In [1]:

    

import numpy as np

# importamos bibliotecas para plotear
import matplotlib
import matplotlib.pyplot as plt

# para desplegar los plots en el notebook
%matplotlib inline

# para cómputo simbólico
from sympy import *
init_printing()

x, y = symbols('x y')


    

In [2]:

    

f = (y + (x/5)) * (1 - x**2)
f


    

    
Out[2]:





$$\left(\frac{x}{5} + y\right) \left(- x^{2} + 1\right)$$

In [3]:

    

g = -x*(1-y**2)
g


    

    
Out[3]:





$$- x \left(- y^{2} + 1\right)$$

In [18]:

    

F = lambdify((x,y), f)
G = lambdify((x,y), g)


    

In [19]:

    

def step(x, y, dt, f, g):
    return (x + dt * f(x, y),
             y + dt * g(x, y))

def trayectoria(x0, y0, f, g, dt=0.01, steps=100):
    x = x0
    y = y0
    t = list()
    for n in range(steps):
        t.append((x, y))
        x, y = step(x, y, dt, f, g)
    return t


    

In [116]:

    

i, j = np.meshgrid(np.linspace(-2, 2.5, 22),
                   np.linspace(-2.5, 2.5, 22))

u = F(i, j)
v = G(i, j)

plt.quiver(i, j, u, v)


    

    
Out[116]:





<matplotlib.quiver.Quiver at 0x7f9429c66da0>






    

In [113]:

    

i, j = np.meshgrid(np.linspace(-5, 2.5, 22),
                   np.linspace(-2.5, 2.5, 22))

u = F(i, j)
v = G(i, j)

plt.quiver(i, j, u, v)

x0 = 0.1
y0 = 0.1
dt = 0.1
steps = 600
x, y = zip(*[coord
           for coord in
           trayectoria(x0, y0, F, G, dt, steps)])
plt.plot(x, y, color='teal')
plt.arrow(x[-2], y[-2],
          x[-1] - x[-2], y[-1] - y[-2],
          fc="teal", head_width=0.1, head_length=0.1)

x0 = 1.5
y0 = 0.6
dt = 0.002
x, y = zip(*[coord
           for coord in
           trayectoria(x0, y0, F, G, dt, steps)])
plt.plot(x, y, color='red')
plt.arrow(x[-2], y[-2],
          x[-1] - x[-2], y[-1] - y[-2],
          fc="red", head_width=0.1, head_length=0.1)

x0 = -1.5
y0 = -0.6
dt = 0.01
x, y = zip(*[coord
           for coord in
           trayectoria(x0, y0, F, G, dt, steps)])
plt.plot(x, y, color='firebrick')
plt.arrow(x[-2], y[-2],
          x[-1] - x[-2], y[-1] - y[-2],
          fc="firebrick", head_width=0.1, head_length=0.1)

x0 = 0.89
y0 = -2
dt = 0.002
x, y = zip(*[coord
           for coord in
           trayectoria(x0, y0, F, G, dt, steps)])
plt.plot(x, y, color='purple')
plt.arrow(x[-2], y[-2],
          x[-1] - x[-2], y[-1] - y[-2],
          fc="purple", head_width=0.1, head_length=0.1)

x0 = 2
y0 = 1.3
dt = 0.0004
x, y = zip(*[coord
           for coord in
           trayectoria(x0, y0, F, G, dt, steps)])
plt.plot(x, y, color='brown')
plt.arrow(x[-2], y[-2],
          x[-1] - x[-2], y[-1] - y[-2],
          fc="brown", head_width=0.1, head_length=0.1)


    

    
Out[113]:





<matplotlib.patches.FancyArrow at 0x7f9429d9b828>






    

In [5]:

    

F(-5, 1), G(-5, 1)


    

    
Out[5]:





$$\left ( -0.0, \quad 0\right )$$

In [6]:

    

J = symbols("J")

J = Matrix([[diff(f, x), diff(f, y)], 
            [diff(g, x), diff(g, y)]])
J


    

    
Out[6]:





$$\left[\begin{matrix}- \frac{x^{2}}{5} - 2 x \left(\frac{x}{5} + y\right) + \frac{1}{5} & - x^{2} + 1\\y^{2} - 1 & 2 x y\end{matrix}\right]$$

In [7]:

    

puntos = []
X = 0
Y = 0
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [8]:

    

X = 5
Y = -1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [9]:

    

X = -5
Y = 1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [10]:

    

X = 1
Y = 1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [11]:

    

X = -1
Y = -1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [12]:

    

X = -1
Y = 1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [13]:

    

X = 1
Y = -1
puntos.append((J.trace().subs({x: X, y:Y}), J.det().subs({x: X, y:Y})))


    

In [14]:

    

puntos


    

    
Out[14]:





$$\left [ \left ( \frac{1}{5}, \quad 1\right ), \quad \left ( - \frac{74}{5}, \quad 48\right ), \quad \left ( - \frac{74}{5}, \quad 48\right ), \quad \left ( - \frac{2}{5}, \quad - \frac{24}{5}\right ), \quad \left ( - \frac{2}{5}, \quad - \frac{24}{5}\right ), \quad \left ( - \frac{2}{5}, \quad - \frac{16}{5}\right ), \quad \left ( - \frac{2}{5}, \quad - \frac{16}{5}\right )\right ]$$

In [15]:

    

plt.scatter([p[0] for p in puntos], [p[1] for p in puntos])


    

    
Out[15]:





<matplotlib.collections.PathCollection at 0x7f942c01f780>