Example-05: Frequency (Jacobian)

# In this example frequency derivative is used as choas indicator

# Frequency estimation based on window weighted average phase advance is a differentiable function
# Hence, it is possible to compute derivaties of it with respect to initial conditions or paraters

# In the case of derivatives with respect to initial condition (frequency vector Jacobin)
# One would expect frequency to be continuous if space for regular initials
# Normally, the norm of Jacobian is expected to be relatively small for regular initials
# For choatic initials this norm is observed to be much large
# Thus, Jacobian norm can be used as a chaos indicator

# Note, in Frequency Map Analysis (FMA) frequency time invariance is used as the idea for chaos detection
# Here, space continuity is used in a similar way

# Import

import numpy

from tqdm import tqdm

import jax
from jax import jit
from jax import vmap
from jax import jacrev
from jax import jacfwd

# Test symplectic mapping and corresponding inverse

from tohubohu.util import forward4D

# Frequency factory

from tohubohu import exponential
from tohubohu import frequency

# Plotting

from matplotlib import pyplot as plt
from matplotlib import colormaps

cmap = colormaps.get_cmap('viridis')
cmap.set_bad(color='lightgray')

# Set data type

jax.config.update("jax_enable_x64", False)

# Set device

device, *_ = jax.devices('gpu')
jax.config.update('jax_default_device', device)

# Set mapping parameters

nux, nuy = 0.168, 0.201
mux, muy = 2*jax.numpy.pi*nux, 2*jax.numpy.pi*nuy
cx, sx, cy, sy = jax.numpy.cos(mux), jax.numpy.sin(mux), jax.numpy.cos(muy), jax.numpy.sin(muy)
mu = 0.0

k = jax.numpy.asarray([cx, sx, cy, sy, mu])
x = jax.numpy.array([0.0, 0.0, 0.0, 0.0])

# Set frequency function

ws = exponential(2**12)
fn = jit(frequency(ws, forward4D))

out = fn(x, k)

# Set frequency based indicator

@jit
def gn(x, k):
    return jax.numpy.log10(1.0E-16 + jax.numpy.linalg.norm(jacrev(fn)(x, k)))

out = gn(x, k)

# Set initial grid in (qx, qy) plane

n = 1001

qx = jax.numpy.linspace(0.0, 0.6, n)
qy = jax.numpy.linspace(0.0, 0.6, n)
qs = jax.numpy.stack(jax.numpy.meshgrid(qx, qy, indexing='ij')).swapaxes(-1, 0).reshape(n*n, -1)
ps = jax.numpy.full_like(qs, 1.0E-12)
xs = jax.numpy.hstack([qs, ps])
xs = jax.numpy.array_split(xs, n)

# Evaluate indicator

xb, *xr = xs
fj = jit(vmap(gn, (0, None)))
out = [fj(xb, k)]

for xb in tqdm(xr):
    out.append(fj(xb, k))
out = jax.numpy.concatenate(out)

# Winsorize data

data = numpy.array(out)
data[data < -16.0] = -16.0
data[data > 16.0] = 16.0
data = data.reshape(n, n)

# Plot

plt.figure(figsize=(8, 8))
plt.imshow(data, aspect='equal', vmin=-16.0, vmax=16.0, origin='lower', cmap=cmap, interpolation='nearest', extent=(0., 0.6, 0., 0.6))
plt.xlabel('qx')
plt.ylabel('qy')
plt.tight_layout()
plt.show()

100%|█████████████████████████████████████████████████████████████████████████████████████████| 1000/1000 [01:33<00:00, 10.68it/s]