I am working on a nonlinear dynamics problem set and I am uncertain of how to go about finding the Jacobian matrix for this set of equations. I know that you can use taylor expansion and I should end up with a matrix of partial derivatives... I think I have an answer, I just want to double check! Thank you so much!
Here are the equations:
$\dot{x} = y - x^3 - 3x^2 + I$
$\dot{y} = 1 - 5x^2 - y$
NB: I think that the jacobian is $\begin{bmatrix}3x^2 - 6x & 1\\ -10x & -1\end{bmatrix}$ and it will be evaluated at some point (x*, y*).
Hint: Your Jacobian is correct.
$$J = \begin{bmatrix}\dfrac{\partial \dot x}{\partial x} & \dfrac{\partial \dot x}{\partial y}\\\dfrac{\partial \dot y}{\partial x} & \dfrac{\partial \dot y}{\partial y}\end{bmatrix} = \begin{bmatrix}3x^2 - 6x & 1\\ -10x & -1\end{bmatrix}$$
Note: The comments state that $I$ is a constant and it can be assumed to be zero.
Now, find the critical points and evaluate the eigenvalues of the Jacobian at each of them.
Update: I find three critical points
$$(x, y) = (-7.98431, -317.746), (-0.361831, 0.345393), (0.346144, 0.400921)$$
Can you proceed?