In [2]:

    

import sympy as sym


    

In [3]:

    

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


    

In [4]:

    

x_1, x_2 ,theta = sym.symbols('x_1 x_2 theta')


    

In [5]:

    

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


    

    
Out[5]:





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

In [6]:

    

f_1 = 2*x_2 + x_1*(x_1**2 + 2*x_2**4)
f_1


    

    
Out[6]:





$$x_{1} \left(x_{1}^{2} + 2 x_{2}^{4}\right) + 2 x_{2}$$

In [7]:

    

f_2 = -2*x_1 + x_2*(x_1**2 + x_2**4)
f_2


    

    
Out[7]:





$$- 2 x_{1} + x_{2} \left(x_{1}^{2} + x_{2}^{4}\right)$$

In [8]:

    

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


    

    
Out[8]:





$$\left[\begin{matrix}x_{1} \left(x_{1}^{2} + 2 x_{2}^{4}\right) + 2 x_{2}\\- 2 x_{1} + x_{2} \left(x_{1}^{2} + x_{2}^{4}\right)\end{matrix}\right]$$

In [9]:

    

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


    

    
Out[9]:





$$\left [ \left \{ x_{1} : 0, \quad x_{2} : 0\right \}\right ]$$

In [10]:

    

A = F.jacobian(X)
A


    

    
Out[10]:





$$\left[\begin{matrix}3 x_{1}^{2} + 2 x_{2}^{4} & 8 x_{1} x_{2}^{3} + 2\\2 x_{1} x_{2} - 2 & x_{1}^{2} + 5 x_{2}^{4}\end{matrix}\right]$$

In [11]:

    

sym.latex(A)


    

    
Out[11]:





'\\left[\\begin{matrix}3 x_{1}^{2} + 2 x_{2}^{4} & 8 x_{1} x_{2}^{3} + 2\\\\2 x_{1} x_{2} - 2 & x_{1}^{2} + 5 x_{2}^{4}\\end{matrix}\\right]'

In [12]:

    

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


    

    
Out[12]:





$$\left[\begin{matrix}0 & 2\\-2 & 0\end{matrix}\right]$$

In [13]:

    

A_1.eigenvals()


    

    
Out[13]:





$$\left \{ - 2 i : 1, \quad 2 i : 1\right \}$$

In [16]:

    

expr = 2*x_1 *F[0] + 2*x_2*F[1]
expr = expr.simplify()


    

In [17]:

    

sym.latex(expr)


    

    
Out[17]:





'2 x_{1}^{4} + 4 x_{1}^{2} x_{2}^{4} + 2 x_{1}^{2} x_{2}^{2} + 2 x_{2}^{6}'

In [18]:

    

expr


    

    
Out[18]:





$$2 x_{1}^{4} + 4 x_{1}^{2} x_{2}^{4} + 2 x_{1}^{2} x_{2}^{2} + 2 x_{2}^{6}$$

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