There are mainly 5 cases of Eigenvalues(from book Elementary Differential Equations and Boundary Value Problems-11th Edition section 9.1):
as it is mentioned above, the equal eigenvalues case mentioned above is CASE 3.
CASE 1: Real, Unequal Eigenvalues of the Same Sign
CASE 2: Real Eigenvalues of Opposite Sign ->saddle point
CASE 3: Equal Eigenvalues
CASE 4: Complex Eigenvalues with Nonzero Real Part
CASE 5: Pure Imaginary Eigenvalues ->center
After memorized there are five cases, CASE 1, CASE 3 and CASE 4 have two branches while the rest of the cases(CASE 2 and CASE 5) only have one:
to be more specific:
CASE 1: Real, Unequal Eigenvalues of the Same Sign separated into:
a)lambda1 >lambda2 >0:
critical point called node/nodal source
a)lambda1 <lambda2 <0:
critical point called node/nodal sink
CASE 3:Equal Eigenvalues separated into:
a)two independent eigenvectors:
critical point called proper node or star point
b)one independent eigenvector:
critical point called improper node or degenerate node
CASE 4:Complex Eigenvalues with Nonzero Real Part separated into:
a)pointing-outward trajectories as lambda > 0:
critical point called spiral source
a)pointing-inward trajectories as lambda < 0:
critical point called spiral sink
For the stability, as long as there is one lambda>0, then it is unstable, and the last one lambda=0 is stable. For the rest of them, asymptotically stable applied.
