5.4 Codimension One Bifurcations
for N-dimensional Systems
Do the generic codimension one bifurcations described above have any relevance
for real systems of possibly high dimension? Fortunately they do. Recall that
bifurcation analysis concerns the qualitative changes in system behavior in
response to parameter variations. Such changes are associated with Jacobian
eigenvalues having a zero real part. Thus, the possible bifurcations depend on the
number of eigenvalues having a zero real part, rather than on the dimension of the
system. If for an n-dimensional system all Jacobian eigenvalues have non-zero real
parts, then no bifurcations of the state x in question will occur when the system
parameters l are perturbed. If one eigenvalue has a zero real part (counting multiplicity), then codimension one bifurcations can occur.
Consider a general nonlinear system of dimension n with a single parameter l
(i.e. all other parameters of the systems kept constant):
_
x ¼ fðx; lÞ; x ¼ xðtÞ 2 R
n
; l 2 R:
ð5:12Þ
Assume that a singular point x ¼ ~ x exists, defined as a solution to the algebraic
set of equations fðx; lÞ ¼ 0. Near x ¼ ~ x the behavior of the system is governed by
the linearization:
_
x ¼ Jð~ x; lÞ x À ~ x
ð
Þ; Jðx; lÞ ¼
@f
@x
:
ð5:13Þ
At the singular point ~ x the Jacobian J(~ x, l) has n eigenvalues k j , j = 1,
n. Bifurcations can occur if one or more of these eigenvalues has a zero real
part. Let l = l 0 be a point for which Re(k j ) = 0 for one or more values of j. That is,
(~ x, l 0 ) can be a bifurcation point. If there is only one eigenvalue with zero real part
(an imaginary pair counting as one) the bifurcation will be of codimension one.
Hence, it will qualitatively resemble one of the generic codimension one bifurcations described in Sect. 5.3. Generally, a simple zero eigenvalue indicates a
saddle-node, a pitchfork or a transcritical bifurcation, whereas a purely imaginary
pair indicates a Hopf bifurcation. However, without taking the nonlinear terms of
(5.12) into account, we can make no useful statements regarding the
post-bifurcational behavior.
Below we state two formal (and formidable) theorems establishing necessary
conditions for, respectively, the saddle-node and the Hopf bifurcation to occur.
Pitchfork and transcritical bifurcations are treated as variants of the saddle-node.
280
5 Bifurcation Analysis
for N-dimensional Systems
Do the generic codimension one bifurcations described above have any relevance
for real systems of possibly high dimension? Fortunately they do. Recall that
bifurcation analysis concerns the qualitative changes in system behavior in
response to parameter variations. Such changes are associated with Jacobian
eigenvalues having a zero real part. Thus, the possible bifurcations depend on the
number of eigenvalues having a zero real part, rather than on the dimension of the
system. If for an n-dimensional system all Jacobian eigenvalues have non-zero real
parts, then no bifurcations of the state x in question will occur when the system
parameters l are perturbed. If one eigenvalue has a zero real part (counting multiplicity), then codimension one bifurcations can occur.
Consider a general nonlinear system of dimension n with a single parameter l
(i.e. all other parameters of the systems kept constant):
_
x ¼ fðx; lÞ; x ¼ xðtÞ 2 R
n
; l 2 R:
ð5:12Þ
Assume that a singular point x ¼ ~ x exists, defined as a solution to the algebraic
set of equations fðx; lÞ ¼ 0. Near x ¼ ~ x the behavior of the system is governed by
the linearization:
_
x ¼ Jð~ x; lÞ x À ~ x
ð
Þ; Jðx; lÞ ¼
@f
@x
:
ð5:13Þ
At the singular point ~ x the Jacobian J(~ x, l) has n eigenvalues k j , j = 1,
n. Bifurcations can occur if one or more of these eigenvalues has a zero real
part. Let l = l 0 be a point for which Re(k j ) = 0 for one or more values of j. That is,
(~ x, l 0 ) can be a bifurcation point. If there is only one eigenvalue with zero real part
(an imaginary pair counting as one) the bifurcation will be of codimension one.
Hence, it will qualitatively resemble one of the generic codimension one bifurcations described in Sect. 5.3. Generally, a simple zero eigenvalue indicates a
saddle-node, a pitchfork or a transcritical bifurcation, whereas a purely imaginary
pair indicates a Hopf bifurcation. However, without taking the nonlinear terms of
(5.12) into account, we can make no useful statements regarding the
post-bifurcational behavior.
Below we state two formal (and formidable) theorems establishing necessary
conditions for, respectively, the saddle-node and the Hopf bifurcation to occur.
Pitchfork and transcritical bifurcations are treated as variants of the saddle-node.
280
5 Bifurcation Analysis