real parts (the codimension) that determines the complexity of the bifurcations. This
is not as strange as it may seem, since motions associated with non-zero eigenvalues
ride on either contracting or expanding state spaces for which the dynamics is simple
and well understood: Motions simply decay or grow exponentially with time.
We explain the codimension one bifurcations in terms of simple
one-dimensional study cases. You may perceive these generic cases as reductions
of full-blown higher dimensional systems possessing a single eigenvalue with a
zero real part (in Sect. 5.5 we provide tools for this reduction).
5.3.1 The Pitchfork Bifurcation
The pitchfork bifurcation can be represented by the generic differential equation
_
x ¼ lx À x
3
;
ð5:2Þ
with singular points (found by letting _
x = 0 and solving for x):
~ x ¼ 0; ~ x ¼ Æ
ffiffiffi
l
p :
ð5:3Þ
If l < 0 there is one singular point, whereas if l > 0 there are three. The
Jacobian of the system is:
Jðx; lÞ ¼
@ _
x
@x
¼ l À 3x
2
:
ð5:4Þ
At the singular point ~ x the eigenvalue k of the Jacobian becomes:
k ¼ l À 3~ x
2
;
ð5:5Þ
or
k ¼ l
for ~ x ¼ 0 ;
k ¼ À2l for ~ x ¼ Æ
ffiffiffi
l
p :
ð5:6Þ
A bifurcation can occur when the Jacobian eigenvalue has a zero real part, that
is, when l = 0. Hence, l = 0 is the (one and only) bifurcation value for this system.
When l < 0 the only singular point is ~ x = 0. This point is stable since Re(k) =
l < 0. When l > 0 the singular point ~ x = 0 becomes unstable (Re(k) > 0), and two
new singular points ~ x ¼ Æ
ffiffiffi
l
p emerge, both stable since Re(k) = −2l < 0.
We are now able to draw the bifurcation diagram of the generic pitchfork
bifurcation (Fig. 5.1). A bifurcation diagram simply shows how singular points
branch out and change stability at a bifurcation point. Indeed, the picture of the
pitchfork bifurcation looks like a pitchfork.
5.3 Codimension One Bifurcations of Equilibriums
275
is not as strange as it may seem, since motions associated with non-zero eigenvalues
ride on either contracting or expanding state spaces for which the dynamics is simple
and well understood: Motions simply decay or grow exponentially with time.
We explain the codimension one bifurcations in terms of simple
one-dimensional study cases. You may perceive these generic cases as reductions
of full-blown higher dimensional systems possessing a single eigenvalue with a
zero real part (in Sect. 5.5 we provide tools for this reduction).
5.3.1 The Pitchfork Bifurcation
The pitchfork bifurcation can be represented by the generic differential equation
_
x ¼ lx À x
3
;
ð5:2Þ
with singular points (found by letting _
x = 0 and solving for x):
~ x ¼ 0; ~ x ¼ Æ
ffiffiffi
l
p :
ð5:3Þ
If l < 0 there is one singular point, whereas if l > 0 there are three. The
Jacobian of the system is:
Jðx; lÞ ¼
@ _
x
@x
¼ l À 3x
2
:
ð5:4Þ
At the singular point ~ x the eigenvalue k of the Jacobian becomes:
k ¼ l À 3~ x
2
;
ð5:5Þ
or
k ¼ l
for ~ x ¼ 0 ;
k ¼ À2l for ~ x ¼ Æ
ffiffiffi
l
p :
ð5:6Þ
A bifurcation can occur when the Jacobian eigenvalue has a zero real part, that
is, when l = 0. Hence, l = 0 is the (one and only) bifurcation value for this system.
When l < 0 the only singular point is ~ x = 0. This point is stable since Re(k) =
l < 0. When l > 0 the singular point ~ x = 0 becomes unstable (Re(k) > 0), and two
new singular points ~ x ¼ Æ
ffiffiffi
l
p emerge, both stable since Re(k) = −2l < 0.
We are now able to draw the bifurcation diagram of the generic pitchfork
bifurcation (Fig. 5.1). A bifurcation diagram simply shows how singular points
branch out and change stability at a bifurcation point. Indeed, the picture of the
pitchfork bifurcation looks like a pitchfork.
5.3 Codimension One Bifurcations of Equilibriums
275