Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't been able to find any good resources online.

For an answer to your question, review the answers to the MSE posting: What is Jacobian Matrix?.

As a practical dynamical systems example, lets look at a system from another problem you posed, we have:

$$f_1 = x'=y+x(1-x^2-y^2)$$ $$f_2 = y'=-x+y(1-x^2-y^2)$$

If we find the critical points for this system, we arrive at:

$$(x, y) = (0,0)$$

We can find the Jacobian matrix of this system as:

$$J_{(x,y)}(f)=\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{pmatrix} = \begin{pmatrix} 1-3x^2-y^2 & 1 - 2 xy \\ -1 -2 xy & 1-x^2-3y^2 \end{pmatrix}$$

We can now evaluate the Jacobian at each critical point we found and look at the eigenvalues, arriving at:

$$J_{(0,0)}(f)=\begin{pmatrix} 1& 1 \\ -1 & 1 \end{pmatrix}$$

The eigenvalues are:

$$\lambda_{1,2} = 1 ~\pm ~i$$

This linearization tells us we have an unstable spiral.

A phase portrait shows (and verifies the linearization): Note: It is important to note that this process only works for robust cases and not for marginal cases!