注釈
Click here to download the full example code
Complexified KAM curve (standard map)¶
\(k=0.9, \nu=0.618034\)
[1]J. M. Greene and I. C. Percival, Hamiltonian Maps in the Complex Plane, Physica D: Nonlinear Phenomena 3, 530 (1981).
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import sys
import glob
k = -0.9
twopi=2.0*np.pi
def Q(coeff, theta):
res = np.copy(theta) + 0.j
for i, m in enumerate(coeff[0]):
res += -1.j*coeff[1][i]*np.exp(1.j*theta*m) + 1.j*coeff[1][i]*np.exp(-1.j*theta*m)
return res
def P(coeff, theta):
return Q(coeff, theta) - Q(coeff, theta + twopi*nu) - np.sin(Q(coeff, theta))*k/2
dirname = "nu_0.618034"
nu = float(dirname.split("_")[1])
path = dirname + "/fourier_coeff_j21.dat"
coeff = np.loadtxt(path).transpose()
fig = plt.figure()
ax = fig.add_subplot(1,1,1,projection='3d')
ax.axis('off')
traj = np.loadtxt(dirname + "/traj.DAT").T
ymin,ymax = 3.5, 4.5
index = (traj[1]>ymin) & (traj[1]<ymax)
index2 = (traj[1]<3.8) & (traj[0]<4)
traj[0][index2] = np.nan
traj[1][index2] = np.nan
index2 = (traj[1]<4.1) & (traj[0]<0.8)
traj[0][index2] = np.nan
traj[1][index2] = np.nan
ax.plot(traj[0][index],traj[1][index],',',c='k')
sample=100
im_max= 0.0
theta = np.linspace(0,twopi,sample) + 1.j*im_max
q = Q(coeff, theta)
p = P(coeff, theta)
ax.plot(q.real, -p.real, q.imag,'-k',lw=2)
sample=3000
im_max= 0.083717354931950
theta = np.linspace(0,twopi,sample) + 1.j*im_max
q = Q(coeff, theta)
p = P(coeff, theta)
ax.plot(q.real, -p.real, q.imag,'-k',lw=2)
sample=200
#im_max = 0.0155717354931950
[re_theta, im_theta] = np.meshgrid(
np.linspace(0,twopi, sample),
np.linspace(0, im_max, sample)
)
theta = re_theta + 1.j*im_theta
q = Q(coeff, theta)
p = P(coeff, theta)
#ax.plot_wireframe(q.real, -p.real, q.imag,rstride=20, cstride=1, alpha=0.1,color='k')
ax.plot_wireframe(q.real, -p.real, q.imag,rstride=20, cstride=2, color='k',alpha=0.5)
ax.set_ylim(ymin,ymax)
ax.set_xlim(0,2*np.pi)
ax.view_init(40,40)
#plt.savefig("KAM.png",transparent=True)
plt.show()
A table of fourier coefficients¶
print(coeff) # fourier_coeff_j21.dat"
Out:
[[ 1.00000000e+00 2.00000000e+00 3.00000000e+00 4.00000000e+00
5.00000000e+00 6.00000000e+00 7.00000000e+00 8.00000000e+00
9.00000000e+00 1.00000000e+01 1.10000000e+01 1.20000000e+01
1.30000000e+01 1.40000000e+01 1.50000000e+01 1.60000000e+01
1.70000000e+01 1.80000000e+01 1.90000000e+01 2.00000000e+01
2.10000000e+01 2.20000000e+01 2.30000000e+01 2.40000000e+01
2.50000000e+01 2.60000000e+01 2.70000000e+01 2.80000000e+01
2.90000000e+01 3.00000000e+01 3.10000000e+01 3.20000000e+01
3.30000000e+01 3.40000000e+01 3.50000000e+01 3.60000000e+01
3.70000000e+01 3.80000000e+01 3.90000000e+01 4.00000000e+01
4.10000000e+01 4.20000000e+01 4.30000000e+01 4.40000000e+01
4.50000000e+01 4.60000000e+01 4.70000000e+01 4.80000000e+01
4.90000000e+01 5.00000000e+01 5.10000000e+01 5.20000000e+01
5.30000000e+01 5.40000000e+01 5.50000000e+01 5.60000000e+01
5.70000000e+01 5.80000000e+01 5.90000000e+01 6.00000000e+01
6.10000000e+01 6.20000000e+01 6.30000000e+01 6.40000000e+01
6.50000000e+01 6.60000000e+01 6.70000000e+01 6.80000000e+01
6.90000000e+01 7.00000000e+01 7.10000000e+01 7.20000000e+01
7.30000000e+01 7.40000000e+01 7.50000000e+01 7.60000000e+01
7.70000000e+01 7.80000000e+01 7.90000000e+01 8.00000000e+01
8.10000000e+01 8.20000000e+01 8.30000000e+01 8.40000000e+01
8.50000000e+01 8.60000000e+01 8.70000000e+01 8.80000000e+01
8.90000000e+01 9.00000000e+01 9.10000000e+01 9.20000000e+01
9.30000000e+01 9.40000000e+01 9.50000000e+01 9.60000000e+01
9.70000000e+01 9.80000000e+01 9.90000000e+01 1.00000000e+02
1.01000000e+02 1.02000000e+02 1.03000000e+02 1.04000000e+02
1.05000000e+02 1.06000000e+02 1.07000000e+02 1.08000000e+02
1.09000000e+02 1.10000000e+02 1.11000000e+02 1.12000000e+02
1.13000000e+02 1.14000000e+02 1.15000000e+02 1.16000000e+02
1.17000000e+02 1.18000000e+02 1.19000000e+02 1.20000000e+02
1.21000000e+02 1.22000000e+02 1.23000000e+02 1.24000000e+02
1.25000000e+02 1.26000000e+02 1.27000000e+02 1.28000000e+02
1.29000000e+02 1.30000000e+02 1.31000000e+02 1.32000000e+02
1.33000000e+02 1.34000000e+02 1.35000000e+02 1.36000000e+02
1.37000000e+02 1.38000000e+02 1.39000000e+02 1.40000000e+02
1.41000000e+02 1.42000000e+02 1.43000000e+02 1.44000000e+02
1.45000000e+02 1.46000000e+02 1.47000000e+02 1.48000000e+02
1.49000000e+02 1.50000000e+02 1.51000000e+02 1.52000000e+02
1.53000000e+02 1.54000000e+02 1.55000000e+02 1.56000000e+02
1.57000000e+02 1.58000000e+02 1.59000000e+02 1.60000000e+02
1.61000000e+02 1.62000000e+02 1.63000000e+02 1.64000000e+02
1.65000000e+02 1.66000000e+02 1.67000000e+02 1.68000000e+02
1.69000000e+02 1.70000000e+02 1.71000000e+02 1.72000000e+02
1.73000000e+02 1.74000000e+02 1.75000000e+02 1.76000000e+02
1.77000000e+02 1.78000000e+02 1.79000000e+02 1.80000000e+02
1.81000000e+02 1.82000000e+02 1.83000000e+02 1.84000000e+02
1.85000000e+02 1.86000000e+02 1.87000000e+02 1.88000000e+02
1.89000000e+02 1.90000000e+02 1.91000000e+02 1.92000000e+02
1.93000000e+02 1.94000000e+02 1.95000000e+02 1.96000000e+02
1.97000000e+02 1.98000000e+02 1.99000000e+02 2.00000000e+02
2.01000000e+02 2.02000000e+02 2.03000000e+02 2.04000000e+02
2.05000000e+02 2.06000000e+02 2.07000000e+02 2.08000000e+02
2.09000000e+02 2.10000000e+02 2.11000000e+02 2.12000000e+02
2.13000000e+02 2.14000000e+02 2.15000000e+02 2.16000000e+02
2.17000000e+02 2.18000000e+02 2.19000000e+02 2.20000000e+02
2.21000000e+02 2.22000000e+02 2.23000000e+02 2.24000000e+02
2.25000000e+02 2.26000000e+02 2.27000000e+02 2.28000000e+02
2.29000000e+02 2.30000000e+02 2.31000000e+02 2.32000000e+02
2.33000000e+02 2.34000000e+02 2.35000000e+02 2.36000000e+02
2.37000000e+02 2.38000000e+02 2.39000000e+02 2.40000000e+02
2.41000000e+02 2.42000000e+02 2.43000000e+02 2.44000000e+02
2.45000000e+02 2.46000000e+02 2.47000000e+02 2.48000000e+02
2.49000000e+02 2.50000000e+02 2.51000000e+02 2.52000000e+02
2.53000000e+02 2.54000000e+02 2.55000000e+02 2.56000000e+02
2.57000000e+02 2.58000000e+02 2.59000000e+02 2.60000000e+02
2.61000000e+02 2.62000000e+02 2.63000000e+02 2.64000000e+02
2.65000000e+02 2.66000000e+02 2.67000000e+02 2.68000000e+02
2.69000000e+02 2.70000000e+02 2.71000000e+02 2.72000000e+02
2.73000000e+02 2.74000000e+02 2.75000000e+02 2.76000000e+02
2.77000000e+02 2.78000000e+02 2.79000000e+02 2.80000000e+02
2.81000000e+02 2.82000000e+02 2.83000000e+02 2.84000000e+02
2.85000000e+02 2.86000000e+02 2.87000000e+02 2.88000000e+02
2.89000000e+02 2.90000000e+02 2.91000000e+02 2.92000000e+02
2.93000000e+02 2.94000000e+02 2.95000000e+02 2.96000000e+02
2.97000000e+02 2.98000000e+02 2.99000000e+02 3.00000000e+02
3.01000000e+02 3.02000000e+02 3.03000000e+02 3.04000000e+02
3.05000000e+02 3.06000000e+02 3.07000000e+02 3.08000000e+02
3.09000000e+02 3.10000000e+02 3.11000000e+02 3.12000000e+02
3.13000000e+02 3.14000000e+02 3.15000000e+02 3.16000000e+02
3.17000000e+02 3.18000000e+02 3.19000000e+02 3.20000000e+02
3.21000000e+02 3.22000000e+02 3.23000000e+02 3.24000000e+02
3.25000000e+02 3.26000000e+02 3.27000000e+02 3.28000000e+02
3.29000000e+02 3.30000000e+02 3.31000000e+02 3.32000000e+02
3.33000000e+02 3.34000000e+02 3.35000000e+02 3.36000000e+02
3.37000000e+02 3.38000000e+02 3.39000000e+02 3.40000000e+02
3.41000000e+02 3.42000000e+02 3.43000000e+02 3.44000000e+02
3.45000000e+02 3.46000000e+02 3.47000000e+02 3.48000000e+02
3.49000000e+02 3.50000000e+02 3.51000000e+02 3.52000000e+02
3.53000000e+02 3.54000000e+02 3.55000000e+02 3.56000000e+02
3.57000000e+02 3.58000000e+02 3.59000000e+02 3.60000000e+02
3.61000000e+02 3.62000000e+02 3.63000000e+02 3.64000000e+02
3.65000000e+02 3.66000000e+02 3.67000000e+02 3.68000000e+02
3.69000000e+02 3.70000000e+02 3.71000000e+02 3.72000000e+02
3.73000000e+02 3.74000000e+02 3.75000000e+02 3.76000000e+02
3.77000000e+02 3.78000000e+02 3.79000000e+02 3.80000000e+02
3.81000000e+02 3.82000000e+02 3.83000000e+02 3.84000000e+02
3.85000000e+02 3.86000000e+02 3.87000000e+02 3.88000000e+02
3.89000000e+02 3.90000000e+02 3.91000000e+02 3.92000000e+02
3.93000000e+02 3.94000000e+02 3.95000000e+02 3.96000000e+02
3.97000000e+02 3.98000000e+02 3.99000000e+02 4.00000000e+02
4.01000000e+02 4.02000000e+02 4.03000000e+02 4.04000000e+02
4.05000000e+02 4.06000000e+02 4.07000000e+02 4.08000000e+02
4.09000000e+02 4.10000000e+02 4.11000000e+02 4.12000000e+02
4.13000000e+02 4.14000000e+02 4.15000000e+02 4.16000000e+02
4.17000000e+02 4.18000000e+02 4.19000000e+02 4.20000000e+02
4.21000000e+02 4.22000000e+02 4.23000000e+02 4.24000000e+02
4.25000000e+02 4.26000000e+02 4.27000000e+02 4.28000000e+02
4.29000000e+02 4.30000000e+02 4.31000000e+02 4.32000000e+02
4.33000000e+02 4.34000000e+02 4.35000000e+02 4.36000000e+02
4.37000000e+02 4.38000000e+02 4.39000000e+02 4.40000000e+02
4.41000000e+02 4.42000000e+02 4.43000000e+02 4.44000000e+02
4.45000000e+02 4.46000000e+02 4.47000000e+02 4.48000000e+02
4.49000000e+02 4.50000000e+02 4.51000000e+02 4.52000000e+02
4.53000000e+02 4.54000000e+02 4.55000000e+02 4.56000000e+02
4.57000000e+02 4.58000000e+02 4.59000000e+02 4.60000000e+02
4.61000000e+02 4.62000000e+02 4.63000000e+02 4.64000000e+02
4.65000000e+02 4.66000000e+02 4.67000000e+02 4.68000000e+02
4.69000000e+02 4.70000000e+02 4.71000000e+02 4.72000000e+02
4.73000000e+02 4.74000000e+02 4.75000000e+02 4.76000000e+02
4.77000000e+02 4.78000000e+02 4.79000000e+02 4.80000000e+02
4.81000000e+02 4.82000000e+02 4.83000000e+02 4.84000000e+02
4.85000000e+02 4.86000000e+02 4.87000000e+02 4.88000000e+02
4.89000000e+02 4.90000000e+02 4.91000000e+02 4.92000000e+02
4.93000000e+02 4.94000000e+02 4.95000000e+02 4.96000000e+02
4.97000000e+02 4.98000000e+02 4.99000000e+02 5.00000000e+02]
[-1.30923453e-01 3.56452081e-02 -2.41351069e-02 4.25617201e-03
-1.08149058e-02 2.35555567e-03 -1.38691088e-03 5.54426960e-03
-7.48850216e-04 7.34701963e-04 -4.93464786e-04 3.33601388e-04
-2.18632563e-03 3.32386775e-04 -1.68928509e-04 2.96589257e-04
-6.00715667e-05 1.53032506e-04 -7.72359230e-05 1.07113688e-04
-7.27102937e-04 9.46203658e-05 -6.18993016e-05 5.12473018e-05
-1.65296139e-05 7.31971149e-05 -1.55929042e-05 1.01956250e-05
-3.57031209e-05 6.90268106e-06 -1.52776812e-05 1.40683532e-05
-2.04246481e-05 1.57194976e-04 -2.16376889e-05 1.12356598e-05
-1.38645589e-05 2.97813726e-06 -8.28524409e-06 2.66171287e-06
-2.31687357e-06 1.28261351e-05 -1.83549012e-06 1.65549397e-06
-1.29469777e-06 7.25814220e-07 -4.60583734e-06 8.57269507e-07
-6.12687164e-07 1.69546286e-06 -4.56644395e-07 1.67331898e-06
-1.36976436e-06 2.48610059e-06 -1.85838261e-05 2.42087387e-06
-1.40405350e-06 1.30931501e-06 -3.41171455e-07 1.22449849e-06
-2.93903811e-07 1.88734654e-07 -7.76776041e-07 1.26545941e-07
-1.68467056e-07 1.31565235e-07 -1.27321730e-07 9.60248840e-07
-1.47304584e-07 7.83522691e-08 -1.25334553e-07 2.68487720e-08
-7.21855840e-08 3.14499150e-08 -3.90066027e-08 2.60298973e-07
-3.65430432e-08 2.91391256e-08 -2.59999117e-08 1.24501313e-08
-7.01442489e-08 1.53027017e-08 -1.54997242e-08 5.79078259e-08
-1.55080967e-08 5.65304420e-08 -5.73708215e-08 1.00312396e-07
-7.75370675e-07 1.03357334e-07 -5.51155239e-08 6.01401514e-08
-1.35878085e-08 4.08594834e-08 -1.15072875e-08 8.54425604e-09
-4.21693305e-08 6.25701237e-09 -6.26547779e-09 4.64987645e-09
-2.99552271e-09 2.05549896e-08 -3.54933458e-09 2.03542689e-09
-4.32809196e-09 9.81755629e-10 -2.87125015e-09 1.80852168e-09
-2.95183072e-09 2.16558513e-08 -2.90497524e-09 1.86456359e-09
-1.65736459e-09 5.01379730e-10 -2.10317269e-09 4.80644067e-10
-3.10224043e-10 1.17570331e-09 -2.13749141e-10 3.82381888e-10
-3.36889688e-10 4.36568318e-10 -3.49922398e-09 5.15697043e-10
-2.77984207e-10 3.91402599e-10 -8.75585124e-11 2.53288280e-10
-9.57887585e-11 1.05929367e-10 -6.46926755e-10 9.37932539e-11
-9.32077592e-11 9.19803778e-11 -6.50162347e-11 4.21176732e-10
-8.66616288e-11 1.00039472e-10 -3.52762607e-10 1.10598876e-10
-4.67860591e-10 4.38690472e-10 -8.15522323e-10 6.20084036e-09
-8.09893287e-10 4.51238706e-10 -4.43073852e-10 1.08101969e-10
-3.59338671e-10 9.07723109e-11 -6.04545939e-11 2.65733344e-10
-4.14486710e-11 4.84916888e-11 -3.64124639e-11 2.97289472e-11
-2.16518913e-10 3.45949922e-11 -1.86027379e-11 3.27261122e-11
-6.97900182e-12 1.85250001e-11 -9.06210178e-12 1.23715204e-11
-8.68435302e-11 1.20437798e-11 -8.66257484e-12 7.45449646e-12
-2.74485679e-12 1.35575878e-11 -2.97214718e-12 2.20385591e-12
-7.97269016e-12 1.75329330e-12 -4.82982741e-12 4.79027861e-12
-7.72812100e-12 6.09243126e-11 -8.44044195e-12 4.50707959e-12
-5.40071330e-12 1.20196899e-12 -3.49958127e-12 1.08598510e-12
-8.99561316e-13 4.86100006e-12 -7.17738367e-13 6.80387205e-13
-5.24592007e-13 3.07968469e-13 -2.04760004e-12 3.76266781e-13
-2.40651485e-13 6.02495504e-13 -1.51745094e-13 5.12383055e-13
-3.88277263e-13 6.97907462e-13 -5.29251890e-12 7.09261180e-13
-4.34586272e-13 4.04279220e-13 -1.13105824e-13 4.37698210e-13
-1.04652427e-13 6.79501905e-14 -2.75012105e-13 4.72522958e-14
-7.06559370e-14 5.83442945e-14 -6.41670249e-14 5.09932690e-13
-7.80636229e-14 4.29360379e-14 -6.80905676e-14 1.60239584e-14
-4.98205168e-14 2.25823254e-14 -3.16692550e-14 2.05435611e-13
-2.92206746e-14 3.00688719e-14 -3.40594183e-14 2.53993981e-14
-1.61815023e-13 3.59972521e-14 -4.94286263e-14 1.91419323e-13
-5.79777798e-14 2.38765740e-13 -2.40150795e-13 4.35630053e-13
-3.34804322e-12 4.41273066e-13 -2.38249322e-13 2.48569571e-13
-5.75029794e-14 1.78751531e-13 -4.80351418e-14 3.39195584e-14
-1.59663452e-13 2.40288961e-14 -2.53062000e-14 1.86862532e-14
-1.30547626e-14 9.16599097e-14 -1.53273340e-14 8.48933193e-15
-1.66659296e-14 3.64657636e-15 -1.00880330e-14 5.72269587e-15
-8.74284790e-15 6.31426206e-14 -8.55821963e-15 5.71160948e-15
-4.98626678e-15 1.60499110e-15 -7.13538152e-15 1.59872141e-15
-1.05451984e-15 3.88327658e-15 -7.45396610e-16 1.54129501e-15
-1.42716333e-15 2.03523556e-15 -1.62135920e-14 2.34035401e-15
-1.25629997e-15 1.65799399e-15 -3.66013651e-16 1.04009651e-15
-3.58374953e-16 3.39270441e-16 -1.98825329e-15 2.91038197e-16
-2.68564223e-16 2.24822296e-16 -1.32416417e-16 8.53547975e-16
-1.66839022e-16 1.40086064e-16 -4.32647611e-16 1.26481451e-16
-5.02878878e-16 4.42507115e-16 -8.26070603e-16 6.30066519e-15
-8.31231475e-16 4.80159890e-16 -4.61849741e-16 1.18617125e-16
-4.18111099e-16 1.03460856e-16 -6.79943025e-17 2.90415292e-16
-4.70447572e-17 6.01213336e-17 -4.66793179e-17 4.27932934e-17
-3.21794862e-16 5.06977858e-17 -2.73215943e-17 4.57276217e-17
-9.83579305e-18 2.63660945e-17 -1.20622899e-17 1.55587062e-17
-1.07468594e-16 1.52052983e-17 -1.16513854e-17 1.01876511e-17
-4.29803168e-18 2.31319687e-17 -5.09154144e-18 4.45798611e-18
-1.65404073e-17 4.13366624e-18 -1.38111861e-17 1.41651368e-17
-2.44403604e-17 1.92764231e-16 -2.63578148e-17 1.42083988e-17
-1.62334816e-17 3.68187305e-18 -1.10431565e-17 3.25856649e-18
-2.54355720e-18 1.31365997e-17 -1.96441562e-18 1.94169138e-18
-1.47276050e-18 9.20297380e-19 -6.30802101e-18 1.12642070e-18
-6.74025066e-19 1.53372257e-18 -3.63289335e-19 1.12616046e-18
-7.68529688e-19 1.31859730e-18 -9.89385371e-18 1.33721744e-18
-8.49681366e-19 7.74441894e-19 -2.28941800e-19 9.38827590e-19
-2.19995073e-19 1.42954284e-19 -5.59845669e-19 1.00187062e-19
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Total running time of the script: ( 0 minutes 2.968 seconds)