Figure 6.24:

Real and imaginary parts of the eigenvalues of the Jacobian matrix near points C--F.

Code for Figure 6.24

Text of the GNU GPL.

main.py

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# Converted from st_st_osc_eigo.m - Oscillatory system eigenvalues (upper/limit-cycle branch)
# Saves rows 441:rowend (the upper/limit-cycle segment)
import numpy as np
from scipy.optimize import fsolve
from misc import save_ascii

p = dict(
    k_m      = 0.004,
    T_m      = 298.,
    E        = 15000.,
    c_Af     = 2.,
    C_p      = 4.,
    rho      = 1000.,
    T_f      = 298.,
    T_a      = 298.,
    DeltaH_R = -2.2e5,
    U        = 340.,
    theta    = 35.,
    T_set    = 321.53,
    T_fs     = 298.,
    Kc       = 0.,
)
p['C_ps'] = p['C_p'] * p['rho']

def st_st_cA(x, c_A, p):
    theta = x[0]; T = x[1]
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    return [p['c_Af'] - (1+k*theta)*c_A,
            p['U']*theta*(p['T_a']-T) + p['C_ps']*(p['T_f']-T) - k*theta*c_A*p['DeltaH_R']]

def st_st_theta(x, theta, p):
    c_A = x[0]; T = x[1]
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    return [p['c_Af'] - (1+k*theta)*c_A,
            p['U']*theta*(p['T_a']-T) + p['C_ps']*(p['T_f']-T) - k*theta*c_A*p['DeltaH_R']]

def compute_lamrow(k, c_A, T, theta, p):
    Jac = np.array([[-1./theta - k, -k*c_A*p['E']/(T*T)],
                    [-k*p['DeltaH_R']/p['C_ps'],
                     -p['U']/p['C_ps'] - 1./theta - k*c_A*p['DeltaH_R']/p['C_ps']*p['E']/(T*T)]])
    lam = np.linalg.eigvals(Jac)
    a = lam[0].real; b = lam[0].imag
    c = lam[1].real; d = lam[1].imag
    return [a,b,c,d] if a >= c else [c,d,a,b]

# Lower branch (logspace in conversion)
npts1 = 200
x0 = [1., p['T_f']]
xvect = np.logspace(np.log10(1e-4), np.log10(0.75), npts1)
c_Avect = (1 - xvect) * p['c_Af']
table = []

for c_A in c_Avect:
    X, _, ier, _ = fsolve(lambda x: st_st_cA(x, c_A, p), x0, full_output=True)
    theta = X[0]; T = X[1]
    conv  = (p['c_Af'] - c_A) / p['c_Af']
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    lamrow = compute_lamrow(k, c_A, T, theta, p)
    table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

c_A_c = c_A; T_c = T; theta_c = theta; conv_c = conv

# Near-extinction patch
npts2 = 50
xvect2 = np.linspace(conv_c, 0.95, npts2)
c_Avect2 = (1 - xvect2) * p['c_Af']

for c_A in c_Avect2:
    X, _, ier, _ = fsolve(lambda x: st_st_cA(x, c_A, p), x0, full_output=True)
    theta = X[0]; T = X[1]
    conv  = (p['c_Af'] - c_A) / p['c_Af']
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    lamrow = compute_lamrow(k, c_A, T, theta, p)
    table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

c_A_c2 = c_A; T_c2 = T; theta_c2 = theta

# Upper branch segment 1 (logspace theta from corner to 20.7)
npts3 = 200
x0 = [c_A_c2, T_c2]
theta_vect3 = np.logspace(np.log10(theta_c2), np.log10(20.7), npts3)

for theta in theta_vect3:
    X, _, ier, _ = fsolve(lambda x: st_st_theta(x, theta, p), x0, full_output=True)
    c_A = X[0]; T = X[1]
    conv = (p['c_Af'] - c_A) / p['c_Af']
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    lamrow = compute_lamrow(k, c_A, T, theta, p)
    table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

theta_c3 = theta; c_A_c3 = X[0]; T_c3 = X[1]

# Fine patch near theta=[20.7, 20.8]
npts4 = 100
x0 = [c_A_c3, T_c3]
theta_vect4 = np.linspace(theta_c3, 20.8, npts4)

for theta in theta_vect4:
    X, _, ier, _ = fsolve(lambda x: st_st_theta(x, theta, p), x0, full_output=True)
    c_A = X[0]; T = X[1]
    conv = (p['c_Af'] - c_A) / p['c_Af']
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    lamrow = compute_lamrow(k, c_A, T, theta, p)
    table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

theta_c4 = theta; c_A_c4 = X[0]; T_c4 = X[1]

# Finish to large theta
npts5 = 200
x0 = [c_A_c4, T_c4]
theta_vect5 = np.logspace(np.log10(theta_c4), np.log10(500.), npts5)

for theta in theta_vect5:
    X, _, ier, _ = fsolve(lambda x: st_st_theta(x, theta, p), x0, full_output=True)
    c_A = X[0]; T = X[1]
    conv = (p['c_Af'] - c_A) / p['c_Af']
    k = p['k_m'] * np.exp(-p['E']*(1./T - 1./p['T_m']))
    lamrow = compute_lamrow(k, c_A, T, theta, p)
    table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

table = np.array(table)
rowend = npts1 + npts2 + npts3 + npts4 + npts5

# Octave saves rows 441:rowend (1-indexed), i.e. Python 440:rowend
tmp = table[440:rowend, :]
save_ascii('st_st_osc_eigo.dat', tmp)