File:PoincareMap Lorenz Runge Lcycle.svg
Summary
| Description |
English: Periodic orbit and Poincaré map. the periodic orbit is stable limit cycle which is generated by Lorenz equation
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| Date | |
| Source | Own work |
| Author | Yapparina |
| Python source | import numpy as np
import matplotlib.pyplot as plt
n = 100000
h = 0.0001
T = h*n
t = np.arange(0,T,h)
PX =[]
PY =[]
PZ =[]
sigma = 10
b = 8/3
r = 220
def dxdt(x,y,z):
return -sigma * x + sigma * y
def dydt(x,y,z):
return r * x - y -x * z
def dzdt(x,y,z):
return -b * z + x * y
Xst = (b*(r-1))**0.5
Yst = (b*(r-1))**0.5
Zst = r - 1
x = np.empty(n)
y = np.empty(n)
z = np.empty(n)
x[0] = 22
y[0] = 70
z[0] = 180
for i in range(n-1):
k_1 = dxdt(x[i] , y[i] , z[i])
j_1 = dydt(x[i] , y[i] , z[i])
m_1 = dzdt(x[i] , y[i] , z[i])
k_2 = dxdt(x[i] + k_1 * h / 2 , y[i] + j_1 * h / 2, z[i] + m_1 * h / 2)
j_2 = dydt(x[i] + k_1 * h / 2 , y[i] + j_1 * h / 2, z[i] + m_1 * h / 2)
m_2 = dzdt(x[i] + k_1 * h / 2 , y[i] + j_1 * h / 2, z[i] + m_1 * h / 2)
k_3 = dxdt(x[i] + k_2 *h / 2 , y[i] + j_2 * h / 2 , z[i] + m_2 * h / 2)
j_3 = dydt(x[i] + k_2 *h / 2 , y[i] + j_2 * h / 2 , z[i] + m_2 * h / 2)
m_3 = dzdt(x[i] + k_2 *h / 2 , y[i] + j_2 * h / 2 , z[i] + m_2 * h / 2)
k_4 = dxdt(x[i] + k_3 *h , y[i] + j_3 * h , z[i] + m_3 * h)
j_4 = dydt(x[i] + k_3 *h , y[i] + j_3 * h , z[i] + m_3 * h)
m_4 = dzdt(x[i] + k_3 *h , y[i] + j_3 * h , z[i] + m_3 * h)
x[i+1] = x[i] + h/6 * (k_1 + 2*k_2 + 2*k_3 + k_4 )
y[i+1] = y[i] + h/6 * (j_1 + 2*j_2 + 2*j_3 + j_4 )
z[i+1] = z[i] + h/6 * (m_1 + 2*m_2 + 2*m_3 + m_4 )
if (x[i+1] - Xst) > 0 and (x[i] - Xst) < 0:
PX.append(Xst)
PY.append(y[i])
PZ.append(z[i])
astime = int(n/2)
asx= x[astime:]
asy= y[astime:]
asz= z[astime:]
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.plot(asx, asy, asz, color='lime', linewidth=0.5)
ax.scatter(PX, PY, PZ, marker=".", color='black')
plt.show()
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Licensing
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