Practica 1 Ejercicio 5

Para cada uno de los siguientes sistemas encontrar todos los puntos de equilibrio y determinar el tipo de cada punto de equilibio aislado.

$$\left\{ \begin{array}{lcc} \dot{x}_{1}=a x_{1}-x_{1}x_{2} \\ \\ \dot{x}_{2}=bx_{1}^{2}-cx_{2} \end{array} \right.$$



In [1]:

    
import sympy as sym



In [2]:

    
#Con esto las salidas van a ser en LaTeX
sym.init_printing(use_latex=True)



In [5]:

    
x_1, x_2, a, b, c= sym.symbols('x_1 x_2 a b c')



In [7]:

    
sym.Symbol("a",positive=True)
sym.Symbol("b",positive=True)
sym.Symbol("c",positive=True)









    Out[7]:





$$c$$



In [8]:

    
X = sym.Matrix([x_1, x_2])
X









    Out[8]:





$$\left[\begin{matrix}x_{1}\\x_{2}\end{matrix}\right]$$



In [18]:

    
f_1 = a * x_1 - x_1 * x_2
f_1









    



  File "<ipython-input-18-a96337569c26>", line 1
    f_1(x_1,x_2) = a * x_1 - x_1 * x_2
SyntaxError: can't assign to function call



In [11]:

    
f_2 = b * x_1 ** 2 - c * x_2
f_2









    Out[11]:





$$b x_{1}^{2} - c x_{2}$$



In [12]:

    
F = sym.Matrix([f_1,f_2])
F









    Out[12]:





$$\left[\begin{matrix}a x_{1} - x_{1} x_{2}\\b x_{1}^{2} - c x_{2}\end{matrix}\right]$$



In [17]:

    
# puntos de equilibrio del sistema
pes = sym.solve([f_1,f_2],[x_1,x_2])
pes









    Out[17]:





$$\begin{bmatrix}\begin{pmatrix}0, & 0\end{pmatrix}, & \begin{pmatrix}- \sqrt{\frac{a c}{b}}, & a\end{pmatrix}, & \begin{pmatrix}\sqrt{\frac{a c}{b}}, & a\end{pmatrix}\end{bmatrix}$$



In [15]:

    
A = F.jacobian(X)
A









    Out[15]:





$$\left[\begin{matrix}a - x_{2} & - x_{1}\\2 b x_{1} & - c\end{matrix}\right]$$



In [23]:

    
A_1 = A.subs({x_1:pes[0][0],x_2:pes[0][0]})
A_1









    Out[23]:





$$\left[\begin{matrix}a & 0\\0 & - c\end{matrix}\right]$$



In [25]:

    
A_2 = A.subs({x_1:pes[1][0],x_2:pes[1][1]})
A_2









    Out[25]:





$$\left[\begin{matrix}0 & \sqrt{\frac{a c}{b}}\\- 2 b \sqrt{\frac{a c}{b}} & - c\end{matrix}\right]$$



In [28]:

    
A_2.eigenvals()









    Out[28]:





$$\begin{Bmatrix}- \frac{c}{2} - \frac{1}{2} \sqrt{- c \left(8 a - c\right)} : 1, & - \frac{c}{2} + \frac{1}{2} \sqrt{- c \left(8 a - c\right)} : 1\end{Bmatrix}$$



In [30]:

    
A_3 = A.subs({x_1:pes[2][0],x_2:pes[2][1]})
A_3









    Out[30]:





$$\left[\begin{matrix}0 & - \sqrt{\frac{a c}{b}}\\2 b \sqrt{\frac{a c}{b}} & - c\end{matrix}\right]$$



In [31]:

    
A_3.eigenvals()









    Out[31]:





$$\begin{Bmatrix}- \frac{c}{2} - \frac{1}{2} \sqrt{- c \left(8 a - c\right)} : 1, & - \frac{c}{2} + \frac{1}{2} \sqrt{- c \left(8 a - c\right)} : 1\end{Bmatrix}$$



In [ ]: