Balachandran (1995) provide a readable and compact treatment with a focus on
applications, as does Seydel (2010). There are algorithms for the numerical
determination and tracking of bifurcations (Kaas-Petersen 1989; Nayfeh and
Balachandran 1995; Dankowicz and Schilder 2013), which may be useful supplements to the mostly analytical methods described here, and ready-to-use software packages exist, e.g. AUTO (in Fortran, for UNIX/Linux), MatCont and
COCO (MATLAB toolboxes), PyCont (Python toolbox).
After some introductory definitions of systems, bifurcations, and bifurcation
conditions, we describe the possible bifurcations of codimension one. These are
described in terms of one-dimensional generic systems. One-dimensional generic
systems have relevance to higher-dimensional systems as well, as described in
subsequent sections on methods for dimensional reduction. An introduction to
continuation techniques is given, useful also for graphing bifurcation diagrams. We
end the chapter by reconsidering, with an emphasis to bifurcations, some of the
physical examples already encountered in Chaps. 3 and 4.
5.2 Systems, Bifurcations, and Bifurcation
Conditions
5.2.1 Systems
We consider a general dynamic system described by a set of n autonomous
first-order differential equations:
_
x ¼ fðx; lÞ; x ¼ xðtÞ 2 R
n
; l 2 R
k
;
ð5:1Þ
where x is an n-vector of state-variables, f is an n-vector of generally nonlinear
functions, and l is a k-vector of control or bifurcation parameter s. We consider as
bifurcation parameters those among the system parameters for which a variation is
concerned.
For example, the equation of motion (3.11) for the pendulum with an oscillating
support, € h þ 2bx 0 _
h þ ðx
2
0 À qX
2 cosðXtÞÞ sin h ¼ 0, can be recast in the form (5.1)
with n = 3, x = {h, v, z}
T
, f ¼ fv; À2bx 0 v À ðx
2
0 À qX
2 cos zÞ sin h; Xg
T , and the
vector l containing one or more of the parameters (b, x 0 , q, X).
5.2.2 Bifurcations
As t ! ±∞ the state x(t) of the system (5.1) will approach a limit set in x-space.
The limit set can be a singular point, a closed orbit, or a chaotic attractor. For the
underlying physical system these limit sets correspond to, respectively: equilibrium,
periodic motion, or chaotic motion. If the parameters l of the system f are changed,
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5 Bifurcation Analysis
applications, as does Seydel (2010). There are algorithms for the numerical
determination and tracking of bifurcations (Kaas-Petersen 1989; Nayfeh and
Balachandran 1995; Dankowicz and Schilder 2013), which may be useful supplements to the mostly analytical methods described here, and ready-to-use software packages exist, e.g. AUTO (in Fortran, for UNIX/Linux), MatCont and
COCO (MATLAB toolboxes), PyCont (Python toolbox).
After some introductory definitions of systems, bifurcations, and bifurcation
conditions, we describe the possible bifurcations of codimension one. These are
described in terms of one-dimensional generic systems. One-dimensional generic
systems have relevance to higher-dimensional systems as well, as described in
subsequent sections on methods for dimensional reduction. An introduction to
continuation techniques is given, useful also for graphing bifurcation diagrams. We
end the chapter by reconsidering, with an emphasis to bifurcations, some of the
physical examples already encountered in Chaps. 3 and 4.
5.2 Systems, Bifurcations, and Bifurcation
Conditions
5.2.1 Systems
We consider a general dynamic system described by a set of n autonomous
first-order differential equations:
_
x ¼ fðx; lÞ; x ¼ xðtÞ 2 R
n
; l 2 R
k
;
ð5:1Þ
where x is an n-vector of state-variables, f is an n-vector of generally nonlinear
functions, and l is a k-vector of control or bifurcation parameter s. We consider as
bifurcation parameters those among the system parameters for which a variation is
concerned.
For example, the equation of motion (3.11) for the pendulum with an oscillating
support, € h þ 2bx 0 _
h þ ðx
2
0 À qX
2 cosðXtÞÞ sin h ¼ 0, can be recast in the form (5.1)
with n = 3, x = {h, v, z}
T
, f ¼ fv; À2bx 0 v À ðx
2
0 À qX
2 cos zÞ sin h; Xg
T , and the
vector l containing one or more of the parameters (b, x 0 , q, X).
5.2.2 Bifurcations
As t ! ±∞ the state x(t) of the system (5.1) will approach a limit set in x-space.
The limit set can be a singular point, a closed orbit, or a chaotic attractor. For the
underlying physical system these limit sets correspond to, respectively: equilibrium,
periodic motion, or chaotic motion. If the parameters l of the system f are changed,
272
5 Bifurcation Analysis