 $$\left\{ \begin{array}{lcc}
            \dot{x}_{1}=x_{1}-x_{2}-(x_{1}^{2}+\frac{3}{2}x_{2}^{2})x_{1} \\
             \\ \dot{x}_{2}=x_{1}+x_{2}-(x_{1}^{2}+\frac{1}{2}x_{2}^{2})x_{2}
             \end{array}
\right.$$
 


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


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


```python
f_1 = x_1 - x_2 - (x_1**2 + sym.Rational(3,2)* x_2**2)*x_1
f_1
```


```python
f_2 = x_1 + x_2 - (x_1**2 + sym.Rational(1,2)* x_2**2)*x_2
```


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


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


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


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


```python
sym.latex(A_1)
```




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




```python
#sym.plot_implicit((f_1 )&(f_2))
```


```python
ine = (f_1 <= 0)&(f_2 <= 0)
```


```python
campo = 2*x_1*F[0] + 2*x_2*F[1]
campo = campo.simplify()
```


```python
sym.plot_implicit(campo < 0,xlabel=r'$x_1$',ylabel=r'$x_2$',title='Zona de existencia de ciclo limite')
```


```python
sym.exp(x_2)
```


```python
F.subs({x_1:2,x_2:-2})
```


```python
#sym.solve(F)
```


```python
expr = sym.cos(theta)**4 + sym.sin(theta)**4 - 5*sym.cos(theta)**2 * sym.sin(theta)**2
expr
```


```python
sym.plot(expr)
```


```python
expr = sym.diff(F[0],x_1) + sym.diff(F[1],x_2)
expr
```


```python

```

a445e04cc0aaa4a98c75d6e4cf9987b96a20c6eb
ipynb
Jupyter Notebook
parcial1_Noblia_ej1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
parcial1_Noblia_ej1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
parcial1_Noblia_ej1.ipynb
elsuizo/Nonlinear_systems
9636d4a450339b8c735934923810c9539ac76042
MIT
Qwen/Qwen-72B
1. YES
2. YES