def newton_raphson(self, ox, oy, direction, step=1e-5):
"""
Computes an iterate of NR
Parameters
----------
ox : float
DESCRIPTION.
oy : float
DESCRIPTION.
direction : +/- 1
DESCRIPTION. 1 for positive branch of unstable manifold.
-1 for negative
step : TYPE, optional
DESCRIPTION. The default is 1e-5.
Returns
-------
PE/PE' : float
DESCRIPTION. Error iterate
"""
bmk = Bmk({'ox': ox, 'oy': oy})
ode = Ode(bmk.ode)
PE = ode.homoclinic_dist(direction, 'rotational')
bmk_incr = Bmk({'ox': ox, 'oy': oy + step})
ode_incr = Ode(bmk_incr.ode)
PE_incr = ode_incr.homoclinic_dist(direction, 'rotational')
return PE / ((PE_incr - PE) / step)
def newton_raphson(ox, oy, direction, step=1e-8):
bmk = Bmk({'ox': ox, 'oy': oy})
ode = Ode(bmk.ode)
PE = ode.homoclinic_dist(direction, 'contractible')
bmk_incr = Bmk({'ox': ox + step, 'oy': oy})
ode_incr = Ode(bmk_incr.ode)
PE_incr = ode_incr.homoclinic_dist(direction, 'contractible')
return PE / ((PE_incr - PE) / step)
def jacobian(PE, ox, oy, step=1e-15):
bmk = Bmk({'ox': ox + step, 'oy': oy})
ode = Ode(bmk.ode)
PE_x = ode.pontryagin_energy('rotational')
bmk = Bmk({'ox': ox, 'oy': oy + step})
ode = Ode(bmk.ode)
PE_y = ode.pontryagin_energy('rotational')
return np.array([[(PE_x[0] - PE[0]) / step, (PE_y[0] - PE[0]) / step],
[(PE_x[1] - PE[1]) / step, (PE_y[1] - PE[1]) / step]])
def periodic_counter(ox, oy, granularity=200):
bmk = Bmk({'ox': ox, 'oy': oy})
ode = Ode(bmk.ode)
centre = ode.get_centre()
saddle = ode.get_saddle_point()
if centre[0]['x'] > saddle[0]['x']:
theta = np.arctan((centre[0]['y'] - saddle[0]['y']) /
(centre[0]['x'] - saddle[0]['x']))
else:
theta = -np.pi + np.arctan((centre[0]['y'] - saddle[0]['y']) /
(centre[0]['x'] - saddle[0]['x']))
c = np.cos(theta)
s = np.sin(theta)
granularity = 80
H_axis_x = np.linspace(centre[0]['x'], saddle[0]['x'], granularity)
H_axis_y = np.linspace(centre[0]['y'], saddle[0]['y'], granularity)
H_axis = list(zip(H_axis_x, H_axis_y))
delta_H = []
for (x_0, y_0) in H_axis:
pt_dict = {'x': x_0, 'y': y_0}
(x_0, y_0) = transform(x_0, y_0, c, s, centre)
curve_x = [x_0]
curve_y = [y_0]
not_crossed = True
looper = 0
while not_crossed and looper < 50000:
looper += 1
pt_dict = ode._euler_iter(pt_dict, h=5e-3)
pt_x, pt_y = transform(pt_dict['x'], pt_dict['y'], c, s, centre)
if curve_y[-1] < -1e-12 and pt_y > 1e-12:
not_crossed = False
curve_x.append(pt_x)
curve_y.append(pt_y)
if np.abs(curve_y[-1]) < 1e-1 and curve_x[-1] < 1e-2:
delta_H.append(x_0 - pt_x)
else:
#print('Escaped bounded region')
break
# plt.plot(curve_x, curve_y)
#plt.show()
po_count = 0
index = []
for H in range(10, len(delta_H) - 1):
if np.sign(delta_H[H]) != np.sign(delta_H[H - 1]):
po_count += 1
index.append(H)
if po_count == 2:
return po_count, delta_H
else:
return po_count, delta_H
PE_x = ode.pontryagin_energy('rotational')
bmk = Bmk({'ox': ox, 'oy': oy + step})
ode = Ode(bmk.ode)
PE_y = ode.pontryagin_energy('rotational')
return np.array([[(PE_x[0] - PE[0]) / step, (PE_y[0] - PE[0]) / step],
[(PE_x[1] - PE[1]) / step, (PE_y[1] - PE[1]) / step]])
par = [-0.00964967, 0.92704312]
progress = [tuple(par)]
PE_progress = []
for i in range(100):
bmk = Bmk({'ox': par[0], 'oy': par[1]})
ode = Ode(bmk.ode)
PE = ode.pontryagin_energy('rotational')
jac = jacobian(PE, par[0], par[1])
par -= 0.1 * np.matmul(np.linalg.inv(jac), np.array(PE))
progress.append(tuple(par))
PE_progress.append(sum(PE))
print("Epoch:", i, " PE:", round(PE_progress[-1], 8))
#pickle.dump(progress, open('progress_08.p','wb'))
#plt.scatter(*zip(*progress))
plt.plot(PE_progress)
plt.show()
# -*- coding: utf-8 -*-
"""
Created on Fri May 29 18:43:12 2020
@author: Francesco
"""
import matplotlib.pyplot as plt
from ode import Ode
from bmk import Bmk
import numpy as np
bmk = Bmk({'ox': -0.009, 'oy': 0.927})
ode = Ode(bmk.ode)
fp = ode.get_fixed_points()
if ode.fp_eigen(fp[0]) in ['hyperbolic repeller', 'hyperbolic attractor']:
centre = fp[0]
saddle = fp[1]
else:
centre = fp[1]
saddle = fp[0]
H_axis_x = np.linspace(centre['x'],saddle['x']+1, 20)
H_axis_y = np.linspace(centre['y'],centre['y'], 20)
H_axis = list(zip(H_axis_x, H_axis_y))
for index, val_pair in enumerate(zip(xs, ys)):
if index != 0:
if np.abs(xs[index] - xs[index - 1]) > break_threshold:
xs_.append(None)
ys_.append(None)
if np.abs(ys[index] - ys[index - 1]) > break_threshold:
xs_.append(None)
ys_.append(None)
xs_.append(val_pair[0])
ys_.append(val_pair[1])
return np.array(xs_), np.array(ys_)
bmk = Bmk({'ox': -0.16, 'oy': 1})
ode = Ode(bmk.ode)
P1, _ = ode.unstable_manifold(1)
P2, _ = ode.unstable_manifold(-1)
x_plt = []
y_plt = []
for i in range(len(P1)):
(x, y) = ode._point_dict_getter(P1[i])
x_plt.append(x)
y_plt.append(y)
x_plt, y_plt = para_plot(np.array(x_plt) % 1, np.array(y_plt) % 1, 0.5)
x_plt_lift = []
y_plt_lift = []
@author: Francesco
"""
import matplotlib.pyplot as plt
from ode import Ode
from bmk import Bmk
import numpy as np
ox = np.linspace(-1e-02, 0, 100)
tr_saddle = []
tr_node = []
#-0.009649961793522382], [0.9270432434144844
for i in ox:
bmk = Bmk({'ox': i, 'oy': 0.9270432434144844})
ode = Ode(bmk.ode)
fp = ode.get_fixed_points()
fp_type = ode.fp_eigen(fp[0])[0]
if fp_type == 'saddle point':
print('case saddle {}'.format(i))
tr_saddle.append(ode.jacobian(fp[0])[0][0] + ode.jacobian(fp[0])[1][1])
tr_node.append(ode.jacobian(fp[1])[0][0] + ode.jacobian(fp[1])[1][1])
else:
print('case node {}'.format(i))
tr_saddle.append(ode.jacobian(fp[1])[0][0] + ode.jacobian(fp[1])[1][1])
tr_node.append(ode.jacobian(fp[0])[0][0] + ode.jacobian(fp[0])[1][1])
#figure, axes = plt.subplots(nrows=2, ncols=1)
plt.plot(ox, tr_saddle)
#plt.plot(ox, [0]*len(ox))
plt.plot(