total += (y2 - y1) / (0.31 * (x2 - x1))
print "Found Lyapunov exponent: {0:.6f}".format(total / count)
plot.render(
*pargs,
xlabel="Iterations",
ylabel="Logarithm of the Stretching Factor",
title="Calculating the Lyapunov Exponent",
file_prefix=suffixed(file_prefix, "_4")
)
def pr6(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, vs = rungekutta.rk4(lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64), 0.001, 30000)
xs = vs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.mutual("-D", 1600, input=data)
tau = first_min(ms)
ws = tispy.delay("-d", int(tau), "-m", 7, input=data)
col1 = 0
col2 = 5
plot.render(
ws[:, col1],
ws[:, col2],
"k.",
markersize=0.6,
title=r"Delay Coordinate Embedding: {0}, $\tau={1}$ s".format(r"lorenz", 0.001 * tau),
xlabel=r"$x(t+{0:.3f})$".format(col1 * 0.001 * tau),
ylabel=r"$x(t+{0:.3f})$".format(col2 * 0.001 * tau),
file_prefix=suffixed(file_prefix, "_6"),
)
return tau
def pr6(file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, vs = rungekutta.rk4(
lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64),
0.001, 30000)
xs = vs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.mutual('-D', 1600, input=data)
tau = first_min(ms)
ws = tispy.delay('-d', int(tau), '-m', 7, input=data)
col1 = 0
col2 = 5
plot.render(ws[:, col1],
ws[:, col2],
'k.',
markersize=0.6,
title=r'Delay Coordinate Embedding: {0}, $\tau={1}$ s'.format(
r'lorenz', 0.001 * tau),
xlabel=r'$x(t+{0:.3f})$'.format(col1 * 0.001 * tau),
ylabel=r'$x(t+{0:.3f})$'.format(col2 * 0.001 * tau),
file_prefix=suffixed(file_prefix, '_6'))
return tau
def pr6x1(tau, file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, vs = rungekutta.rk4(lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64), 0.001, 30000)
xs = vs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.lyap_k("-d", int(tau), "-m", 7, "-M", 7, input=data)
pargs = cut(ms[:, 0], ms[:, 1])
total = 0.0
count = len(pargs) / 2
for i in xrange(0, len(pargs), 2):
xs = 0.31 * pargs[i][:50]
ys = pargs[i + 1][:50]
ps = numpy.polyfit(xs, ys, 1)
total += ps[0]
print "Lorenz lyapunov exponent: {0}".format(total / count)
plot.render(
*pargs,
xlabel="Iterations",
ylabel="Logarithm of the Stretching Factor",
title="Calculating the Lyapunov Exponent",
file_prefix=suffixed(file_prefix, "_7")
)
def pr6x1(tau, file_prefix=None):
lfunc = lorenz.lorenz(16, 45, 4)
ts, vs = rungekutta.rk4(
lfunc, 0.0, numpy.array([-13.0, -12.0, 52.0], dtype=numpy.float64),
0.001, 30000)
xs = vs.transpose()[:, 0]
data = numpy.array(zip(xs, ts), dtype=numpy.float64)
ms = tispy.lyap_k('-d', int(tau), '-m', 7, '-M', 7, input=data)
pargs = cut(ms[:, 0], ms[:, 1])
total = 0.0
count = len(pargs) / 2
for i in xrange(0, len(pargs), 2):
xs = 0.31 * pargs[i][:50]
ys = pargs[i + 1][:50]
ps = numpy.polyfit(xs, ys, 1)
total += ps[0]
print 'Lorenz lyapunov exponent: {0}'.format(total / count)
plot.render(*pargs,
xlabel='Iterations',
ylabel='Logarithm of the Stretching Factor',
title='Calculating the Lyapunov Exponent',
file_prefix=suffixed(file_prefix, '_7'))
def pr3(file_prefix):
ic = lambda y: numpy.array(
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, y, 0, 0, -0.15, 0],
dtype=numpy.float64)
df = threebody(1.0, 0.5, 0.5, 0.5)
for i in xrange(8):
ts, xs = rungekutta.rk4(df, 0, ic(0.3 * i + 20.0), 0.005, 8000)
xs = xs.transpose()
plot.render(xs[:, 0],
xs[:, 1],
'b',
xs[:, 6],
xs[:, 7],
'r',
xs[:, 12],
xs[:, 13],
'g',
xlabel='x (normalized AU)',
ylabel='y (normalized AU)',
title=r'3-body Orbital Interaction ($x_{14}=%.1f$)' %
(0.3 * i + 20.0),
legend=('Star A', 'Star B', 'Star C'),
file_prefix=suffixed(file_prefix,
'_3{0}'.format(chr(97 + i))))
def pr2c(xs, file_prefix=None):
x0 = numpy.array([-30, -40, 4], dtype=numpy.float64)
d_cap = find_loglog_slope(xs[100000:, :], x0, file_prefix=suffixed(file_prefix, "_2c"))
print "Lorenz\tlong\td_cap = {0:.6f}".format(d_cap)
def pr2b(file_prefix, delay, dim):
embed_run(dset='data3',
delay=delay,
vdim=dim,
ydim=5,
file_prefix=suffixed(file_prefix, '_2b'))
def pr2a(file_prefix=None):
embed_run(dset='data2', delay=75, ydim=2, file_prefix=suffixed(file_prefix, '_2a'))
def pr2c(xs, file_prefix=None):
x0 = numpy.array([-30, -40, 4], dtype=numpy.float64)
d_cap = find_loglog_slope(xs[100000:, :],
x0,
file_prefix=suffixed(file_prefix, '_2c'))
print 'Lorenz\tlong\td_cap = {0:.6f}'.format(d_cap)
def pr2a(file_prefix=None):
embed_run(dset='data2',
delay=75,
ydim=2,
file_prefix=suffixed(file_prefix, '_2a'))