19.13 Final Remarks on Normal Forms 355
where
A(t) ≡ Df(¯x(t)).
In applying the method of normal forms to (19.13.2), the fact that
A(t) is time dependent causes problems. If ¯x(t) is periodic in t, then
A(t) is periodic. Hence, Floquet theory can be used to transform
(19.13.2) to a vector ﬁeld where the linear part is constant (this is
described in Arnold [1983]). In this case the method of normal forms
as developed in this chapter can then be applied. Recently, Floquet
theory has been generalized to the quasiperiodic case by Johnson
[1986, 1987]; using these ideas, the normal form theory can be applied
in this case also. Recent results along these lines have also been
obtained by Jorba and Simo [1992], Treshchev [1995], and Jorba et
al [1997]. Their results also apply to linear systems with quasiperiodically
varying coeﬃcients, and are computationally very eﬃcient.
Concerning center manifold theory, Sell [1978] has proved existence
theorems for stable, unstable, and center manifolds in nonautonomous
systems.
Some very interesting work of Siegmund [2002] has recently appeared
that develops the method of normal forms for very general nonautonomous
vector ﬁelds.
Smooth Linearization. There exists a number of results concerning differentiable
coordinate changes that linearize a dynamical system (vector
ﬁeld or diﬀeomorphism) in the neighborhood of an invariant manifold.
A recent review of these results can be found in Bronstein and
Kopanskii [1994].
Real Normal Forms and Complex Coordinates. We have seen numerous
examples in this chapter where the use of complex coordinates
simpliﬁes normal form calculations. A systematic development
of this approach can be found in Menck [1993].
Normal Forms for Stochastic Systems. Normal form theory for stochastic
dynamical systems has been worked out in Namachchivaya
and Leng [1990] and Namachchivaya and Lin [1991].
20
Bifurcation of Fixed Points of
Vector Fields
Consider the parameterized vector ﬁeld
˙y = g(y, λ), y ∈ Rn
, λ ∈ Rp
, (20.0.1)
where g is a Cr
function on some open set in Rn
× Rp
. The degree of
diﬀerentiability will be determined by our need to Taylor expand (20.0.1).
Usually C5
will be suﬃcient.
Suppose (20.0.1) has a ﬁxed point at (y, λ) = (y0, λ0), i.e.,
g(y0, λ0) = 0. (20.0.2)
Two questions immediately arise.
1. Is the ﬁxed point stable or unstable?
2. How is the stability or instability aﬀected as λ is varied?
To answer Question 1, the ﬁrst step to take is to examine the linear vector
ﬁeld obtained by linearizing (20.0.1) about the ﬁxed point (y, λ) = (y0, λ0).
This linear vector ﬁeld is given by
˙ξ = Dyg(y0, λ0)ξ, ξ ∈ Rn
. (20.0.3)
If the ﬁxed point is hyperbolic (i.e., none of the eigenvalues of Dyg(y0, λ0)
lie on the imaginary axis), we know that the stability of (y0, λ0) in (20.0.1)
is determined by the linear equation (20.0.3) (cf. Chapter 1). This also
enables us to answer Question 2, because since hyperbolic ﬁxed points are
structurally stable (cf. Chapter 12), varying λ slightly does not change the
nature of the stability of the ﬁxed point. This should be clear intuitively,
but let us belabor the point slightly.
We know that
g(y0, λ0) = 0, (20.0.4)
and that
Dyg(y0, λ0) (20.0.5)
has no eigenvalues on the imaginary axis. Therefore, Dyg(y0, λ0) is invertible.
By the implicit function theorem, there thus exists a unique Cr
function, y(λ), such that
g(y(λ), λ) = 0 (20.0.6)
20.1 A Zero Eigenvalue 357
for λ suﬃciently close to λ0 with
y(λ0) = y0. (20.0.7)
Now, by continuity of the eigenvalues with respect to parameters, for λ
suﬃciently close to λ0,
Dyg(y(λ), λ) (20.0.8)
has no eigenvalues on the imaginary axis. Therefore, for λ suﬃciently close
to λ0, the hyperbolic ﬁxed point (y0, λ0) of (20.0.1) persists and its stability
type remains unchanged. To summarize, in a neighborhood of λ0 an isolated
ﬁxed point of (20.0.1) persists and always has the same stability type.
The real fun starts when the ﬁxed point (y0, λ0) of (20.0.1) is not hyperbolic,
i.e., when Dyg(y0, λ0) has some eigenvalues on the imaginary axis.
In this case, for λ very close to λ0 (and for y close to y0), radically new
dynamical behavior can occur. For example, ﬁxed points can be created or
destroyed and time-dependent behavior such as periodic, quasiperiodic, or
even chaotic dynamics can be created. In a certain sense (to be clariﬁed
later), the more eigenvalues on the imaginary axis, the more exotic the
dynamics will be.
We will begin our study by considering the simplest way in which
Dyg(y0, λ0) can be nonhyperbolic. This is the case where Dyg(y0, λ0) has
a single zero eigenvalue with the remaining eigenvalues having nonzero real
parts. The question we ask in this situation is what is the nature of this
nonhyperbolic ﬁxed point for λ close to λ0? It is under these circumstances
where the real power of the center manifold theory becomes apparent, since
we know that this question can be answered by studying the vector ﬁeld
(20.0.1) restricted to the associated center manifold (cf. Section 18.2). In
this case the vector ﬁeld on the center manifold will be a p-parameter family
of one-dimensional vector ﬁelds. This represents a vast simpliﬁcation of
(20.0.1).
20.1 A Zero Eigenvalue
Suppose that Dyg(y0, λ0) has a single zero eigenvalue with the remaining
eigenvalues having nonzero real parts; then the orbit structure near (y0, λ0)
is determined by the associated center manifold equation, which we write
as
˙x = f(x, µ), x ∈ R1
, µ ∈ Rp
, (20.1.1)
where µ = λ − λ0. Furthermore, we know that (20.1.1) must satisfy
f(0, 0) = 0, (20.1.2)
∂f
∂x
(0, 0) = 0. (20.1.3)
358 20. Bifurcation of Fixed Points of Vector Fields
Equation (20.1.2) is simply the ﬁxed point condition and (20.1.3) is the
zero eigenvalue condition. We remark that (20.1.1) is Cr
if (20.0.1) is Cr
.
Let us begin by studying a few speciﬁc examples. In these examples we will
assume
µ ∈ R1
.
If there are more parameters in the problem (i.e., µ ∈ Rp
, p > 1), we
will consider all, except one, as ﬁxed. Later we will consider more carefully
the role played by the number of parameters in the problem. We remark
also that we have not yet precisely deﬁned what we mean by the term
“bifurcation.” We will consider this after the following series of examples.
20.1a Examples
Example 20.1.1. Consider the vector ﬁeld
˙x = f(x, µ) = µ − x2
, x ∈ R1
, µ ∈ R1
. (20.1.4)
It is easy to verify that
f(0, 0) = 0 (20.1.5)
and
∂f
∂x
(0, 0) = 0, (20.1.6)
but in this example we can determine much more. The set of all ﬁxed points of
(20.1.4) is given by
µ − x2
= 0
or
µ = x2
. (20.1.7)
This represents a parabola in the µ − x plane as shown in Figure 20.1.1.
FIGURE 20.1.1.
In the ﬁgure the arrows along the vertical lines represent the ﬂow generated
by (20.1.4) along the x-direction. Thus, for µ < 0, (20.1.4) has no ﬁxed points,
20.1 A Zero Eigenvalue 359
and the vector ﬁeld is decreasing in x. For µ > 0, (20.1.4) has two ﬁxed points.
A simple linear stability analysis shows that one of the ﬁxed points is stable
(represented by the solid branch of the parabola), and the other ﬁxed point is
unstable (represented by the broken branch of the parabola). However, we hope
that it is obvious to the reader that, given a Cr
(r ≥ 1) vector ﬁeld on R1
having
only two hyperbolic ﬁxed points, one must be stable and the other unstable.
This is an example of bifurcation. We refer to (x, µ) = (0, 0) as a bifurcation
point and the parameter value µ = 0 as a bifurcation value.
Figure 20.1.1 is referred to as a bifurcation diagram. This particular type of
bifurcation (i.e., where on one side of a parameter value there are no ﬁxed points
and on the other side there are two ﬁxed points) is referred to as a saddlenode
bifurcation. Later on we will worry about seeking precise conditions on
the vector ﬁeld on the center manifold that deﬁne the saddle-node bifurcation
unambiguously.
End of Example 20.1.1
Example 20.1.2. Consider the vector ﬁeld
˙x = f(x, µ) = µx − x2
, x ∈ R1
, µ ∈ R1
. (20.1.8)
It is easy to verify that
f(0, 0) = 0 (20.1.9)
and
∂f(0, 0)
∂x
= 0. (20.1.10)
Moreover, the ﬁxed points of (20.1.8) are given by
x = 0 (20.1.11)
and
x = µ (20.1.12)
FIGURE 20.1.2.
360 20. Bifurcation of Fixed Points of Vector Fields
and are plotted in Figure 20.1.2. Hence, for µ < 0, there are two ﬁxed points;
x = 0 is stable and x = µ is unstable. These two ﬁxed points coalesce at µ = 0
and, for µ > 0, x = 0 is unstable and x = µ is stable. Thus, an exchange of
stability has occurred at µ = 0. This type of bifurcation is called a transcritical
bifurcation.
End of Example 20.1.2
Example 20.1.3. Consider the vector ﬁeld
˙x = f(x, µ) = µx − x3
, x ∈ R1
, µ ∈ R1
. (20.1.13)
It is clear that we have
f(0, 0) = 0, (20.1.14)
FIGURE 20.1.3.
∂f
∂x
(0, 0) = 0. (20.1.15)
Moreover, the ﬁxed points of (20.1.13) are given by
x = 0 (20.1.16)
and
x2
= µ (20.1.17)
and are plotted in Figure 20.1.3.
Hence, for µ < 0, there is one ﬁxed point, x = 0, which is stable. For µ > 0,
x = 0 is still a ﬁxed point, but two new ﬁxed points have been created at µ = 0
and are given by x2
= µ. In the process, x = 0 has become unstable for µ > 0,
with the other two ﬁxed points stable. This type of bifurcation is called a pitchfork
bifurcation.
End of Example 20.1.3
20.1 A Zero Eigenvalue 361
Example 20.1.4. Consider the vector ﬁeld
˙x = f(x, µ) = µ − x3
, x ∈ R1
, µ ∈ R1
. (20.1.18)
It is trivial to verify that
f(0, 0) = 0 (20.1.19)
and
∂f
∂x
(0, 0) = 0. (20.1.20)
FIGURE 20.1.4.
Moreover, all ﬁxed points of (20.1.18) are given by
µ = x3
(20.1.21)
and are shown in Figure 20.1.4.
However in this example, despite (20.1.19) and (20.1.20), the dynamics of
(20.1.18) are qualitatively the same for µ > 0 and µ < 0. Namely, (20.1.18)
possesses a unique, stable ﬁxed point.
End of Example 20.1.4
20.1b What Is A “Bifurcation of a Fixed Point”?
The term “bifurcation” is extremely general. We will begin to learn its
uses in dynamical systems by understanding its use in describing the orbit
structure near nonhyperbolic ﬁxed points. Let us consider what we learned
from the previous examples.
In all four examples we had
f(0, 0) = 0
362 20. Bifurcation of Fixed Points of Vector Fields
and
∂f
∂x
(0, 0) = 0,
and yet the orbit structure near µ = 0 was diﬀerent in all four cases.
Hence, knowing that a ﬁxed point has a zero eigenvalue for µ = 0 is not
suﬃcient to determine the orbit structure for λ near zero. Let us consider
each example individually.
1. (Example 20.1.1). In this example a unique curve (or branch) of ﬁxed
points passed through the origin. Moreover, the curve lay entirely on
one side of µ = 0 in the µ − x plane.
2. (Example 20.1.2). In this example two curves of ﬁxed points intersected
at the origin in the µ − x plane. Both curves existed on either
side of µ = 0. However, the stability of the ﬁxed point along a given
curve changed on passing through µ = 0.
3. (Example 20.1.3). In this example two curves of ﬁxed points intersected
at the origin in the µ − x plane. Only one curve (x = 0)
existed on both sides of µ = 0; however, its stability changed on
passing through µ = 0. The other curve of ﬁxed points lay entirely
to one side of µ = 0 and had a stability type that was the opposite
of x = 0 for µ > 0.
4. (Example 20.1.4). This example had a unique curve of ﬁxed points
passing through the origin in the µ − x plane and existing on both
sides of µ = 0. Moreover, all ﬁxed points along the curve had the same
stability type. Hence, despite the fact that the ﬁxed point (x, µ) =
(0, 0) was nonhyperbolic, the orbit structure was qualitatively the
same for all µ.
We want to apply the term “bifurcation” to Examples 20.1.1, 20.1.2, and
20.1.3 but not to Example 20.1.4 to describe the change in orbit structure
as µ passes through zero. We are therefore led to the following deﬁnition.
Deﬁnition 20.1.1 (Bifurcation of a Fixed Point) A ﬁxed point (x, µ)
= (0, 0) of a one-parameter family of one-dimensional vector ﬁelds is said
to undergo a bifurcation at µ = 0 if the ﬂow for µ near zero and x near
zero is not qualitatively the same as the ﬂow near x = 0 at µ = 0.
Several remarks are now in order concerning this deﬁnition.
Remark 1. The phrase “qualitatively the same” is a bit vague. It can be
made precise by substituting the term “C0
-equivalent” (cf. Section 19.12),
and this is perfectly adequate for the study of the bifurcation of ﬁxed
points of one-dimensional vector ﬁelds. However, we will see that as we
explore higher dimensional phase spaces and global bifurcations, how to
20.1 A Zero Eigenvalue 363
make mathematically precise the statement “two dynamical systems have
qualitatively the same dynamics” becomes more and more ambiguous.
Remark 2. Practically speaking, a ﬁxed point (x0, µ0) of a one-dimensional
vector ﬁeld is a bifurcation point if either more than one curve of ﬁxed
points passes through (x0, µ0) in the µ − x plane or if only one curve of
ﬁxed points passes (x0, µ0) in the µ − x plane; then it (locally) lies entirely
on one side of the line µ = µ0 in the µ − x plane.
Remark 3. It should be clear from Example 20.1.4 that the condition that
a ﬁxed point is nonhyperbolic is a necessary but not suﬃcient condition for
bifurcation to occur in one-parameter families of vector ﬁelds.
We next turn to deriving general conditions on one-parameter families
of one-dimensional vector ﬁelds which exhibit bifurcations exactly as in
Examples 20.1.1, 20.1.2, and 20.1.3.
20.1c The Saddle-Node Bifurcation
We now want to derive conditions under which a general one-parameter
family of one-dimensional vector ﬁelds will undergo a saddle-node bifurcation
exactly as in Example 20.1.1. These conditions will involve derivatives
of the vector ﬁeld evaluated at the bifurcation point and are obtained by
a consideration of the geometry of the curve of ﬁxed points in the µ − x
plane in a neighborhood of the bifurcation point.
Let us recall Example 20.1.1. In this example a unique curve of ﬁxed
points, parameterized by x, passed through (µ, x) = (0, 0). We denote the
curve of ﬁxed points by µ(x). The curve of ﬁxed points satisﬁed two prop-
erties.
1. It was tangent to the line µ = 0 at x = 0, i.e.,
dµ
dx
(0) = 0. (20.1.22)
2. It lay entirely to one side of µ = 0. Locally, this will be satisﬁed if we
have
d2
µ
dx2
(0) = 0. (20.1.23)
Now let us consider a general, one-parameter family of one-dimensional
vector ﬁelds.
˙x = f(x, µ), x ∈ R1
, µ ∈ R1
. (20.1.24)
Suppose (20.1.24) has a ﬁxed point at (x, µ) = (0, 0), i.e.,
f(0, 0) = 0. (20.1.25)
364 20. Bifurcation of Fixed Points of Vector Fields
Furthermore, suppose that the ﬁxed point is not hyperbolic, i.e.,
∂f
∂x
(0, 0) = 0. (20.1.26)
Now, if we have
∂f
∂µ
(0, 0) = 0, (20.1.27)
then, by the implicit function theorem, there exists a unique function
µ = µ(x), µ(0) = 0 (20.1.28)
deﬁned for x suﬃciently small such that f(x, µ(x)) = 0. (Note: the reader
should check that (20.1.27) holds in Example 20.1.1.) Now we want to
derive conditions in terms of derivatives of f evaluated at (µ, x) = (0, 0) so
that we have
dµ
dx
(0) = 0, (20.1.29)
d2
µ
dx2
(0) = 0. (20.1.30)
Equations (20.1.29) and (20.1.30), along with (20.1.25), (20.1.26), and
(20.1.27), imply that (µ, x) = (0, 0) is a bifurcation point at which a saddlenode
bifurcation occurs.
We can derive expressions for (20.1.29) and (20.1.30) in terms of derivatives
of f at the bifurcation point by implicitly diﬀerentiating f along the
curve of ﬁxed points.
Using (20.1.27), we have
f(x, µ(x)) = 0. (20.1.31)
Diﬀerentiating (20.1.31) with respect to x gives
df
dx
(x, µ(x)) = 0 =
∂f
∂x
(x, µ(x)) +
∂f
∂µ
(x, µ(x))
dµ
dx
(x). (20.1.32)
Evaluating (20.1.32) at (µ, x) = (0, 0), we obtain
dµ
dx
(0) =
−
∂f
∂x
(0, 0)
∂f
∂µ
(0, 0)
; (20.1.33)
thus we see that (20.1.26) and (20.1.27) imply that
dµ
dx
(0) = 0, (20.1.34)
i.e., the curve of ﬁxed points is tangent to the line µ = 0 at x = 0.
20.1 A Zero Eigenvalue 365
FIGURE 20.1.5.
a) −∂2
f
∂x2 (0, 0)/∂f
∂µ
(0, 0) > 0; b) −∂2
f
∂x2 (0, 0)/∂f
∂µ
(0, 0) < 0.
Next, let us diﬀerentiate (20.1.32) once more with respect to x to obtain
d2
f
dx2
(x, µ(x)) = 0 =
∂2
f
∂x2
(x, µ(x)) + 2
∂2
f
∂x∂µ
(x, µ(x))
dµ
dx
(x)
+
∂2
f
∂µ2
(x, µ(x))
dµ
dx
(x)
2
+
∂f
∂µ
(µ, µ(x))
d2
µ
dx2
(x). (20.1.35)
Evaluating (20.1.35) at (µ, x) = (0, 0) and using (20.1.33) gives
∂2
f
∂x2
(0, 0) +
∂f
∂µ
(0, 0)
d2
µ
dx2
(0) = 0
or
d2
µ
dx2
(0) =
−
∂2
f
∂x2
(0, 0)
∂f
∂µ
(0, 0)
. (20.1.36)
Hence, (20.1.36) is nonzero provided we have
∂2
f
∂x2
(0, 0) = 0. (20.1.37)
366 20. Bifurcation of Fixed Points of Vector Fields
Let us summarize. In order for (20.1.24) to undergo a saddle-node bifurcation
we must have
f(0, 0) = 0
∂f
∂x
(0, 0) = 0



nonhyperbolic ﬁxed point (20.1.38)
and
∂f
∂µ
(0, 0) = 0, (20.1.39)
∂2
f
∂x2
(0, 0) = 0. (20.1.40)
Equation (20.1.39) implies that a unique curve of ﬁxed points passes
through (µ, x) = (0, 0), and (20.1.40) implies that the curve lies locally on
one side of µ = 0. It should be clear that the sign of (20.1.36) determines
on which side of µ = 0 the curve lies. In Figure 20.1.5 we show both cases
without indicating stability and leave it as an exercise for the reader to
verify the stability types of the diﬀerent branches of ﬁxed points emanating
from the bifurcation point.
Let us end our discussion of the saddle-node bifurcation with the following
remark. Consider a general one-parameter family of one-dimensional
vector ﬁelds having a nonhyperbolic ﬁxed point at (x, µ) = (0, 0). The
Taylor expansion of this vector ﬁeld is given as follows
f(x, µ) = a0µ + a1x2
+ a2µx + a3µ2
+ O(3). (20.1.41)
Our computations show that the dynamics of (20.1.41) near (µ, x) = (0, 0)
are qualitatively the same as one of the following vector ﬁelds
˙x = µ ± x2
. (20.1.42)
Hence, (20.1.42) can be viewed as the normal form for saddle-node bifur-
cations.
This brings up another important point. In applying the method of normal
forms there is always the question of truncation of the normal form;
namely, how are the dynamics of the normal form including only the O(k)
terms modiﬁed when the higher order terms are included? We see that,
in the study of the saddle-node bifurcation, all terms of O(3) and higher
could be neglected and the dynamics would not be qualitatively changed.
The implicit function theorem was the tool that enabled us to verify this
fact.
20.1d The Transcritical Bifurcation
We want to follow the same strategy as in our discussion and derivation
of general conditions for the saddle-node bifurcation given in the previous
20.1 A Zero Eigenvalue 367
section, namely, to use the implicit function theorem to characterize the
geometry of the curves of ﬁxed points passing through the bifurcation point
in terms of derivatives of the vector ﬁeld evaluated at the bifurcation point.
For the example of transcritical bifurcation discussed in Example 20.1.2,
the orbit structure near the bifurcation point was characterized as follows.
1. Two curves of ﬁxed points passed through (x, µ) = (0, 0), one given
by x = µ, the other by x = 0.
2. Both curves of ﬁxed points existed on both sides of µ = 0.
3. The stability along each curve of ﬁxed points changed on passing
through µ = 0.
Using these three points as a guide, let us consider a general one-parameter
family of one-dimensional vector ﬁelds
˙x = f(x, µ), x ∈ R1
, µ ∈ R1
. (20.1.43)
We assume that at (x, µ) = (0, 0), (20.1.43) has a nonhyperbolic ﬁxed point,
i.e.,
f(0, 0) = 0 (20.1.44)
and
∂f
∂x
(0, 0) = 0. (20.1.45)
Now, in Example 20.1.2 we had two curves of ﬁxed points passing through
(µ, x) = (0, 0). In order for this to occur it is necessary to have
∂f
∂µ
(0, 0) = 0, (20.1.46)
or else, by the implicit function theorem, only one curve of ﬁxed points
could pass through the origin.
Equation (20.1.46) presents a problem if we wish to proceed as in the
case of the saddle-node bifurcation; in that situation we used the condition
∂f
∂µ (0, 0) = 0 in order to conclude that a unique curve of ﬁxed points,
µ(x), passed through the bifurcation point. We then evaluated the vector
ﬁeld on the curve of ﬁxed points and used implicit diﬀerentiation to derive
local characteristics of the geometry of the curve of ﬁxed points based on
properties of the derivatives of the vector ﬁeld evaluated at the bifurcation
point. However, if we use Example 20.1.2 as a guide, we can extricate
ourselves from this diﬃculty.
In Example 20.1.2, x = 0 was a curve of ﬁxed points passing through the
bifurcation point. We will require that to be the case for (20.1.43), so that
(20.1.43) has the form
˙x = f(x, µ) = xF(x, µ), x ∈ R1
, µ ∈ R1
, (20.1.47)
368 20. Bifurcation of Fixed Points of Vector Fields
where, by deﬁnition, we have
F(x, µ) ≡



f(x,µ)
x , x = 0
∂f
∂x (0, µ), x = 0



. (20.1.48)
Since x = 0 is a curve of ﬁxed points for (20.1.47), in order to obtain an
additional curve of ﬁxed points passing through (µ, x) = (0, 0) we need
to seek conditions on F whereby F has a curve of zeros passing through
(µ, x) = (0, 0) (that is not given by x = 0). These conditions will be in terms
of derivatives of F which, using (20.1.48), can be expressed as derivatives
of f.
Using (20.1.48), it is easy to verify the following
F(0, 0) = 0, (20.1.49)
∂F
∂x
(0, 0) =
∂2
f
∂x2
(0, 0), (20.1.50)
∂2
F
∂x2
(0, 0) =
∂3
f
∂x3
(0, 0), (20.1.51)
and (most importantly)
∂F
∂µ
(0, 0) =
∂2
f
∂x∂µ
(0, 0). (20.1.52)
Now let us assume that (20.1.52) is not zero; then by the implicit function
theorem there exists a function, µ(x), deﬁned for x suﬃciently small, such
that
F(x, µ(x)) = 0. (20.1.53)
Clearly, µ(x) is a curve of ﬁxed points of (20.1.47). In order for µ(x) to not
coincide with x = 0 and to exist on both sides of µ = 0, we must require
that
0 <
dµ
dx
(0) < ∞.
Implicitly diﬀerentiating (20.1.53) exactly as in the case of the saddle-node
bifurcation we obtain
dµ
dx
(0) =
−∂F
∂x (0, 0)
∂F
∂µ (0, 0)
. (20.1.54)
Using (20.1.49), (20.1.50), (20.1.51), and (20.1.52), (20.1.54) becomes
dµ
dx
(0) =
−∂2
f
∂x2 (0, 0)
∂2f
∂x∂µ (0, 0)
. (20.1.55)
20.1 A Zero Eigenvalue 369
FIGURE 20.1.6.
a) −∂2
f
∂x2 (0, 0)/ ∂2
f
∂x∂µ
(0, 0) > 0; b) −∂2
f
∂x2 (0, 0)/ ∂2
f
∂x∂µ
(0, 0) < 0.
We now summarize our results. In order for a vector ﬁeld
˙x = f(x, µ), x ∈ R1
, µ ∈ R1
, (20.1.56)
to undergo a transcritical bifurcation, we must have
f(0, 0) = 0
∂f
∂x
(0, 0) = 0



nonhyperbolic ﬁxed point (20.1.57)
and
∂f
∂µ
(0, 0) = 0, (20.1.58)
∂2
f
∂x∂µ
(0, 0) = 0, (20.1.59)
∂2
f
∂x2
(0, 0) = 0. (20.1.60)
We note that the slope of the curve of ﬁxed points not equal to x = 0 is
given by (20.1.55). These two cases are shown in Figure 20.1.6; however, we
do not indicate stabilities of the diﬀerent branches of ﬁxed points. We leave
it as an exercise to the reader to verify the stability types of the diﬀerent
curves of ﬁxed points emanating from the bifurcation point.
370 20. Bifurcation of Fixed Points of Vector Fields
Thus, (20.1.57), (20.1.58), (20.1.59), and (20.1.60) show that the orbit
structure near (x, µ) = (0, 0) is qualitatively the same as the orbit structure
near (x, µ) = (0, 0) of
˙x = µx ∓ x2
. (20.1.61)
Equation (20.1.61) can be viewed as a normal form for the transcritical
bifurcation.
20.1e The Pitchfork Bifurcation
The discussion and derivation of conditions under which a general oneparameter
family of one-dimensional vector ﬁelds will undergo a bifurcation
of the type shown in Example 20.1.3 follows very closely our discussion of
the transcritical bifurcation.
The geometry of the curves of ﬁxed points associated with the bifurcation
in Example 20.1.3 had the following characteristics.
1. Two curves of ﬁxed points passed through (µ, x) = (0, 0), one given
by x = 0, the other by µ = x2
.
2. The curve x = 0 existed on both sides of µ = 0; the curve µ = x2
existed on one side of µ = 0.
3. The ﬁxed points on the curve x = 0 had diﬀerent stability types on
opposite sides of µ = 0. The ﬁxed points on µ = x2
all had the same
stability type.
Now we want to consider conditions on a general one-parameter family
of one-dimensional vector ﬁelds having two curves of ﬁxed points passing
through the bifurcation point in the µ − x plane that have the properties
given above.
We denote the vector ﬁeld by
˙x = f(x, µ), x ∈ R1
, µ ∈ R1
, (20.1.62)
and we suppose
f(0, 0) = 0, (20.1.63)
∂f
∂x
(0, 0) = 0. (20.1.64)
As in the case of the transcritical bifurcation, in order to have more than
one curve of ﬁxed points passing through (µ, x) = (0, 0) we must have
∂f
∂µ
(0, 0) = 0. (20.1.65)
Proceeding further along these lines, we require x = 0 to be a curve of ﬁxed
points for (20.1.62) by assuming the vector ﬁeld (20.1.62) has the form
˙x = xF(x, µ), x ∈ R1
, µ ∈ R1
, (20.1.66)
20.1 A Zero Eigenvalue 371
where
F(x, µ) ≡



f(x,µ)
x , x = 0
∂f
∂x (0, µ), x = 0



. (20.1.67)
In order to have a second curve of ﬁxed points passing through (µ, x) =
(0, 0) we must have
F(0, 0) = 0 (20.1.68)
with
∂F
∂µ
(0, 0) = 0. (20.1.69)
Equation (20.1.69) insures that only one additional curve of ﬁxed points
passes through (µ, x) = (0, 0). Also, using (20.1.69), the implicit function
theorem implies that for x suﬃciently small there exists a unique function
µ(x) such that
F(x, µ(x)) = 0. (20.1.70)
In order for the curve of ﬁxed points, µ(x), to satisfy the above-mentioned
characteristics, it is suﬃcient to have
dµ
dx
(0) = 0 (20.1.71)
and
d2
µ
dx2
(0) = 0. (20.1.72)
The conditions for (20.1.71) and (20.1.72) to hold in terms of the derivatives
of F evaluated at the bifurcation point can be obtained via implicit
diﬀerentiation of (20.1.70) along the curve of ﬁxed points exactly as in the
case of the saddle-node bifurcation. They are given by
dµ
dx
(0) =
−∂F
∂x (0, 0)
∂F
∂µ (0, 0)
= 0 (20.1.73)
and
d2
µ
dx2
(0) =
−∂2
F
∂x2 (0, 0)
∂F
∂µ (0, 0)
= 0. (20.1.74)
Using (20.1.67), (20.1.73) and (20.1.74) can be expressed in terms of derivatives
of f as follows
dµ
dx
(0) =
−∂2
f
∂x2 (0, 0)
∂2f
∂x∂µ (0, 0)
= 0 (20.1.75)
and
d2
µ
dx2
(0) =
−∂3
f
∂x3 (0, 0)
∂2f
∂x∂µ (0, 0)
= 0. (20.1.76)
372 20. Bifurcation of Fixed Points of Vector Fields
We summarize as follows. In order for the vector ﬁeld
˙x = f(x, µ), x ∈ R1
, µ ∈ R1
, (20.1.77)
to undergo a pitchfork bifurcation at (x, µ) = (0, 0), it is suﬃcient to have
f(0, 0) = 0
∂f
∂x
(0, 0) = 0



nonhyperbolic ﬁxed point (20.1.78)
FIGURE 20.1.7.
a) −∂3
f
∂x3 (0, 0)/ ∂2
f
∂x∂µ
(0, 0) > 0; b) −∂3
f
∂x3 (0, 0)/ ∂2
f
∂x∂µ
(0, 0) < 0.
with
∂f
∂µ
(0, 0) = 0, (20.1.79)
∂2
f
∂x2
(0, 0) = 0, (20.1.80)
∂2
f
∂x∂µ
(0, 0) = 0, (20.1.81)
∂3
f
∂x3
(0, 0) = 0. (20.1.82)
There are two possibilities for the disposition of the two branches of ﬁxed
points depending on the sign of (20.1.76). These two possibilities are shown
20.1 A Zero Eigenvalue 373
in Figure 20.1.7 without indicating stabilities. We leave it as an exercise for
the reader to verify the stability types for the diﬀerent branches of ﬁxed
points emanating from the bifurcation point.
We conclude by noting that (20.1.78), (20.1.79), (20.1.80), (20.1.81), and
(20.1.82) imply that the orbit structure near (x, µ) = (0, 0) is qualitatively
the same as the orbit structure near (x, µ) = (0, 0) in the vector ﬁeld
˙x = µx ∓ x3
. (20.1.83)
Thus, (20.1.83) can be viewed as a normal form for the pitchfork bifurca-
tion.
20.1f Exercises
1. In our development of the transcritical and pitchfork bifurcations we assumed that
x = 0 was a trivial solution. Was this necessary? In particular, would the conditions
for transcritical and pitchfork bifurcations change if this were not the case?
2. Consider a Cr
(r ≥ 1) autonomous vector ﬁeld on R1
having precisely two hyperbolic
ﬁxed points. Can you infer the nature of the stability of the two ﬁxed points? How
does the situation change if one of the ﬁxed points is not hyperbolic? Can both ﬁxed
points be nonhyperbolic? Construct explicit examples illustrating each situation.
3. Consider the saddle-node bifurcation for vector ﬁelds and Figure 20.1.5. For the case
− ∂2f
∂x2 (0, 0)/ ∂f
∂µ (0, 0) > 0, give conditions under which the upper part of the curve
of ﬁxed points is stable and the lower part is unstable. Alternatively, give conditions
under which the upper part of the curve of ﬁxed points is unstable and the lower part
is stable.
Repeat the exercise for the case − ∂2f
∂x2 (0, 0)/ ∂f
∂µ (0, 0) < 0.
4. Consider the transcritical bifurcation for vector ﬁelds and Figure 20.1.6. For the case
− ∂2f
∂x2 (0, 0)/ ∂2f
∂x∂µ (0, 0) > 0, give conditions for x = 0 to be stable for µ > 0 and
unstable for µ < 0. Alternatively, give conditions for x = 0 to be unstable for µ > 0
and stable for µ < 0.
Repeat the exercise for the case − ∂2f
∂x2 (0, 0)/ ∂2f
∂x∂µ (0, 0) < 0.
5. Consider the pitchfork bifurcation for vector ﬁelds and Figure 20.1.7. For the case
− ∂3f
∂x3 (0, 0)/ ∂2f
∂x∂µ (0, 0) > 0, give conditions for x = 0 to be stable for µ > 0 and
unstable for µ < 0. Alternatively, give conditions for x = 0 to be unstable for µ > 0
and stable for µ < 0.
Repeat the exercise for the case − ∂3f
∂x3 (0, 0)/ ∂2f
∂x∂µ (0, 0) < 0.
6. In Exercise 4 following Chapter 18 we computed center manifolds near the origin for
the following one-parameter families of vector ﬁelds. Describe the bifurcations of the
origin. In, for example, a) and a′
) the parameter ε multiplies a linear and nonlinear
term, respectively. In terms of bifurcations, is there a qualitative diﬀerence in the two
cases? What kinds of general statements can you make?
a)
˙θ = −θ + εv + v
2
,
˙v = − sin θ,
(θ, v) ∈ S
1
× R
1
.
a′
)
˙θ = −θ + v
2
+ εv
2
,
˙v = − sin θ.
374 20. Bifurcation of Fixed Points of Vector Fields
b)
˙x =
1
2
x + y + x
2
y,
˙y = x + 2y + εy + y
2
,
(x, y) ∈ R
2
.
b′
)
˙x =
1
2
x + y + x
2
y,
˙y = x + 2y + y
2
+ εy
2
.
d)
˙x = 2x + 2y + εy,
˙y = x + y + x
4
,
(x, y) ∈ R
2
.
d′
)
˙x = 2x + 2y,
˙y = x + y + x
4
+ εy
2
.
f)
˙x = −2x + 3y + εx + y
3
,
˙y = 2x − 3y + x
3
,
(x, y) ∈ R
2
.
f′
)
˙x = −2x + 3y + y
3
+ εx
2
,
˙y = 2x − 3y + x
3
.
h)
˙x = −x + y,
˙y = −e
x
+ e
−x
+ 2x + εy,
(x, y) ∈ R
2
.
h′
)
˙x = −x + y + εx
2
,
˙y = −e
x
+ e
−x
+ 2x.
i)
˙x = −2x + y + z + εx + y
2
z,
˙y = x − 2y + z + εx + xz
2
,
˙z = x + y − 2z + εx + x
2
y,
(x, y, z) ∈ R
3
.
i′
)
˙x = −2x + y + z + εx
2
+ y
2
z,
˙y = x − 2y + z + εxy + xz
2
,
˙z = x + y − 2z + x
2
y.
j)
˙x = −x − y + z
2
,
˙y = 2x + y + εy − z
2
,
˙z = x + 2y − z,
(x, y, z) ∈ R
3
.
j′
)
˙x = −x − y + εx
2
+ z
2
,
˙y = 2x + y − z
2
+ εy
2
,
˙z = x + 2y − z.
k)
˙x = −x − y − z + εx − yz,
˙y = −x − y − z − xz,
˙z = −x − y − z − yz,
(x, y, z) ∈ R
3
.
k′
)
˙x = −x − y − z − yz + εx
2
,
˙y = −x − y − z − xz,
˙z = −x − y − z − xy.
l)
˙x = y + x
2
+ εy,
˙y = −y − x
2
,
(x, y) ∈ R
2
.
l′
)
˙x = y + x
2
+ εy
2
,
˙y = −y − x
2
.
m)
˙x = x
2
+ εy,
˙y = −y − x
2
,
(x, y) ∈ R
2
.
m′
)
˙x = x
2
+ εy
2
,
˙y − −y − x
2
.
20.1 A Zero Eigenvalue 375
7. Center Manifolds at a Saddle-node Bifurcation Point for Vector Fields
In developing the center manifold theory for parametrized families of vector ﬁelds, we
dealt with equations of the following form
˙x = Ax + f(x, y, ε),
˙y = By + g(x, y, ε),
(x, y, ε) ∈ R
c
× R
s
× R
p
, (20.1.84)
where A is a c × c matrix whose eigenvalues all have zero real parts, B is an s × s
matrix whose eigenvalues all have negative real parts, and
f(0, 0, 0) = 0, Df(0, 0, 0) = 0,
g(0, 0, 0) = 0, Dg(0, 0, 0) = 0.
(20.1.85)
The conditions Df(0, 0, 0) = 0, Dg(0, 0, 0) = 0 do not allow for terms that are linear in
the parameter ε. Clearly, this may not be the case at a saddle-node bifurcation point,
and we want to consider this issue in this exercise. Although this could have been
done in Chapter 18, in that chapter we were introducing only center manifold theory
and were not really concerned with bifurcations. In this case the form of the equations
given by (20.1.84) and (20.1.85) was the “cleanest and quickest” way to introduce the
notion of parametrized families of center manifolds.
We will start at a very basic level. Consider the Cr
(r as large as necessary) vector
ﬁeld
˙z = F (z, ε), (z, ε) ∈ R
c+s
× R
p
. (20.1.86)
Suppose that (z, ε) = (0, 0) is a ﬁxed point of (20.1.86) at which the matrix
DzF (0, 0) (20.1.87)
has c eigenvalues with zero real parts and s eigenvalues with negative real parts.
Our goal is to apply the center manifold theory in order to examine the dynamics
of (20.1.86) near (z, ε) = (0, 0).
We rewrite Equation (20.1.86) as follows
˙z = DzF (0, 0)z + DεF (0, 0)ε + G(z, ε), (20.1.88)
where
G(z, ε) = F (z, ε) − DzF (0, 0)z − DεF (0, 0)ε = O(2) (20.1.89)
in z and ε. Note that the term “DεF (0, 0)ε” in (20.1.88) is the new wrinkle—it was
zero under our previous assumptions. For notational purposes we let
DzF (0, 0) ≡ M −(c + s) × (c + s) matrix,
DεF (0, 0) ≡ Λ −(c + s) × p matrix,
so that (20.1.88) becomes
˙z = Mz + Λε + G(z, ε). (20.1.90)
Now let T be the (s +c) ×(s +c) matrix that puts M into the following block diagonal
form
T
−1
MT =
A 0
0 B
, (20.1.91)
where A is a (c × c) matrix with all eigenvalues having zero real parts and B is an
(s × s) matrix with all eigenvalues having negative real parts. If we let
z = T w, (x, y) ∈ R
c
× R
s
, (20.1.92)
where w = (x, y), and apply this linear transformation to (20.1.90), we obtain
˙x
˙y
=
A 0
0 B
x
y
+ Λε +
f(x, y, ε)
g(x, y, ε)
, (20.1.93)
where
Λ ≡ T
−1
Λ,
376 20. Bifurcation of Fixed Points of Vector Fields
f(x, y, ε)
g(x, y, ε)
≡ T
−1
G(T (x, y), ε).
Note that f(0, 0, 0) = 0, g(0, 0, 0) = 0, Df(0, 0, 0) = 0, and Dg(0, 0, 0) = 0. Next, let
Λ = Λc
Λs
,
where Λc corresponds to the ﬁrst c rows of Λ, and Λs corresponds to the last s rows
of Λ. Then (20.1.93) can be rewritten as
˙x
˙ε
˙y
=


A Λc 0
0 0 0
0 Λs B


x
ε
y
+
f(x, y, ε)
0
g(x, y, ε)
. (20.1.94)
The reader should recognize that (20.1.94) is “almost” in the standard normal form for
application of the center manifold theory. The ﬁnal step would be to introduce a linear
transformation that block diagonalizes the linear part of (20.1.94) into a (c+p)×(c+p)
matrix with eigenvalues all having zero real parts (and p identically zero) and an (s×s)
matrix with all eigenvalues having negative real parts.
a) Carry out this ﬁnal step and discuss applying the center manifold theorem to
the resulting system. In particular, do the relevant theorems from Chapter 18
go through?
Before we work out some speciﬁc problems, let us ﬁrst answer an example.
Consider the vector ﬁeld
˙x = ε + x
2
+ y
2
,
˙y = −y + x
2
,
(x, y, ε) ∈ R
3
. (20.1.95)
It should be clear that (x, y, ε) = (0, 0, 0) is a ﬁxed point of (20.1.95). We want to
study the orbit structure near this ﬁxed point for ε small. Rewriting (20.1.95) in the
form of (20.1.94) gives
˙x
˙ε
˙y
=
0 1 0
0 0 0
0 0 −1
x
ε
y
+
x2
+ y2
0
x2
. (20.1.96)
We seek a center manifold of the form
h(x, ε) = ax
2
+ bxε + cε
2
+ O(3).
Utilizing the usual procedure for calculating the center manifold, we obtain
h(x, ε) = x
2
− 2xε + 2ε
2
+ O(3).
The vector ﬁeld restricted to the center manifold is then given by
˙x = ε + x
2
+ O(4),
˙ε = 0.
Hence, a saddle-node bifurcation occurs at ε = 0.
Now consider the following vector ﬁelds
b)
˙x = ε + x
4
+ y
2
,
˙y = −y + x
3
,
(x, y, ε) ∈ R
3
.
c)
˙x = ε + x
2
− y
3
,
˙y = ε − y + x
2
.
d)
˙x = ε + εx + x
2
,
˙y = −y + x
2
.
20.1 A Zero Eigenvalue 377
e)
˙x = ε + εx + x
2
,
˙y = ε − y + x
2
.
f) ˙x = ε +
1
2
x + y + x
3
,
˙y = x + 2y − xy.
g)
˙x = 2ε + 2x + 2y,
˙y = ε + x + y + y
2
.
h) ˙x = ε − 2x + 2y − x
4
,
˙y = 2x − 2y.
i)
˙x = ε − 2x + y + z + yz,
˙y = x − 2y + z + zx,
˙z = x + y − 2z + xy,
(x, y, z, ε) ∈ R
4
.
For each vector ﬁeld, construct the center manifold and discuss the dynamics near the
origin for ε small. What types of bifurcations occur?
8. Consider the vector ﬁeld
˙x = ε + x
2
+ y
2
,
˙y = −y + x
2
,
(x, y, ε) ∈ R
3
.
For this vector ﬁeld the tangent space approximation is suﬃcient for approximating
the center manifold of the origin. Verify this statement and discuss conditions under
which the tangent space approximation might work in general. Consider your ideas in
the context of the following examples.
a)
˙x = εx + x
2
+ y
2
,
˙y = −y + x
2
.
b)
˙x = ε + x
2
+ xy,
˙y = −y + x
2
.
c)
˙x = ε + y
2
,
˙y = −y + x
2
.
d)
˙x = ε + xy + y
2
,
˙y = −y + x
2
.
9. Consider the block diagonal “normal form” of (20.1.84) to which we ﬁrst transformed
the vector ﬁeld in order to apply the center manifold theory. Discuss why (or why
not) this preliminary transformation was necessary. Is this preliminary transformation
necessary for equations of the form of (20.1.94) in order to apply the center manifold
theory? Work out several examples to support your views and illustrate the relevant
points. (Hint: consider the coordinatization of the center manifold and how the invariance
condition is manifested in those coordinates.)
10. Consider the following one-parameter family of two-dimensional Cr
(r as large as
necessary) vector ﬁelds
˙x = f(x; µ), (x, µ) ∈ R
2
× R
1
,
where f(0; 0) = 0 and Dxf(0, 0) has a zero eigenvalue and a negative eigenvalue.
Suppose the vector ﬁeld has the following symmetry
f(x, µ) = −f(−x, µ).
What can you then conclude concerning the symmetry of the vector ﬁeld restricted
to the center manifold for x and µ small? Can the vector ﬁeld undergo a saddle-node
bifurcation at (x, µ) = (0, 0)? Can the vector ﬁeld undergo a saddle-node bifurcation
at other points (x, µ) ∈ R2
× R1
?
378 20. Bifurcation of Fixed Points of Vector Fields
20.2 A Pure Imaginary Pair of Eigenvalues: The
Poincare-Andronov-Hopf Bifurcation
We now turn to the next most simple way that a ﬁxed point can be nonhyperbolic;
namely, that the matrix associated with the vector ﬁeld linearized
about the ﬁxed point has a pair of purely imaginary eigenvalues, with the
remaining eigenvalues having nonzero real parts. Let us be more precise.
Recall (20.0.1), which we restate here;
˙y = g(y, λ), y ∈ Rn
, λ ∈ Rp
, (20.2.1)
where g is Cr
(r ≥ 5) on some suﬃciently large open set containing the
ﬁxed point of interest. The ﬁxed point is denoted by (y, λ) = (y0, λ0), i.e.,
0 = g(y0, λ0). (20.2.2)
We are interested in how the orbit structure near y0 changes as λ is varied.
In this situation the ﬁrst thing to examine is the linearization of the vector
ﬁeld about the ﬁxed point, which is given by
˙ξ = Dyg(y0, λ0)ξ, ξ ∈ Rn
. (20.2.3)
Suppose that Dyg(y0, λ0) has two purely imaginary eigenvalues with the
remaining n − 2 eigenvalues having nonzero real parts. We know (cf.
the remarks at the beginning of this chapter) that since the ﬁxed point
is not hyperbolic, the orbit structure of the linearized vector ﬁeld near
(y, λ) = (y0, λ0) may reveal little (and, possibly, even incorrect) information
concerning the nature of the orbit structure of the nonlinear vector
ﬁeld (20.2.1) near (y, λ) = (y0, λ0).
Fortunately, we have a systematic procedure for analyzing this problem.
By the center manifold theorem, we know that the orbit structure near
(y, λ) = (y0, λ0) is determined by the vector ﬁeld (20.2.1) restricted to the
center manifold. This restriction gives us a p-parameter family of vector
ﬁelds on a two-dimensional center manifold. For now we will assume that
we are dealing with a single, scalar parameter, i.e., p = 1. If there is more
than one parameter in the problem, we will consider all but one of them as
ﬁxed.
On the center manifold the vector ﬁeld (20.2.1) has the following form
˙x
˙y
=
Re λ(µ) −Im λ(µ)
Im λ(µ) Re λ(µ)
x
y
+
f1
(x, y, µ)
f2
(x, y, µ)
,
(x, y, µ) ∈ R1
× R1
× R1
, (20.2.4)
where f1
and f2
are nonlinear in x and y and λ(µ), λ(µ) are the eigenvalues
of the vector ﬁeld linearized about the ﬁxed point at the origin.
Equation (20.2.4) was ﬁrst discussed in Section 19.2. The reader should
recall that in performing the center manifold reduction to obtain (20.2.4),
20.2 The Poincare-Andronov-Hopf Bifurcation 379
several preliminary steps were ﬁrst implemented. Namely, ﬁrst we transformed
the ﬁxed point to the origin and, then, if necessary, performed a
linear transformation of the coordinates so that the vector ﬁeld (20.2.1)
was in the form of (20.2.4). We further remark that the eigenvalue, denoted
λ(µ), should not be confused with the general vector of parameters
in (20.2.1), denoted λ ∈ Rp
, which we subsequently restricted to a scalar
and labeled µ. We will henceforth denote
λ(µ) = α(µ) + iω(µ), (20.2.5)
and note that by our assumptions we have
α(0) = 0,
ω(0) = 0. (20.2.6)
The next step is to transform (20.2.4) into normal form. This was done in
Section 19.2. The normal form was found to be
˙x = α(µ)x − ω(µ)y + (a(µ)x − b(µ)y)(x2
+ y2
) + O(|x|5
, |y|5
),
˙y = ω(µ)x + α(µ)y + (b(µ)x + a(µ)y)(x2
+ y2
) + O(|x|5
, |y|5
).
(20.2.7)
We will ﬁnd it more convenient to work with (20.2.7) in polar coordinates.
In polar coordinates (20.2.7) is given by
˙r = α(µ)r + a(µ)r3
+ O(r5
),
˙θ = ω(µ) + b(µ)r2
+ O(r4
). (20.2.8)
Because we are interested in the dynamics near µ = 0, it is natural to
Taylor expand the coeﬃcients in (20.2.8) about µ = 0. Equation (20.2.8)
thus becomes
˙r = α′
(0)µr + a(0)r3
+ O(µ2
r, µr3
, r5
),
˙θ = ω(0) + ω′
(0)µ + b(0)r2
+ O(µ2
, µr2
, r4
), (20.2.9)
where “ ′ ” denotes diﬀerentiation with respect to µ and we have used the
fact that α(0) = 0.
Our goal is to understand the dynamics of (20.2.9) for r small and µ
small. This will be accomplished in two steps.
Step 1. Neglect the higher order terms of (20.2.9) and study the resulting
“truncated” normal form.
Step 2. Show that the dynamics exhibited by the truncated normal form
are qualitatively unchanged when one considers the inﬂuence of the
previously neglected higher order terms.
380 20. Bifurcation of Fixed Points of Vector Fields
Step 1. Neglecting the higher order terms in (20.2.9) gives
˙r = dµr + ar3
,
˙θ = ω + cµ + br2
, (20.2.10)
where, for ease of notation, we deﬁne
α′
(0) ≡ d,
a(0) ≡ a,
ω(0) ≡ ω,
ω′
(0) ≡ c,
b(0) ≡ b. (20.2.11)
In analyzing the dynamics of vector ﬁelds we have always started with the
simplest situation; namely, we have found the ﬁxed points and studied the
nature of their stability. In regard to (20.2.10), however, we proceed slightly
diﬀerently because of the nature of the coordinate system. To be precise,
values of r > 0 and µ for which ˙r = 0, but ˙θ = 0, correspond to periodic
orbits of (20.2.10). We highlight this in the following lemma.
Lemma 20.2.1 For −∞ < µd
a < 0 and µ suﬃciently small
(r(t), θ(t)) =
−µd
a
, ω + c −
bd
a
µ t + θ0 (20.2.12)
is a periodic orbit for (20.2.10).
Proof: In order to interpret (20.2.12) as a periodic orbit, we need only
to insure that ˙θ is not zero. Since ω is a constant independent of µ, this
immediately follows by taking µ suﬃciently small. ⊓⊔
We address the question of stability in the following lemma.
Lemma 20.2.2 The periodic orbit is
i) asymptotically stable for a < 0;
ii) unstable for a > 0.
Proof: The way to prove this lemma is to construct a one-dimensional
Poincar´e map along the lines of Chapter 10 (and in particular, Example
10.1.1), from which the results of this lemma follow. ⊓⊔
We note that since we must have r > 0, (20.2.12) is the only periodic
orbit possible for (20.2.10). Hence, for µ = 0, (20.2.10) possesses a unique
periodic orbit having amplitude O(
√
µ). Concerning the details of stability
of the periodic orbit and whether it exists for µ > 0 or µ < 0, from (20.2.12)
it is easy to see that there are four possibilities:
20.2 The Poincare-Andronov-Hopf Bifurcation 381
FIGURE 20.2.1. d > 0, a > 0.
FIGURE 20.2.2. d > 0, a < 0.
1. d > 0, a > 0;
2. d > 0, a < 0;
3. d < 0, a > 0;
382 20. Bifurcation of Fixed Points of Vector Fields
4. d < 0, a < 0.
We will examine each case individually; however, we note that in all cases
the origin is a ﬁxed point which is
stable at µ = 0 for a < 0,
unstable at µ = 0 for a > 0.
Case 1: d > 0, a > 0. In this case the origin is an unstable ﬁxed point for
µ > 0 and an asymptotically stable ﬁxed point for µ < 0, with an unstable
periodic orbit for µ < 0 (note: the reader should realize that if the origin
is stable for µ < 0, then the periodic orbit should be unstable); see Figure
20.2.1.
Case 2: d > 0, a < 0. In this case the origin is an asymptotically stable
ﬁxed point for µ < 0 and an unstable ﬁxed point for µ > 0, with an
asymptotically stable periodic orbit for µ > 0; see Figure 20.2.2.
Case 3: d < 0, a > 0. In this case the origin is an unstable ﬁxed point for
µ < 0 and an asymptotically stable ﬁxed point for µ > 0, with an unstable
FIGURE 20.2.3. d < 0, a > 0.
periodic orbit for µ > 0; see Figure 20.2.3.
Case 4: d < 0, a < 0. In this case the origin is an asymptotically stable
ﬁxed point for µ < 0 and an unstable ﬁxed point for µ > 0, with an
asymptotically stable periodic orbit for µ < 0; see Figure 20.2.4.
20.2 The Poincare-Andronov-Hopf Bifurcation 383
FIGURE 20.2.4. d < 0, a < 0.
From these four cases we can make the following general remarks.
Remark 1. For a < 0 it is possible for the periodic orbit to exist for either
µ > 0 (Case 2) or µ < 0 (Case 4); however, in each case the periodic orbit
is asymptotically stable. Similarly, for a > 0 it is possible for the periodic
orbit to exist for either µ > 0 (Case 3) or µ < 0 (Case 1); however, in each
case the periodic orbit is unstable. Thus, the number a tells us whether
the bifurcating periodic orbit is stable (a < 0) or unstable (a > 0). The
case a < 0 is referred to as a supercritical bifurcation, and the case a > 0
is referred to as a subcritical bifurcation.
Remark 2. Recall that
d =
d
dµ
(Reλ(µ))
µ=0
.
Hence, for d > 0, the eigenvalues cross from the left half-plane to the right
half-plane as µ increases and, for d < 0, the eigenvalues cross from the right
half-plane to the left half-plane as µ increases. For d > 0, it follows that the
origin is asymptotically stable for µ < 0 and unstable for µ > 0. Similarly,
for d < 0, the origin is unstable for µ < 0 and asymptotically stable for
µ > 0.
Step 2. At this point we have a fairly complete analysis of the orbit structure
of the truncated normal form near (r, µ) = (0, 0). We now must consider
Step 2 in our analysis of the normal form (20.2.9); namely, are the dynamics
that we have found in the truncated normal form changed when the eﬀects
384 20. Bifurcation of Fixed Points of Vector Fields
of the neglected higher order term are considered? Fortunately, the answer
to this question is no and is the content of the following theorem.
Theorem 20.2.3 (Poincar´e-Andronov-Hopf Bifurcation) Consider
the full normal form (20.2.9). Then, for µ suﬃciently small, Case 1, Case
2, Case 3, and Case 4 described above hold.
Proof: We will outline a proof that uses the Poincar´e-Bendixson Theorem.
We begin by considering the truncated normal form (20.2.10) and the case
a < 0, d > 0. In this case the periodic orbit is stable and exists for µ > 0,
and the r coordinate is given by
r =
−dµ
a
.
FIGURE 20.2.5.
We next choose µ > 0 suﬃciently small and consider the annulus in the
plane, A, given by
A = {(r, θ)| r1 ≤ r ≤ r2} ,
where r1 and r2 are chosen such that
0 < r1 <
−dµ
a
< r2.
By (20.2.10), it is easy to verify that on the boundary of A, the vector
ﬁeld given by the truncated normal form (20.2.10) is pointing strictly into
the interior of A. Hence, A is a positive invariant region (cf. Deﬁnition
3.0.3, Chapter 3); see Figure 20.2.5.
It is also easy to verify that A contains no ﬁxed points so, by the Poincar´eBendixson
theorem, A contains a stable periodic orbit. Of course we already
20.2 The Poincare-Andronov-Hopf Bifurcation 385
knew this; our goal is to show that this situation still holds when the full
normal form (20.2.9) is considered.
Now consider the full normal form (20.2.9). By taking µ and r suﬃciently
small, the O(µ2
r, µr3
, r5
) terms can be made much smaller than the rest
of the normal form (i.e., the truncated normal form (20.2.10). Therefore,
by taking r1 and r2 suﬃciently small, A is still a positive invariant region
containing no ﬁxed points. Hence, by the Poincar´e-Bendixson theorem, A
contains a stable periodic orbit. The remaining three cases can be treated
similarly; however, in the cases where a > 0, the time-reversed ﬂow (i.e.,
letting t → −t) must be considered. ⊓⊔
To apply this theorem to speciﬁc systems, we need to know d (which is
easy) and a. In principle, a is relatively straightforward to calculate. We
simply carefully keep track of the coeﬃcients in the normal form transformation
in terms of our original vector ﬁeld. However, in practice, the algebraic
manipulations are horrendous. The explicit calculation can be found
in Hassard, Kazarinoﬀ, and Wan [1980], Marsden and McCracken[1976],
and Guckenheimer and Holmes [1983]; here we will just state the result.
At bifurcation (i.e., µ = 0), (20.2.4) becomes
˙x
˙y
=
0 −ω
ω 0
x
y
+
f1
(x, y, 0)
f2
(x, y, 0)
, (20.2.13)
and the coeﬃcient a(0) ≡ a is given by
a =
1
16
f1
xxx + f1
xyy + f2
xxy + f2
yyy
+
1
16ω
f1
xy f1
xx + f1
yy − f2
xy f2
xx + f2
yy
− f1
xxf2
xx + f1
yyf2
yy , (20.2.14)
where all partial derivatives are evaluated at the bifurcation point, i.e.,
(x, y, µ) = (0, 0, 0).
We end this section with some historical remarks. Usually Theorem
20.2.3 goes by the name of the “Hopf bifurcation theorem.” However, as has
been pointed out repeatedly by V. Arnold [1983], this is inaccurate, since
examples of this type of bifurcation can be found in the work of Poincar´e
[1892]. The ﬁrst speciﬁc study and formulation of a theorem was due to
Andronov [1929]. However, this is not to say that E. Hopf did not make
an important contribution; while the work of Poincar´e and Andronov was
concerned with two-dimensional vector ﬁelds, the theorem due to E. Hopf
[1942] is valid in n dimensions (note: this was before the discovery of the
center manifold theorem). For these reasons we refer to Theorem 20.2.3 as
the Poincar´e-Andronov-Hopf bifurcation theorem.
386 20. Bifurcation of Fixed Points of Vector Fields
20.2a Exercises
1. This exercise comes from Marsden and McCracken [1976]. Consider the following vector
ﬁelds
a)
˙r = −r(r − µ)
2
,
˙θ = 1,
(r, θ) ∈ R
+
× S
1
.
b) ˙r = r(µ − r
2
)(2µ − r
2
)
2
,
˙θ = 1.
c)
˙r = r(r + µ)(r − µ),
˙θ = 1.
d) ˙r = µr(r
2
− µ),
˙θ = 1.
FIGURE 20.2.6.
e) ˙r = −µ
2
r(r + µ)
2
(r − µ)
2
,
˙θ = 1.