##Practica 2 Ejercicio 1
Mostrar que cada uno de los siguientes sistemas tiene una orbita periódica:
  
  * a) 
 
 $$\left\{ \begin{array}{lcc}
            \dot{x}_{1}=x_{2} \\
             \\ \dot{x}_{2}=-x_{1}+x_{2}(1-3x_{1}^{2}-2x_{2}^{2})
             \end{array}
\right.$$
 
  * b) 
         
$$
\ddot{y}+y=\epsilon \dot{y}(1-y^{2}-(\dot{y})^{2}) \:; \epsilon > 0
$$ 
 


```python
import sympy as sym
```


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


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


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


```python
f_1 = x_2
```


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


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


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


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


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


```python
A_1.eigenvals()
```


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


```python
eq.factor()
```


```python
sym.plot_implicit(eq)
```




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




```python

```

e561d0da286509573c32a05b287b6e9e4c3cdb5b
ipynb
Jupyter Notebook
practica2_eje1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
practica2_eje1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
practica2_eje1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
Qwen/Qwen-72B
1. YES
2. YES