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I have to draw the phase space of this modification of Duffing oscillator:\begin{align} \dot x &= y,\\ \dot y &= x - x^3 - ay - (x^2)y \quad \text{ when } \quad a=1 \; \text{ and } \;a=5. \end{align}
I know that critical points are $(0, 0)$, $(-1, 0)$, and $(1, 0),$ and I have demonstrated that when $a =1$ I have a spiral the form $(\pm1,0)$ and when $a = 5$ not.
Does someone know how to draw it?
Thank you.
$\endgroup$ 31 Answer
$\begingroup$This may help you
generated with python:
import numpy as np
import matplotlib.pyplot as plt
xmin, xmax = -2.0, 2.0
ymin, ymax = -1.0, 1.0
x, y = np.meshgrid(np.linspace(xmin, xmax, 30, endpoint = True), np.linspace(ymin, ymax, 30, endpoint = True))
a = 1
u = y
v = x - x**3 - a * y - x**2 * y
plt.streamplot(x, y, u, v)
plt.show()
