uðtÞ ¼ a cos Xt À w
ð
ÞþOðeÞ;
ð5:52Þ
where the slowly varying amplitude a(t) and phase w(t) are solutions of
a
0
¼ Àbx 0 a þ
q
2x 0
sin w;
aw
0
¼ ðX À x 0 Þa À
3c
8x 0
a
3
þ
q
2x 0
cos w:
ð5:53Þ
The frequency response, describing stationary values of a as a function of X, was
obtained by letting a′ = w′ = 0, resulting in (3.189). A typical response is shown in
Fig. 5.13(a) (adopted from Fig. 3.17). Qualitative changes in the response occur at
points A and C. We can now identify these as saddle-node bifurcations
(cf. Fig. 5.3).
From Fig. 5.13(a) it should appear why saddle-nodes are also called fold or
turning point or tangent bifurcations. The figure further illustrates why
saddle-nodes belong to the class of dangerous bifurcations (Sect. 5.8): For example,
beyond point A, and below C, the response escapes to ‘distant attractors’. For the
present case these attractors happen to be equilibriums (for the amplitude), though
this would not be revealed by a local bifurcation analysis. (The ‘local eye’ cannot
see what happens on the empty side of a saddle-node bifurcation.)
Forgetting for a moment that Fig. 5.13(a) has already been computed, we can
rather easily check (5.53) for the presence of bifurcating equilibriums. At the
equilibriums (defined by a′ = w′ = 0 in (5.53)) the Jacobian of the system becomes:
Fig. 5.13 (a) Typical near-resonant frequency response a(X/x 0 ) for the Duffing equation (5.51),
showing saddle-node bifurcations. (c = 0.5, q = 0.2, b = 0.05). (b) Example of a physical system
displaying this response: a hinged-hinged beam subjected to time-harmonic loading and midplane
stretching, with transverse beam deformations being described by u(t) = acos(Xt − w) + O(e)
5.14 Examples of Bifurcating Systems
311
ð
ÞþOðeÞ;
ð5:52Þ
where the slowly varying amplitude a(t) and phase w(t) are solutions of
a
0
¼ Àbx 0 a þ
q
2x 0
sin w;
aw
0
¼ ðX À x 0 Þa À
3c
8x 0
a
3
þ
q
2x 0
cos w:
ð5:53Þ
The frequency response, describing stationary values of a as a function of X, was
obtained by letting a′ = w′ = 0, resulting in (3.189). A typical response is shown in
Fig. 5.13(a) (adopted from Fig. 3.17). Qualitative changes in the response occur at
points A and C. We can now identify these as saddle-node bifurcations
(cf. Fig. 5.3).
From Fig. 5.13(a) it should appear why saddle-nodes are also called fold or
turning point or tangent bifurcations. The figure further illustrates why
saddle-nodes belong to the class of dangerous bifurcations (Sect. 5.8): For example,
beyond point A, and below C, the response escapes to ‘distant attractors’. For the
present case these attractors happen to be equilibriums (for the amplitude), though
this would not be revealed by a local bifurcation analysis. (The ‘local eye’ cannot
see what happens on the empty side of a saddle-node bifurcation.)
Forgetting for a moment that Fig. 5.13(a) has already been computed, we can
rather easily check (5.53) for the presence of bifurcating equilibriums. At the
equilibriums (defined by a′ = w′ = 0 in (5.53)) the Jacobian of the system becomes:
Fig. 5.13 (a) Typical near-resonant frequency response a(X/x 0 ) for the Duffing equation (5.51),
showing saddle-node bifurcations. (c = 0.5, q = 0.2, b = 0.05). (b) Example of a physical system
displaying this response: a hinged-hinged beam subjected to time-harmonic loading and midplane
stretching, with transverse beam deformations being described by u(t) = acos(Xt − w) + O(e)
5.14 Examples of Bifurcating Systems
311