Jacobian matrix of a nonlinear system

Question

 

is a Jacobian matrix of a nonlinear system   with a list of critical points.

From below choices, select all critical points  that are classified as stable using the above Jacobian matrix.

A.)

B.)

C.)

D.)

Answer 1

A critical point is stable if the eigenvalues of Jacobian are both real and negative or complex and the real parts are negative.

At point A, the Jacobian is

|-π 0|

|0 π|.

The eigenvalues of this Jacobian are ±π. One eigenvalue is positive. Therefore, point A is not stable.

For the rest points, you can do it in a similar way.