17.6 Concluding thoughts

To summarize, let’s say we have the following system of equations

\[\begin{equation} \begin{split} \frac{dx}{dt} &= f(x,y) \\ \frac{dy}{dt} &= g(x,y) \end{split} \end{equation}\]

Assuming we have an equibrium solution at \((x,y)=(a,b)\), the Jacobian matrix at that solution is:

\[\begin{equation} J_{(a,b)} =\begin{pmatrix} f_{x}(a,b) & f_{y}(a,b) \\ g_{x}(a,b) & g_{y}(a,b) \end{pmatrix} \end{equation}\]

While we don’t discuss it here, the Jacobian matrix also extends to higher order systems as well.