def test_differential_correction(self):
""" Test solution obtained using differential correction """
MASS_A = 1.0
MASS_B = 1.0
SADDLE_ENERGY = 0.0 # Energy of the saddle
ALPHA = 1.00
BETA = 1.00
OMEGA = 1
EPSILON = 0.00 # uncoupled
parameters = np.array([MASS_A, MASS_B, SADDLE_ENERGY, \
ALPHA, BETA, OMEGA, EPSILON])
deltaE_vals = [0.1]
linecolors = ['b']
save_final_plot = False
show_final_plot = True
diff_corr_unc.upo(deltaE_vals, linecolors, \
save_final_plot, show_final_plot)
TSPAN = [0, 30] # arbitrary range, just to finish the integration
RelTol = 3.e-12
AbsTol = 1.e-12
for deltaE_val in deltaE_vals:
# po_fam_file = "x0_diffcorr_deltaE%s_uncoupled.dat" %(deltaE_val)
with open(path_to_data + "x0_diffcorr_deltaE%s_uncoupled.dat" %
(deltaE_val)) as po_fam_file:
x0podata = np.loadtxt(po_fam_file.name)
x0po_diffcorr = x0podata[0:4]
# print(x0po_diffcorr)
f = lambda t, x: uncoupled.ham2dof_uncoupled(t, x, parameters)
soln = solve_ivp(f, TSPAN, x0po_diffcorr,method='RK45',dense_output=True, \
events = lambda t,x : uncoupled.half_period_uncoupled(t,x,parameters), \
rtol=RelTol, atol=AbsTol)
te = soln.t_events[0]
tt = [0, te[2]]
t,x_diffcorr,phi_t1,PHI = diffcorr.state_transit_matrix(
tt, x0po_diffcorr, parameters, \
uncoupled.variational_eqns_uncoupled)
total_energy = deltaE_val + parameters[2]
y, py = uncoupled.upo_analytical(total_energy, t, parameters)
numerical_orbit = np.array([x_diffcorr[:, 1], x_diffcorr[:, 3]])
analytical_orbit = np.array([y, py])
np.testing.assert_array_almost_equal(numerical_orbit,
analytical_orbit)
hausd_dist = hausd_dist_numeric_analytic(x_diffcorr, t, deltaE_val,
parameters)
self.assertLessEqual(hausd_dist, 1e-8)
def upo(deltaE_vals, linecolor, \
save_final_plot = True, show_final_plot = False):
eqNum = 1
eqPt = tp.get_eq_pts(eqNum, uncoupled.init_guess_eqpt_uncoupled, \
uncoupled.grad_pot_uncoupled, parameters)
for i in range(len(deltaE_vals)):
e = deltaE_vals[i] # total energy
n = 12 # number of intervals we want to divide
n_turn = 1 # nth turning point we want to choose
deltaE = e - parameters[2]
#Trial initial Condition s.t. one initial condition is on the LHS of the UPO and the
#other one is on the RHS of the UPO
state0_2 = [-0.1 , -math.sqrt(2*e+0.1**2-0.5*0.1**4),0.0,0.0]
state0_3 = [0.11 , -math.sqrt(2*e+0.11**2-0.5*0.11**4),0.0,0.0]
with open("x0_turningpoint_deltaE%s_uncoupled.dat" %(deltaE),'a+') as po_fam_file:
[x0po_1, T_1,energyPO_1] = tp.turningPoint(
state0_2, state0_3, uncoupled.get_coord_uncoupled, \
uncoupled.guess_coords_uncoupled, uncoupled.ham2dof_uncoupled, \
uncoupled.half_period_uncoupled, uncoupled.variational_eqns_uncoupled, \
uncoupled.pot_energy_uncoupled, uncoupled.plot_iter_orbit_uncoupled, \
parameters, e, n, n_turn, show_itrsteps_plots, po_fam_file)
#%% Load periodic orbit data from ascii files
x0po = np.zeros((4,len(deltaE_vals))) #each column is a different initial condition
for i in range(len(deltaE_vals)):
deltaE = deltaE_vals[i]
with open("x0_turningpoint_deltaE%s_uncoupled.dat" %(deltaE),'a+') as po_fam_file:
print('Loading the periodic orbit family from data file',po_fam_file.name,'\n')
x0podata = np.loadtxt(po_fam_file.name)
x0po[:,i] = x0podata[-1,0:4]
#%% Plotting the Family
TSPAN = [0,30]
plt.close('all')
axis_fs = 15
RelTol = 3.e-10
AbsTol = 1.e-10
f = lambda t,x : uncoupled.ham2dof_uncoupled(t,x,parameters)
ax = plt.gca(projection='3d')
for i in range(len(deltaE_vals)):
soln = solve_ivp(f, TSPAN, x0po[:,i], method='RK45', dense_output=True, \
events = lambda t,x : uncoupled.half_period_uncoupled(t,x,parameters), \
rtol=RelTol, atol=AbsTol)
te = soln.t_events[0]
tt = [0,te[2]]
t,x,phi_t1,PHI = tp.state_transit_matrix(tt, x0po[:,i], parameters, \
uncoupled.variational_eqns_uncoupled)
ax.plot(x[:,0],x[:,1],x[:,3],'-', color=linecolor[i], \
label='$\Delta E$ = %.2f'%(deltaE_vals[i]))
ax.scatter(x[0,0],x[0,1],x[0,3], s=10, marker='*')
ax.plot(x[:,0], x[:,1], zs=0, zdir='z')
resX = 100
xVec = np.linspace(-4,4,resX)
yVec = np.linspace(-4,4,resX)
xMat, yMat = np.meshgrid(xVec, yVec)
cset1 = ax.contour(
xMat, yMat, tp.get_pot_surf_proj(
xVec, yVec, uncoupled.pot_energy_uncoupled, parameters), \
[0.01,0.1,1,2,4], zdir='z', offset=0, \
linewidths = 1.0, cmap=cm.viridis, \
alpha = 0.8)
ax.scatter(eqPt[0], eqPt[1], s = 50, c = 'r', marker = 'X')
ax.set_xlabel('$x$', fontsize=axis_fs)
ax.set_ylabel('$y$', fontsize=axis_fs)
ax.set_zlabel('$p_y$', fontsize=axis_fs)
legend = ax.legend(loc='upper left')
ax.set_xlim(-4, 4)
ax.set_ylim(-4, 4)
ax.set_zlim(-4, 4)
plt.grid()
if show_final_plot:
plt.show()
if save_final_plot:
plt.savefig( path_to_saveplot + 'tp_uncoupled_upos.pdf', \
format='pdf', bbox_inches='tight')
def upo(deltaE_vals, linecolor, \
save_final_plot = True, show_final_plot = False):
#%% Setting up parameters and global variables
eqPt = diffcorr.get_eq_pts(eqNum, uncoupled.init_guess_eqpt_uncoupled, \
uncoupled.grad_pot_uncoupled, \
parameters)
#energy of the saddle eq pt
eSaddle = diffcorr.get_total_energy([eqPt[0],eqPt[1],0,0], \
uncoupled.pot_energy_uncoupled, \
parameters)
#%%
nFam = 100 # use nFam = 10 for low energy
# first two amplitudes for continuation procedure to get p.o. family
Ax1 = 2.e-5 # initial amplitude (1 of 2) values to use: 2.e-3
Ax2 = 2 * Ax1 # initial amplitude (2 of 2)
t = time.time()
# get the initial conditions and periods for a family of periodic orbits
with open("x0_diffcorr_fam_eqPt%s_uncoupled.dat" % eqNum,
'a+') as po_fam_file:
[po_x0Fam,po_tpFam] = diffcorr.get_po_fam(
eqNum, Ax1, Ax2, nFam, po_fam_file, uncoupled.init_guess_eqpt_uncoupled, \
uncoupled.grad_pot_uncoupled, uncoupled.jacobian_uncoupled, \
uncoupled.guess_lin_uncoupled, uncoupled.diffcorr_setup_uncoupled, \
uncoupled.conv_coord_uncoupled, uncoupled.diffcorr_acc_corr_uncoupled, \
uncoupled.ham2dof_uncoupled, uncoupled.half_period_uncoupled, \
uncoupled.pot_energy_uncoupled, uncoupled.variational_eqns_uncoupled, \
uncoupled.plot_iter_orbit_uncoupled, parameters)
poFamRuntime = time.time() - t
x0podata = np.concatenate((po_x0Fam, po_tpFam), axis=1)
#%%
for i in range(len(deltaE_vals)):
deltaE = deltaE_vals[i]
with open("x0_diffcorr_fam_eqPt%s_uncoupled.dat" % eqNum,
'a+') as po_fam_file:
eTarget = eSaddle + deltaE
print('Loading the periodic orbit family from data file',
po_fam_file.name, '\n')
x0podata = np.loadtxt(po_fam_file.name)
#%
with open(
"x0po_T_energyPO_eqPt%s_brac%s_uncoupled.dat" %
(eqNum, deltaE), 'a+') as po_brac_file:
t = time.time()
x0poTarget,TTarget = diffcorr.po_bracket_energy(
eTarget, x0podata, po_brac_file, \
uncoupled.diffcorr_setup_uncoupled, uncoupled.conv_coord_uncoupled, \
uncoupled.diffcorr_acc_corr_uncoupled, uncoupled.ham2dof_uncoupled, \
uncoupled.half_period_uncoupled, uncoupled.pot_energy_uncoupled, \
uncoupled.variational_eqns_uncoupled, uncoupled.plot_iter_orbit_uncoupled, \
parameters)
poTarE_runtime = time.time() - t
with open(
"model_parameters_eqPt%s_DelE%s_uncoupled.dat" %
(eqNum, deltaE), 'a+') as model_parameters_file:
np.savetxt(model_parameters_file.name,
parameters,
fmt='%1.16e')
# target specific periodic orbit
# Target PO of specific energy with high precision does not work for the
# model
with open("x0_diffcorr_deltaE%s_uncoupled.dat" % (deltaE),
'a+') as po_target_file:
[x0po, T,energyPO] = diffcorr.po_target_energy(
x0poTarget,eTarget, po_target_file, \
uncoupled.diffcorr_setup_uncoupled, uncoupled.conv_coord_uncoupled, \
uncoupled.diffcorr_acc_corr_uncoupled, \
uncoupled.ham2dof_uncoupled, uncoupled.half_period_uncoupled, \
uncoupled.pot_energy_uncoupled, uncoupled.variational_eqns_uncoupled, \
uncoupled.plot_iter_orbit_uncoupled, parameters)
#%% Load periodic orbit data from ascii files
x0po = np.zeros((4, len(deltaE_vals)))
for i in range(len(deltaE_vals)):
deltaE = deltaE_vals[i]
with open("x0_diffcorr_deltaE%s_uncoupled.dat" % (deltaE),
'a+') as po_fam_file:
print('Loading the periodic orbit family from data file',
po_fam_file.name, '\n')
x0podata = np.loadtxt(po_fam_file.name)
x0po[:, i] = x0podata[0:4]
#%% Plotting the unstable periodic orbits at the specified energies
TSPAN = [0, 30] # arbitrary range, just to finish the integration
plt.close('all')
axis_fs = 15
RelTol = 3.e-10
AbsTol = 1.e-10
for i in range(len(deltaE_vals)):
deltaE = deltaE_vals[i]
f = lambda t, x: uncoupled.ham2dof_uncoupled(t, x, parameters)
soln = solve_ivp(f, TSPAN, x0po[:,i],method='RK45',dense_output=True, \
events = lambda t,x : uncoupled.half_period_uncoupled(
t,x,parameters), \
rtol=RelTol, atol=AbsTol)
te = soln.t_events[0]
tt = [0, te[1]]
t,x,phi_t1,PHI = diffcorr.state_transit_matrix(tt, x0po[:,i], parameters, \
uncoupled.variational_eqns_uncoupled)
ax = plt.gca(projection='3d')
ax.plot(x[:,0],x[:,1],x[:,3],'-',color = linecolor[i], \
label = '$\Delta E$ = %.2f'%(deltaE))
ax.plot(x[:, 0], x[:, 1], -x[:, 3], '-', color=linecolor[i])
ax.scatter(x[0, 0], x[0, 1], x[0, 3], s=10, marker='*')
ax.scatter(x[0, 0], x[0, 1], -x[0, 3], s=10, marker='o')
ax.plot(x[:, 0], x[:, 1], zs=0, zdir='z')
ax = plt.gca(projection='3d')
resX = 100
xVec = np.linspace(-4, 4, resX)
yVec = np.linspace(-4, 4, resX)
xMat, yMat = np.meshgrid(xVec, yVec)
cset1 = ax.contour(
xMat, yMat, diffcorr.get_pot_surf_proj(
xVec, yVec, uncoupled.pot_energy_uncoupled, \
parameters), \
[0.01,0.1,1,2,4], zdir='z', offset=0, linewidths = 1.0, \
cmap=cm.viridis, alpha = 0.8)
ax.scatter(eqPt[0], eqPt[1], s=50, c='r', marker='X')
ax.set_xlabel('$x$', fontsize=axis_fs)
ax.set_ylabel('$y$', fontsize=axis_fs)
ax.set_zlabel('$p_y$', fontsize=axis_fs)
legend = ax.legend(loc='upper left')
ax.set_xlim(-4, 4)
ax.set_ylim(-4, 4)
ax.set_zlim(-4, 4)
plt.grid()
if show_final_plot:
plt.show()
if save_final_plot:
plt.savefig(path_to_saveplot + 'diff_corr_uncoupled_upos.pdf', \
format='pdf', bbox_inches='tight')