Nullcline Plot for Nonlinear System of ODEs

问题

I am attempting to plot the nullcline (steady state) curves of the Oregonator model to assert the existence of a limit cycle by applying the Poincare-Bendixson Theorem. I am close, but for some reason the plot that is produced shows two straight lines. I think it has something to do with the plotting stage. Any ideas?

Also any hints for how to construct a quadrilateral to apply the theorem with would be most appreciated.

Code:

import numpy as np
import matplotlib.pyplot as plt

# Dimensionless parameters
eps = 0.04
q = 0.0008
f = 1


# Oregonator model as numpy array
def Sys(Y, t = 0):
    return np.array((Y[0] * (1 - Y[0] - ((Y[0] - q) * f * Y[1]) / (Y[0] + q)) / eps, Y[0] - Y[1] ))



# Oregonator model steady states
def g(x,z):
    return (x * (1 - x) + ((q - x) * f * z) / (q + x)) / eps
def h(x,z):
    return x - z

# Initial lists containing values
x = []
z = []

def sys(iv1, iv2, dt, time):
    # initial values:
    x.append(iv1)
    z.append(iv2)
    # Compute and fill lists
    for i in range(time):
        x.append(x[i] + (g(x[i],z[i])) * dt)
        z.append(z[i] + (h(x[i],z[i])) * dt)
    return x, z

sys(1, 0.5, 0.01, 30)

# Locate and find equilibrium points
eqp = []

def find_fixed_points(r):
    for x in range(r):
        for z in range(r):
            if ((g(x, z) == 0) and (h(x, z) == 0)):
                eqp.append((x,z))

    return eqp


# Plot nullclines
plt.plot([0,2],[2,0], 'r-', lw=2, label='x-nullcline')
plt.plot([1,1],[0,2], 'b-', lw=2, label='z-nullcline')

# Plot equilibrium points
for point in eqp:
    plt.plot(point[0],point[1],"red", marker = "o", markersize = 10.0)

plt.legend(loc='best')

x = np.linspace(0, 2, 20)
z = np.linspace(0, 2, 20)

X1 , Z1  = np.meshgrid(x, z)                    # Create a grid
DX1, DZ1 = Sys([X1, Z1])                        # Compute reaction rate on the grid
M = (np.hypot(DX1, DZ1))                        # Norm reaction rate
M[ M == 0] = 1.                                 # Avoid zero division errors
DX1 /= M                                        # Normalise each arrows
DZ1 /= M

plt.quiver(X1, Z1, DX1, DZ1, M, pivot='mid')
plt.xlabel("x(\u03C4)")
plt.ylabel("z(\u03C4)")
plt.legend()
plt.grid()
plt.show()

来源：https://stackoverflow.com/questions/59009381/nullcline-plot-for-nonlinear-system-of-odes