Chapter 2
Bifurcations
2.1
Types of Bifurcations
2.1.1
Saddlenode bifurcation
We remarked above how
f
′
(
u
) is in general nonzero when
f
(
u
) itself vanishes, since two
equations in a single unknown is an overdetermined set.
However, consider the function
F
(
x, α
), where
α
is a control parameter.
If we demand
F
(
x, α
) = 0 and
∂
x
F
(
x, α
) = 0,
we have two equations in two unknowns, and in general there will be a zerodimensional
solution set consisting of points (
x
c
, α
c
). The situation is depicted in Fig. 2.1.
Let’s expand
F
(
x, α
) in the vicinity of such a point (
x
c
, α
c
):
F
(
x, α
) =
F
(
x
c
, α
c
) +
∂F
∂x
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(
x
c
,α
c)
(
x
−
x
c
) +
∂F
∂α
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(
x
c
,α
c)
(
α
−
α
c
) +
1
2
∂
2
F
∂x
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(
x
c
,α
c)
(
x
−
x
c
)
2
+
∂
2
F
∂x ∂α
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(
x
c
,α
c)
(
x
−
x
c
) (
α
−
α
c
) +
1
2
∂
2
F
∂α
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(
x
c
,α
c)
(
α
−
α
c
)
2
+
. . .
(2.1)
=
A
(
α
−
α
c
) +
B
(
x
−
x
c
)
2
+
. . . ,
(2.2)
where we keep terms of lowest order in the deviations
δx
and
δα
. If we now rescale
u
≡
radicalbig
B/A
(
x
−
x
c
),
r
≡
α
−
α
c
, and
τ
=
At
, we have, neglecting the higher order terms, we
obtain the ‘normal form’ of the saddlenode bifurcation,
du
dτ
=
r
+
u
2
.
(2.3)
The evolution of the flow is depicted in Fig. 2.2. For
r <
0 there are two fixed points – one
stable (
u
∗
=
−
√
−
r
) and one unstable (
u
= +
√
−
r
). At
r
= 0 these two nodes coalesce and
annihilate each other. (The point
u
∗
= 0 is halfstable precisely at
r
= 0.) For
r >
0 there
are no longer any fixed points in the vicinity of
u
= 0. In the left panel of Fig. 2.3 we show
the flow in the extended (
r, u
) plane. The unstable and stable nodes annihilate at
r
= 0.
1
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CHAPTER 2.
BIFURCATIONS
Figure 2.1: Evolution of
F
(
x, α
) as a function of the control parameter
α
.
2.1.2
Transcritical bifurcation
Another situation which arises frequently is the
transcritical bifurcation
.
Consider the
equation ˙
x
=
f
(
x
) in the vicinity of a fixed point
x
∗
.
dx
dt
=
f
′
(
x
∗
) (
x
−
x
∗
) +
1
2
f
′′
(
x
∗
)(
x
−
x
∗
)
2
+
. . . .
(2.4)
We rescale
u
≡
β
(
x
−
x
∗
) with
β
=
−
1
2
f
′′
(
x
∗
) and define
r
≡
f
′
(
x
∗
) as the control parameter,
to obtain, to order
u
2
,
du
dt
=
ru
−
u
2
.
(2.5)
Note that the sign of the
u
2
term can be reversed relative to the others by sending
u
→ −
u
and
r
→ −
r
.
Consider a crude model of a laser threshold. Let
n
be the number of photons in the laser
cavity, and
N
the number of excited atoms in the cavity.
The dynamics of the laser are
approximated by the equations
˙
n
=
GNn
−
kn
(2.6)
N
=
N
0
−
αn .
(2.7)
Here
G
is the gain coefficient and
k
the photon decay rate.
N
0
is the pump strength, and
α
is a numerical factor. The first equation tells us that the number of photons in the cavity
grows with a rate
GN
−
k
; gain is proportional to the number of excited atoms, and the
loss rate is a constant cavitydependent quantity (typically through the ends, which are
semitransparent). The second equation says that the number of excited atoms is equal to
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 Fall '11
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 Critical Point, Optimization, Phase transition, Bifurcations

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