Someone please help non-linear system!

Question

Given the following non-linear system:

 

find its critical points and classify the solution behavior of the non-linear system near each critical point using the linearization method.

In other words, is the solution Stable? Unstable? Unstable (near a saddle point)? Spiral sink (stable)? Stable but tracing out an ellipse?

Answer 1

Compute the eigenvalues of the Jacobian matrix evaluated at the critical points.