Type and stability of critical point at (0,0)

[lionsgirl12]

Consider the following system (where B is a real number, B is not equal to 0),

x₁' = -2x₁ + (B+2)x₂
x₂' = Bx₂

Depending on the value of B, the critical point at (0,0) can be of different type and/or stability. Describe the possible type/stability of the critical point for the different values of B, and give explicitly the corresponding range of values of B in each case.
-------------------------------------------------------------------------------------------
My work:

2x2 matrix:
(-2 B+2, 0 B)

This is known as a triangular matrix which means that the eigenvalues are r = -2, B

Since at least one of the eigenvalues are less than 0, we know it is moving directly
toward and will converge to the critical point.


If B is a positive value, then it will be an unstable saddle point.

If B = -2, then it will be an asymptotically stable proper node (or star point).

I am not sure how to determine what B will be.

 

[kurt.physics]

I think it would depend on the initial conditions and the direction of the vector field. Is that correct?