Phase portrait for a system with a strange critical point.

phase_portrait.png

phase_portrait.py

# -*- coding: utf-8 -*-
"""
Phase portrait for the system

  x '= y
  y' = -(x**2 - a**2)*(1 + x)

@author: Nicolás Guarín-Zapata
@date: April 2021
"""

import numpy as np
import matplotlib.pyplot as plt

plt.rcParams["mathtext.fontset"] = "cm"

a = 0.0
y, x = np.mgrid[-1:1:201j, -1.5:0.5:201j]

u = y
v = -(x**2 - a**2)*(1 + x)

plt.streamplot(x, y, u, v, color="#111111")

V = y**2/2 + (3*x**4 + 4*x**3 - 6*a**2*x**2 - 12*a**2*x)/12

levels = np.linspace(-0.1, 0.7, 21)
plt.contourf(x, y, V, levels)
plt.contour(x, y, V, [0], colors="#eeeeee")
plt.axis("image")
plt.xlim(-1.5, 0.5)
plt.ylim(-1, 1)
plt.xlabel(r"$x$", size=14)
plt.ylabel(r"$y$", size=14)
plt.yticks([-1, -0.5, 0.0, 0.5, 1.0])
plt.xticks([-1.5, -1.0, -0.5, 0.0, 0.5])
plt.savefig("phase_portrait.png", dpi=300)
plt.show()