roots k = ±ix, while the remaining roots have negative real parts. The Hopf
conditions establish conditions for a Hopf bifurcation to occur, and this is just the
kind of bifurcation we are looking for – turning a stable equilibrium unstable in
favor of stable periodic oscillations (we shall return to Hopf in Chap. 5).
One can rather easily show that the Hopf conditions are met on the part of the
curve D 3 = 0 that is restricted by H 3 > 0 and D 2 > 0. In Fig. 4.12 the Hopf
bifurcation set is located on D 3 = 0, a a < a < a b .
To examine periodic motions we perform a perturbation analysis for values of
the loading parameters (a, p) near the Hopf bifurcation set. First introduce into
(4.52) a nondimensional parameter e to indicate the assumed smallness of nonlinear
terms:
_
x ¼ Aða; pÞx þ efða; p; xÞ:
ð4:59Þ
Then assume that, for a fixed value of loading ‘conservativeness’ a, the loading
magnitude p is given a slight perturbation ep 1 from the critical value p 0 :
p ¼ p 0 þ ep 1 ;
ð4:60Þ
where (a, p 0 ) belongs to the Hopf bifurcation set, and p 1 is a free perturbation
parameter (playing a role similar to that of the detuning parameters in previous
examples).
Using the method of multiple scales we seek an approximate solution to (4.59) in
form of a uniformly valid expansion:
xðs; eÞ ¼ x 0 ðT 0 ; T 1 Þ þ ex 1 ðT 0 ; T 1 Þ þ Oðe
2
Þ;
ð4:61Þ
where x 0 and x 1 are the unknown functions to be determined, and where T 0 = s and
T 1 = es are the fast and slow time-scales, respectively. By virtue of (4.60) and
(4.61) we may Taylor-expand f and A in terms of e in the vicinity of the bifurcation
point (a, p 0 ), as follows:
fða; p; xÞ ¼ fða; p 0 ; x 0 Þ þ OðeÞ;
Aða; pÞ ¼ A 0 þ ep 1 A 1 þ Oðe
2
Þ;
ð4:62Þ
where
A 0 Aða; p 0 Þ; A 1
@A
@p
p¼p 0
:
ð4:63Þ
Substituting (4.61), (4.62) and (4.53) into (4.59), by noting that d/ds = ∂/∂T 0 +
e∂/∂T 1 , and equating then coefficients to like powers e one obtains, to order e
0 :
234
4 Nonlinear Multiple-DOF Systems: Local Analysis
conditions establish conditions for a Hopf bifurcation to occur, and this is just the
kind of bifurcation we are looking for – turning a stable equilibrium unstable in
favor of stable periodic oscillations (we shall return to Hopf in Chap. 5).
One can rather easily show that the Hopf conditions are met on the part of the
curve D 3 = 0 that is restricted by H 3 > 0 and D 2 > 0. In Fig. 4.12 the Hopf
bifurcation set is located on D 3 = 0, a a < a < a b .
To examine periodic motions we perform a perturbation analysis for values of
the loading parameters (a, p) near the Hopf bifurcation set. First introduce into
(4.52) a nondimensional parameter e to indicate the assumed smallness of nonlinear
terms:
_
x ¼ Aða; pÞx þ efða; p; xÞ:
ð4:59Þ
Then assume that, for a fixed value of loading ‘conservativeness’ a, the loading
magnitude p is given a slight perturbation ep 1 from the critical value p 0 :
p ¼ p 0 þ ep 1 ;
ð4:60Þ
where (a, p 0 ) belongs to the Hopf bifurcation set, and p 1 is a free perturbation
parameter (playing a role similar to that of the detuning parameters in previous
examples).
Using the method of multiple scales we seek an approximate solution to (4.59) in
form of a uniformly valid expansion:
xðs; eÞ ¼ x 0 ðT 0 ; T 1 Þ þ ex 1 ðT 0 ; T 1 Þ þ Oðe
2
Þ;
ð4:61Þ
where x 0 and x 1 are the unknown functions to be determined, and where T 0 = s and
T 1 = es are the fast and slow time-scales, respectively. By virtue of (4.60) and
(4.61) we may Taylor-expand f and A in terms of e in the vicinity of the bifurcation
point (a, p 0 ), as follows:
fða; p; xÞ ¼ fða; p 0 ; x 0 Þ þ OðeÞ;
Aða; pÞ ¼ A 0 þ ep 1 A 1 þ Oðe
2
Þ;
ð4:62Þ
where
A 0 Aða; p 0 Þ; A 1
@A
@p
p¼p 0
:
ð4:63Þ
Substituting (4.61), (4.62) and (4.53) into (4.59), by noting that d/ds = ∂/∂T 0 +
e∂/∂T 1 , and equating then coefficients to like powers e one obtains, to order e
0 :
234
4 Nonlinear Multiple-DOF Systems: Local Analysis