def sine_timeplot_fig(mu_vals = [-1e-3, -0.1, -0.3, -0.40]):
# set up the model parameters for this figure
# mu = -0.2
alpha = 0.23333
k = 1
n_phis = np.linspace(0, 2*math.pi, 4*20+1)
dx = 1e-4
dy = 0.
mag = math.sqrt(dx**2 + dy**2)
phasescale = 4 / (2 * math.pi) # convert from (0,2 \pi) to (0,4)
# create a new figure
fig = plt.figure(figsize=(6,6))
width = 1./len(mu_vals)
padding = 0.2*width
for i in range(len(mu_vals)):
mu = mu_vals[i]
axes = plt.axes((2*padding, 1-(i+1) * width+padding,
1 - width - 2*padding, width - 1.5*padding))
# draw the trajectory components vs. time
T, y0, error = iris.sine_limit_cycle(mu, alpha, k)
ts = np.linspace(0, 3*T, 1000);
vals = integrate.odeint(iris.sine_system,
[0, -y0],
ts,
args=(mu, alpha, k))
axes.plot(ts, vals[:,1], '-', color='0.8', lw=2)
axes.plot(ts, vals[:,0], 'k-', lw=2)
axes.set_xlim(0, 3*T)
# make the y-axis symmetric around zero
#ymaxabs = np.max(np.abs(axes.get_ylim()))
ymaxabs = 0.6*math.pi
axes.set_ylim(-ymaxabs, ymaxabs)
# draw the phase plot for reference
axes = plt.axes((1-width+padding, 1-(i+1) * width + padding,
width - 1.5 * padding, width - 1.5 * padding))
iris.draw_fancy_sine_system(axes, mu, alpha, k, scale=3.)
# center the plot and clean up the scale bars
border = 0.2
axes.set_xlim(-math.pi/2-border, math.pi/2+border)
axes.set_ylim(-math.pi/2-border, math.pi/2+border)
axes.set_xticks([])
axes.set_yticks([])
axes.set_frame_on(False)
return fig
def sine_prc_fig(mu_vals = [-1e-3, -0.1, -0.3, -0.40]):
# set up the model parameters for this figure
# mu = -0.2
alpha = 0.23333
k = 1
n_phis = np.linspace(0, 2*math.pi, 4*20+1)
dx = 1e-4
dy = 0.
mag = math.sqrt(dx**2 + dy**2)
phasescale = 4 / (2 * math.pi) # convert from (0,2 \pi) to (0,4)
# create a new figure
fig = plt.figure(figsize=(6,6))
width = 1./len(mu_vals)
padding = 0.2*width
for i in range(len(mu_vals)):
mu = mu_vals[i]
axes = plt.axes((2*padding, 1-(i+1) * width+padding,
1 - width - 2*padding, width - 1.5*padding))
# draw the orthogonal prc found numerically
T, y0, error = iris.sine_limit_cycle(mu, alpha, k)
n_prc_o = np.array([
iris.sine_phase_reset(phi, dx=-dy, dy=dx, mu=mu, alpha=alpha, k=k,
y0=y0, T=T,
steps_per_cycle=100000)
for phi in n_phis
])
orthogonal_color = '0.8'
axes.plot(n_phis, n_prc_o/mag*phasescale, '--', markersize=3,
color=orthogonal_color)
axes.plot(n_phis, n_prc_o/mag*phasescale, 'bo', markersize=3,
color=orthogonal_color, markeredgecolor=orthogonal_color)
axes.set_xlim(0, 2*math.pi)
# draw the prc found numerically
T, y0, error = iris.sine_limit_cycle(mu, alpha, k)
n_prc = np.array([
iris.sine_phase_reset(phi, dx=dx, dy=dy, mu=mu, alpha=alpha, k=k,
y0=y0, T=T,
steps_per_cycle=100000)
for phi in n_phis
])
axes.plot(n_phis, n_prc/mag*phasescale, '--k', markersize=3)
axes.plot(n_phis, n_prc/mag*phasescale, 'bo', markersize=3)
axes.set_xlim(0, 2*math.pi)
plt.xticks(np.arange(5.)*math.pi/2,
#['$0$', '$\\pi/2$', '$\\pi$',
# '$3\\pi/2$', '$2\\pi$'])
# phase rescaled from 0 to 4 to match analysis
['$0$', '$1$', '$2$', '$3$', '$4$'])
# make the y-axis symmetric around zero
ymaxabs = np.max(np.abs(axes.get_ylim()))
axes.set_ylim(-ymaxabs, ymaxabs)
# draw the phase plot for reference
axes = plt.axes((1-width+padding, 1-(i+1) * width + padding,
width - 1.5 * padding, width - 1.5 * padding))
iris.draw_fancy_sine_system(axes, mu, alpha, k, scale=3.)
# center the plot and clean up the scale bars
border = 0.2
axes.set_xlim(-math.pi/2-border, math.pi/2+border)
axes.set_ylim(-math.pi/2-border, math.pi/2+border)
axes.set_xticks([])
axes.set_yticks([])
axes.set_frame_on(False)
return fig
def sine_fig(mu = -0.1):
alpha = 0.23333
k = 1
# create a new figure
fig = plt.figure(figsize=(3.5,3.5))
#axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])
#axes = fig.add_axes([0.1, 0.1, 0.89, 0.89])
axes = fig.add_axes([0.07, 0.07, 0.92, 0.92])
# draw the vector field
X, Y = np.meshgrid(
np.linspace(-math.pi, math.pi, 6*4+1)+ math.pi/(6*4),
np.linspace(-math.pi, math.pi, 6*4+1)+ math.pi/(6*4)
)
(U,V) = iris.sine_system(np.asarray([X, Y]), 0., mu, alpha, k)
axes.quiver(X, Y, U, V)
# draw the x nullclines
ys = xs = np.linspace(-math.pi, math.pi, 4*200)
# Each x nullcline exists within a diagonal band;
# calculate the band's height from the intersection of the nullclines
# when mu = 0.
h1 = 2 * math.asin(2*alpha) + math.pi
h2 = math.pi - 2 * math.asin(2*alpha)
nullx1 = np.array([
optimize.brentq(
lambda y : iris.sine_system(np.asarray([x,y]), 0., mu, alpha, k)[0],
x + h1/2, x - h1/2
) for x in xs
])
axes.plot(xs, nullx1, 'k')
axes.plot(xs, nullx1 - 2*math.pi, 'k')
axes.plot(xs, nullx1 + 2*math.pi, 'k')
nullx2 = np.array([
optimize.brentq(
lambda y : iris.sine_system(np.asarray([x,y]), 0., mu, alpha, k)[0],
x + math.pi + h2/2, x + math.pi - h2/2
) for x in xs
])
axes.plot(xs, nullx2, 'k')
axes.plot(xs, nullx2 - 2*math.pi, 'k')
axes.plot(xs, nullx2 + 2*math.pi, 'k')
# plot the y nullclines
nully1 = np.array([
optimize.brentq(
lambda x : iris.sine_system(np.asarray([x,y]), 0., mu, alpha, k)[1],
-y + h1/2, -y - h1/2
) for y in ys
])
axes.plot(nully1, ys, 'k', dashes=(2,1))
axes.plot(nully1 - 2*math.pi, ys, 'k', dashes=(2,1))
axes.plot(nully1 + 2*math.pi, ys, 'k', dashes=(2,1))
nully2 = np.array([
optimize.brentq(
lambda x : iris.sine_system(np.asarray([x,y]), 0., mu, alpha, k)[1],
-y + math.pi + h2/2, -y + math.pi - h2/2
) for y in ys
])
axes.plot(nully2, ys, 'k', dashes=(2,1))
axes.plot(nully2 - 2*math.pi, ys, 'k', dashes=(2,1))
axes.plot(nully2 + 2*math.pi, ys, 'k', dashes=(2,1))
# draw the limit cycle and limit points
iris.draw_fancy_sine_system(axes, mu, alpha, k, scale=1.)
# add a trajectory
x0 = [0, 0.1]
vals = integrate.odeint(iris.sine_system, x0,
np.linspace(0, 200, 20000),
args=(mu, alpha, k))
axes.plot(vals[:,0], vals[:,1], 'b', lw=1)
# center the plot and clean up the scale bars
border = 0.0
axes.set_xlim(-math.pi/2-border, math.pi/2+border)
axes.set_ylim(-math.pi/2-border, math.pi/2+border)
plt.xticks(np.linspace(-2, 2, 5)*math.pi/2,
[r'$-\pi$', r'$-\frac{\pi}{2}$', '$0$', r'$\frac{\pi}{2}$',
r'$\pi$'])
#[r'$-\pi$', r'$-\pi/2$', '$0$', r'$\pi/2$', r'$\pi$'])
plt.yticks(np.linspace(-2, 2, 5)*math.pi/2,
[r'$-\pi$', r'$-\frac{\pi}{2}$', '$0$', r'$\frac{\pi}{2}$',
r'$\pi$'])
return fig
