17C - Saddle, Node, Spiral (Stable/Unstable)

det(A) < 0

saddle point

if det(A) does not equal 0

only critical point is (0,0)

det(A) > 0 & (tr(A))² > 4det(A)

node

det(A) > 0 & tr(A) < 0

stable node/spiral

det(A) > 0 & tr(A) > 0

unstable node/spiral

det(A) > 0 & (tr(A))² < 4det(A)

spiral

Locally stable (asymptotically stable)

All eigenvalues have strictly negative real parts ($\lambda_1 < 0$ and $\lambda_2 < 0$).

Unstable

At least one eigenvalue has a strictly positive real part ($\lambda > 0$).

Inconclusive

Any eigenvalue has a zero real part ($\text{Re}(\lambda) = 0$) while no eigenvalues have positive real parts.