The diagonal Jordan form (3.41) applies when |[a 1 a 2 ]| 6 ¼ 0. The system (3.37)
then becomes, writing out the components:
_
u 1 ¼ k 1 u 1 ;
_
u 2 ¼ k 2 u 2 ;
ð3:44Þ
with solution
u 1 ðtÞ ¼ u 10 e
k 1 t
;
u 2 ðtÞ ¼ u 20 e
k 2 t
;
ð3:45Þ
or, upon eliminating the time-variable t:
u 2 ¼ u 20 u 1 =u 10
ð
Þ
k 2 =k 1
ð3:46Þ
where u 10 = u 1 (0) and u 20 = u 2 (0) define the initial conditions.
If the eigenvalues k 1,2 are real with equal sign, the arrangement of orbits (u 1 (t),
u 2 (t)) is as depicted in Fig. 3.8(a) for k 1 6 ¼ k 2 , and in Fig. 3.8(c) for k 1 = k 2 . The
singular point (0,0) is in this case a node; in fact a stable node for k 1,2 < 0 and an
unstable node for k 1 > 0 or k 2 > 0. The orbits in Fig. 3.8(a) are tangent to the u 1 -
axis (except the one on the u 2 -axis). This occurs for k 2 > k 1 , whereas for k 2 < k 1
the u 2 -axis becomes the tangent.
If the eigenvalues are real with different sign, then k 2 /k 1 < 0, and the orbits are
as depicted in Fig. 3.8(b). The singular point is a saddle. A saddle point is always
unstable, though occasionally we will refer to the stable and unstable directions or
branches of a saddle (in Fig. 3.8(b), the u 2 - and u 1 -axis, respectively).
If the eigenvalues are complex they will be complex conjugates, that is, if k 1 = k
then k 2 = k. The solution (3.46) remains valid, though u 1 and u 2 now turn complex
with u 2 = ū 1 . Introducing polar coordinates:
u 1 ¼ re
iu
; u 2 ¼ u 1 ¼ re
Àiu
; r ¼ rðtÞ; u ¼ uðtÞ;
ð3:47Þ
equation (3.46) becomes
re
Àiu
¼ r 0 e
Àiu 0
re
iu
r 0 e iu 0
k=k
;
ð3:48Þ
which can be reduced to
r ¼ r 0 exp
ReðkÞ
ImðkÞ
u À u 0
ð
Þ
:
ð3:49Þ
3.4 Qualitative Analysis of the Unforced Response
113