D
2
0 u 1 þ x
2
0 u 1 ¼
1
2
qe
irT 1 À i2x 0 A
0
þ bx 0 A
ð
ÞÀ3cA
2 A
h
i
e
ix 0 T 0
À cA
3 e
i3x 0 T 0 þ cc:
ð3:186Þ
Any particular solution u 1 to this equation will contain secular terms, unless the
function A(T 1 ) meets the solvability condition:
1
2
qe
irT 1 À i2x 0 A
0
þ bx 0 A
ð
ÞÀ3cA
2 A ¼ 0:
ð3:187Þ
With this condition fulfilled a particular solution of (3.186) becomes:
u 1 ¼
c
8x 2
0
A
3 e
i3x 0 T 0 þ cc;
ð3:188Þ
so that, by (3.179) and (3.183):
uðt; eÞ ¼ Ae
ix 0 T 0 þ e
c
8x 2
0
A
3 e
i3x 0 T 0 þ Oðe
2
Þ þ cc:
ð3:189Þ
To solve (3.187) for A(T 1 ) we let
A ¼
1
2
ae
iu
; aðT 1 Þ; uðT 1 Þ 2 R:
ð3:190Þ
Substituting into (3.187) one obtains, upon separating real and imaginary parts,
the following pair of modulation equations:
a
0
¼ Àbx 0 a þ
q
2x 0
sin rT 1 À u
ð
Þ ;
au
0
¼
3c
8x 0
a
3
À
q
2x 0
cos rT 1 À u
ð
Þ :
ð3:191Þ
To transform these into autonomous equations (i.e. with no explicit dependence
on T 1 ) a new dependent variable w is introduced, by
w ¼ rT 1 À u ð) w
0
¼ r À u
0
Þ:
ð3:192Þ
Slow modulations of amplitudes a(T 1 ) and phases w(T 1 ) are then governed by:
a
0
¼ Àbx 0 a þ
q
2x 0
sin w;
aw
0
¼ ra À
3c
8x 0
a
3
þ
q
2x 0
cos w:
ð3:193Þ
Stationary solutions are defined by having constant-valued amplitudes and
phases. Thus, for seeking stationary solutions we locate the singular points of
3.7 Externally Excited Duffing Systems
155