Jacobiana

$$\frac{dx}{dt}=a_{1}x-b_{1}x^{2}+c_{1}xy$$$$\frac{dy}{dt}=a_{2}y-b_{2}y^{2}+c_{1}xy$$

$$\frac{dx}{dt}=(1-x-y)x$$$$\frac{dy}{dt}=(4-7x-3y)y$$



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 = (1-x-y)*x
f









    Out[2]:





$$x \left(- x - y + 1\right)$$



In [3]:

    
g = (4-7*x-3*y)*y
g









    Out[3]:





$$y \left(- 7 x - 3 y + 4\right)$$

Equilibrios



In [4]:

    
solve(f, x)









    Out[4]:





$$\left [ 0, \quad - y + 1\right ]$$



In [5]:

    
solve(g, y)









    Out[5]:





$$\left [ 0, \quad - \frac{7 x}{3} + \frac{4}{3}\right ]$$



In [6]:

    
Y = solve(g, y)[1]
solve(f.subs(y, Y),x)









    Out[6]:





$$\left [ 0, \quad \frac{1}{4}\right ]$$



In [7]:

    
solve(g.subs(x, -y + 1), y)









    Out[7]:





$$\left [ 0, \quad \frac{3}{4}\right ]$$

Jacobiana



In [11]:

    
J = symbols("J")

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









    Out[11]:





$$\left[\begin{matrix}- 2 x - y + 1 & - x\\- 7 y & - 7 x - 6 y + 4\end{matrix}\right]$$

Evaluada en un punto de equilibrio



In [15]:

    
J = J.subs({x: 1/4, y:3/4})
J









    Out[15]:





$$\left[\begin{matrix}-0.25 & -0.25\\-5.25 & -2.25\end{matrix}\right]$$



In [16]:

    
J.det(), J.trace()









    Out[16]:





$$\left ( -0.75, \quad -2.5\right )$$