segment of H 4 = 0 implies that, for loadings just below this segment, the zero
solution is stable only to small disturbances. A strong disturbance may throw the
state of the system beyond the unstable branches of the bifurcation. Indeed,
numerical analysis show that the stable zero solution may explode into a large and
complicated limit cycle, which is approached only after long periods of transient
chaos (we shall return to this phenomenon in Chap. 6).
The qualitative aspects of the above results could have been obtained by using
bifurcation analysis. Sect. 4.5.2 here provides the analysis of Jacobian eigenvalues,
which is necessary for a subsequent bifurcation analysis. One-dimensional center
manifold reductions could then be employed for qualitatively describing the
codimension one bifurcations shown in Fig. 5.16(a).
Recall that center manifold reductions can alternatively be obtained by using
perturbation analysis, as illustrated also by the present example: The full set of
equations governing motions of the double pendulum is given by (4.52), which is
four-dimensional, and rather complicated. The multiple scales analysis of
Sect. 4.5.3, performed near the Hopf bifurcation set, reduces the system to (4.73)–
(4.74), which is two-dimensional, and very simple. This is the minimal dimension
required for capturing Hopf bifurcations. And the reduced system is a center
manifold reduction, as indicated by the comment ‘disregarding damped terms’ (just
above (4.66)), which corresponds to ignoring dimensions needed only to describe
motions on the stable manifold.
5.15 Problems
Problem 5.1 For each of the systems below, examine and sketch the equilibriums as l varies near l = 0. Identify the bifurcations, in case there are any.
(a) _
x ¼ lx þ x
5
(b) _
x ¼ l À x
5
(c) _
x ¼ l
2 x À x
3
(d) _
x ¼ Ày þ x l þ ðx
2
þ y
2
Þ
ð
Þ ; _
y ¼ x þ y l þ ðx
2
þ y
2
Þ
ð
Þ
(e) _
x ¼ lx À xy þ ðax À byÞðx
2
þ y
2
Þ; _
y ¼ xx þ ly þ ðbx þ ayÞðx
2
þ y
2
Þ
Problem 5.2 Consider the system:
€ u þ b _
u À u þ u
2
¼ 0:
ð5:69Þ
(a) Discuss possible bifurcations for the equilibriums of this system.
(b) Sketch the phase plane orbits for b = 0, b < 0, and b > 0, respectively.
318
5 Bifurcation Analysis
solution is stable only to small disturbances. A strong disturbance may throw the
state of the system beyond the unstable branches of the bifurcation. Indeed,
numerical analysis show that the stable zero solution may explode into a large and
complicated limit cycle, which is approached only after long periods of transient
chaos (we shall return to this phenomenon in Chap. 6).
The qualitative aspects of the above results could have been obtained by using
bifurcation analysis. Sect. 4.5.2 here provides the analysis of Jacobian eigenvalues,
which is necessary for a subsequent bifurcation analysis. One-dimensional center
manifold reductions could then be employed for qualitatively describing the
codimension one bifurcations shown in Fig. 5.16(a).
Recall that center manifold reductions can alternatively be obtained by using
perturbation analysis, as illustrated also by the present example: The full set of
equations governing motions of the double pendulum is given by (4.52), which is
four-dimensional, and rather complicated. The multiple scales analysis of
Sect. 4.5.3, performed near the Hopf bifurcation set, reduces the system to (4.73)–
(4.74), which is two-dimensional, and very simple. This is the minimal dimension
required for capturing Hopf bifurcations. And the reduced system is a center
manifold reduction, as indicated by the comment ‘disregarding damped terms’ (just
above (4.66)), which corresponds to ignoring dimensions needed only to describe
motions on the stable manifold.
5.15 Problems
Problem 5.1 For each of the systems below, examine and sketch the equilibriums as l varies near l = 0. Identify the bifurcations, in case there are any.
(a) _
x ¼ lx þ x
5
(b) _
x ¼ l À x
5
(c) _
x ¼ l
2 x À x
3
(d) _
x ¼ Ày þ x l þ ðx
2
þ y
2
Þ
ð
Þ ; _
y ¼ x þ y l þ ðx
2
þ y
2
Þ
ð
Þ
(e) _
x ¼ lx À xy þ ðax À byÞðx
2
þ y
2
Þ; _
y ¼ xx þ ly þ ðbx þ ayÞðx
2
þ y
2
Þ
Problem 5.2 Consider the system:
€ u þ b _
u À u þ u
2
¼ 0:
ð5:69Þ
(a) Discuss possible bifurcations for the equilibriums of this system.
(b) Sketch the phase plane orbits for b = 0, b < 0, and b > 0, respectively.
318
5 Bifurcation Analysis