Practica 2 Ejercicio 1

Mostrar que cada uno de los siguientes sistemas tiene una orbita periódica:

$$\left{ \begin{array}{lcc}

    \dot{x}_{1}=x_{2} \\
     \\ \dot{x}_{2}=-x_{1}+x_{2}(1-3x_{1}^{2}-2x_{2}^{2})
     \end{array}

\right.$$

$$ \ddot{y}+y=\epsilon \dot{y}(1-y^{2}-(\dot{y})^{2}) \:; \epsilon > 0 $$



In [1]:

    
import sympy as sym



In [2]:

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



In [3]:

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



In [4]:

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









    Out[4]:





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



In [5]:

    
f_1 = x_2



In [6]:

    
f_2 = -x_1 + x_2 * (1 - 3 * x_1 ** 2 - 2 * x_2 ** 2)
f_2









    Out[6]:





$$- x_{1} + x_{2} \left(- 3 x_{1}^{2} - 2 x_{2}^{2} + 1\right)$$



In [7]:

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









    Out[7]:





$$\left[\begin{matrix}x_{2}\\- x_{1} + x_{2} \left(- 3 x_{1}^{2} - 2 x_{2}^{2} + 1\right)\end{matrix}\right]$$



In [8]:

    
A = F.jacobian(X)
#A.simplify()
A









    Out[8]:





$$\left[\begin{matrix}0 & 1\\- 6 x_{1} x_{2} - 1 & - 3 x_{1}^{2} - 6 x_{2}^{2} + 1\end{matrix}\right]$$



In [18]:

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









    Out[18]:





$$\begin{bmatrix}\begin{pmatrix}0, & 0\end{pmatrix}\end{bmatrix}$$



In [20]:

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









    Out[20]:





$$\left[\begin{matrix}0 & 1\\-1 & 1\end{matrix}\right]$$



In [21]:

    
A_1.eigenvals()









    Out[21]:





$$\begin{Bmatrix}\frac{1}{2} - \frac{\sqrt{3} i}{2} : 1, & \frac{1}{2} + \frac{\sqrt{3} i}{2} : 1\end{Bmatrix}$$



In [28]:

    
eq =2 * x_2 ** 2 - 6 * x_1 * x_2 ** 2 - 4 * x_2 ** 4



In [29]:

    
eq.factor()









    Out[29]:





$$- 2 x_{2}^{2} \left(3 x_{1} + 2 x_{2}^{2} - 1\right)$$



In [27]:

    
sym.plot_implicit(eq)









    Out[27]:





<sympy.plotting.plot.Plot at 0x7fca80f266d0>



In [ ]: