lyapunov spectrum of lorenz oscillator
0
Here is my try to calculate lyapunov spectrum of lorenz oscillator. I followed step by step from boost library program, but I can not reproduce the results.
import numpy as np
import pylab as pl
from copy import copy
from numpy import dot, log
from numpy.linalg import norm
from scipy.integrate import odeint
from sys import exit
# system with out perturbation----------------------------#
def lorenz(x0, t):
x, y, z = x0
dxdt = s * (y - x)
dydt = x * (r - z) - y
dzdt = x * y - b * z
return [dxdt, dydt, dzdt]
#system with perturbation---------------------------------#
def lorenz_with_lyap(x0, t):
x,y,z = x0[:3]
dxdt = [0.0]*12
dxdt[0] = s * (y - x)
dxdt[1] = x * (r - z) - y
dxdt[2] = x * y - b * z
# this is jacobian*Y
for l in range(3):
m = 3*l+3
dxdt[m] = -s * x0[m] + s * x0[m+1]
dxdt[m+1] = (r - z) * x0[m] -1.0 * x0[m+1] -1.0 * x * x0[m+2]
dxdt[m+2] = y * x0[m] + x * x0[m+1] -b * x0[m+2]
return dxdt
#---------------------------------------------------------#
def gram_schmidt(x0, lyap, t):
# vectors
a = x0[3: 6]
b = x0[6: 9]
c = x0[9:12]
v1 = copy(a)
v2 = b - dot(v1, b)/dot(v1, v1) * v1
v3 = c - dot(v1, c)/dot(v1, v1) * v1 -dot(v2, c)/dot(v2, v2) * v2
znorm = [0] * 3
znorm[0] = norm(v1)
znorm[1] = norm(v2)
znorm[2] = norm(v3)
# normalize vectors
x0[3:6 ] = v1/znorm[0]
x0[6:9 ] = v2/znorm[1]
x0[9:12] = v3/znorm[2]
for i in range(3):
lyap[i] += log(znorm[i])/t
# parameters
s = 10.0
r = 28.0
b = 8.0/3.0
stept = 1.0
dt = 0.01
n = 3
lyap = [0.0]*n
x0 = np.zeros(12)
initial_condition = [10.0, 10.0, 5.0]
x0[:3] = copy(initial_condition)
for i in range(3):
x0[n+n*i+i]=1.0
# integrate trantient time
t = np.arange(0, 10, 0.1)
sol = odeint(lorenz, initial_condition, t)
x0[:3] = copy(sol[-1,:]) # new initial condition
ti = t[-1]
tf = ti + stept
for i in range(10):
t_interval = np.arange(ti, tf, dt)
sol = odeint(lorenz_with_lyap, x0, t_interval)
x0 = copy(sol[-1,:])
gram_schmidt(x0, lyap, stept)
for xx in lyap:
print "%15.9f" % xx,
print ""
ti = tf
tf = ti+stept
I think the problem is in gram_schmidt function. Is the question clear?
Thank you for any guide.
python c++
asked 5 mins ago
Abolfazl
285
add a comment |
0
Here is my try to calculate lyapunov spectrum of lorenz oscillator. I followed step by step from boost library program, but I can not reproduce the results.
import numpy as np
import pylab as pl
from copy import copy
from numpy import dot, log
from numpy.linalg import norm
from scipy.integrate import odeint
from sys import exit
# system with out perturbation----------------------------#
def lorenz(x0, t):
x, y, z = x0
dxdt = s * (y - x)
dydt = x * (r - z) - y
dzdt = x * y - b * z
return [dxdt, dydt, dzdt]
#system with perturbation---------------------------------#
def lorenz_with_lyap(x0, t):
x,y,z = x0[:3]
dxdt = [0.0]*12
dxdt[0] = s * (y - x)
dxdt[1] = x * (r - z) - y
dxdt[2] = x * y - b * z
# this is jacobian*Y
for l in range(3):
m = 3*l+3
dxdt[m] = -s * x0[m] + s * x0[m+1]
dxdt[m+1] = (r - z) * x0[m] -1.0 * x0[m+1] -1.0 * x * x0[m+2]
dxdt[m+2] = y * x0[m] + x * x0[m+1] -b * x0[m+2]
return dxdt
#---------------------------------------------------------#
def gram_schmidt(x0, lyap, t):
# vectors
a = x0[3: 6]
b = x0[6: 9]
c = x0[9:12]
v1 = copy(a)
v2 = b - dot(v1, b)/dot(v1, v1) * v1
v3 = c - dot(v1, c)/dot(v1, v1) * v1 -dot(v2, c)/dot(v2, v2) * v2
znorm = [0] * 3
znorm[0] = norm(v1)
znorm[1] = norm(v2)
znorm[2] = norm(v3)
# normalize vectors
x0[3:6 ] = v1/znorm[0]
x0[6:9 ] = v2/znorm[1]
x0[9:12] = v3/znorm[2]
for i in range(3):
lyap[i] += log(znorm[i])/t
# parameters
s = 10.0
r = 28.0
b = 8.0/3.0
stept = 1.0
dt = 0.01
n = 3
lyap = [0.0]*n
x0 = np.zeros(12)
initial_condition = [10.0, 10.0, 5.0]
x0[:3] = copy(initial_condition)
for i in range(3):
x0[n+n*i+i]=1.0
# integrate trantient time
t = np.arange(0, 10, 0.1)
sol = odeint(lorenz, initial_condition, t)
x0[:3] = copy(sol[-1,:]) # new initial condition
ti = t[-1]
tf = ti + stept
for i in range(10):
t_interval = np.arange(ti, tf, dt)
sol = odeint(lorenz_with_lyap, x0, t_interval)
x0 = copy(sol[-1,:])
gram_schmidt(x0, lyap, stept)
for xx in lyap:
print "%15.9f" % xx,
print ""
ti = tf
tf = ti+stept
I think the problem is in gram_schmidt function. Is the question clear?
Thank you for any guide.
python c++
asked 5 mins ago
Abolfazl
285
add a comment |
0
0
0
Here is my try to calculate lyapunov spectrum of lorenz oscillator. I followed step by step from boost library program, but I can not reproduce the results.
import numpy as np
import pylab as pl
from copy import copy
from numpy import dot, log
from numpy.linalg import norm
from scipy.integrate import odeint
from sys import exit
# system with out perturbation----------------------------#
def lorenz(x0, t):
x, y, z = x0
dxdt = s * (y - x)
dydt = x * (r - z) - y
dzdt = x * y - b * z
return [dxdt, dydt, dzdt]
#system with perturbation---------------------------------#
def lorenz_with_lyap(x0, t):
x,y,z = x0[:3]
dxdt = [0.0]*12
dxdt[0] = s * (y - x)
dxdt[1] = x * (r - z) - y
dxdt[2] = x * y - b * z
# this is jacobian*Y
for l in range(3):
m = 3*l+3
dxdt[m] = -s * x0[m] + s * x0[m+1]
dxdt[m+1] = (r - z) * x0[m] -1.0 * x0[m+1] -1.0 * x * x0[m+2]
dxdt[m+2] = y * x0[m] + x * x0[m+1] -b * x0[m+2]
return dxdt
#---------------------------------------------------------#
def gram_schmidt(x0, lyap, t):
# vectors
a = x0[3: 6]
b = x0[6: 9]
c = x0[9:12]
v1 = copy(a)
v2 = b - dot(v1, b)/dot(v1, v1) * v1
v3 = c - dot(v1, c)/dot(v1, v1) * v1 -dot(v2, c)/dot(v2, v2) * v2
znorm = [0] * 3
znorm[0] = norm(v1)
znorm[1] = norm(v2)
znorm[2] = norm(v3)
# normalize vectors
x0[3:6 ] = v1/znorm[0]
x0[6:9 ] = v2/znorm[1]
x0[9:12] = v3/znorm[2]
for i in range(3):
lyap[i] += log(znorm[i])/t
# parameters
s = 10.0
r = 28.0
b = 8.0/3.0
stept = 1.0
dt = 0.01
n = 3
lyap = [0.0]*n
x0 = np.zeros(12)
initial_condition = [10.0, 10.0, 5.0]
x0[:3] = copy(initial_condition)
for i in range(3):
x0[n+n*i+i]=1.0
# integrate trantient time
t = np.arange(0, 10, 0.1)
sol = odeint(lorenz, initial_condition, t)
x0[:3] = copy(sol[-1,:]) # new initial condition
ti = t[-1]
tf = ti + stept
for i in range(10):
t_interval = np.arange(ti, tf, dt)
sol = odeint(lorenz_with_lyap, x0, t_interval)
x0 = copy(sol[-1,:])
gram_schmidt(x0, lyap, stept)
for xx in lyap:
print "%15.9f" % xx,
print ""
ti = tf
tf = ti+stept
I think the problem is in gram_schmidt function. Is the question clear?
Thank you for any guide.
python c++
asked 5 mins ago
Abolfazl
285
Here is my try to calculate lyapunov spectrum of lorenz oscillator. I followed step by step from boost library program, but I can not reproduce the results.
import numpy as np
import pylab as pl
from copy import copy
from numpy import dot, log
from numpy.linalg import norm
from scipy.integrate import odeint
from sys import exit
# system with out perturbation----------------------------#
def lorenz(x0, t):
x, y, z = x0
dxdt = s * (y - x)
dydt = x * (r - z) - y
dzdt = x * y - b * z
return [dxdt, dydt, dzdt]
#system with perturbation---------------------------------#
def lorenz_with_lyap(x0, t):
x,y,z = x0[:3]
dxdt = [0.0]*12
dxdt[0] = s * (y - x)
dxdt[1] = x * (r - z) - y
dxdt[2] = x * y - b * z
# this is jacobian*Y
for l in range(3):
m = 3*l+3
dxdt[m] = -s * x0[m] + s * x0[m+1]
dxdt[m+1] = (r - z) * x0[m] -1.0 * x0[m+1] -1.0 * x * x0[m+2]
dxdt[m+2] = y * x0[m] + x * x0[m+1] -b * x0[m+2]
return dxdt
#---------------------------------------------------------#
def gram_schmidt(x0, lyap, t):
# vectors
a = x0[3: 6]
b = x0[6: 9]
c = x0[9:12]
v1 = copy(a)
v2 = b - dot(v1, b)/dot(v1, v1) * v1
v3 = c - dot(v1, c)/dot(v1, v1) * v1 -dot(v2, c)/dot(v2, v2) * v2
znorm = [0] * 3
znorm[0] = norm(v1)
znorm[1] = norm(v2)
znorm[2] = norm(v3)
# normalize vectors
x0[3:6 ] = v1/znorm[0]
x0[6:9 ] = v2/znorm[1]
x0[9:12] = v3/znorm[2]
for i in range(3):
lyap[i] += log(znorm[i])/t
# parameters
s = 10.0
r = 28.0
b = 8.0/3.0
stept = 1.0
dt = 0.01
n = 3
lyap = [0.0]*n
x0 = np.zeros(12)
initial_condition = [10.0, 10.0, 5.0]
x0[:3] = copy(initial_condition)
for i in range(3):
x0[n+n*i+i]=1.0
# integrate trantient time
t = np.arange(0, 10, 0.1)
sol = odeint(lorenz, initial_condition, t)
x0[:3] = copy(sol[-1,:]) # new initial condition
ti = t[-1]
tf = ti + stept
for i in range(10):
t_interval = np.arange(ti, tf, dt)
sol = odeint(lorenz_with_lyap, x0, t_interval)
x0 = copy(sol[-1,:])
gram_schmidt(x0, lyap, stept)
for xx in lyap:
print "%15.9f" % xx,
print ""
ti = tf
tf = ti+stept
I think the problem is in gram_schmidt function. Is the question clear?
Thank you for any guide.
python c++
python c++
asked 5 mins ago
Abolfazl
285
asked 5 mins ago
Abolfazl
285
asked 5 mins ago
Abolfazl
285
asked 5 mins ago
Abolfazl
285
asked 5 mins ago
Abolfazl
285
285
add a comment |
add a comment |
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