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8.1 Bifurcations of Equilibria
Bifurcation theory studies qualitative changes in solutions as a parameter varies. In general,
one could study the bifurcation theory of ODEs, PDEs, integro-differential equations, discrete
mappings etc. Of course, we are concerned with ODEs.
Local bifurcations refer to qualitative changes occurring in a neighborhood of an equilibrium
point of a differential equation or a fixed point of an associated Poincaré map. These can be
studied by expanding the equations of motion in power series about the point.
Consider an ODE depending on a parameter :
,
,
.
Assume this system has an equilibrium point
that is a sink for
critical value of the parameter. Graph the evls as a function of .
where
is a
Real and imaginary axes, region of evls with negative real parts for
that the real parts of some evls are increasing with 
, arrows showing
A real evl or a cc pair may traverse the imaginary axis as increases through . The sink
becomes a source or a saddle. Or the equilibrium point may simply disappear when it has a
zero evl!
Example. Fishery model with constant harvesting.
Recall the logistic model of population growth with an additional constant term.
Interpret as a model of a fishery with the number of fish, time, the fish growth rate,
carrying capacity, and a constant rate of harvesting. The model may be simplified by
measuring the quantity of fish and the harvest rate in units of the carrying capacity:
and
.
Further simplify by introducing a dimensionless time variable:
.
Equilibria correspond to
the
. Then obtain
[1]
. Analyze graphically
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sketch
-,
-,

,
The direction of the arrows on the -axis now depends on the sign of
there are two equilibrium points, when
there is one, and when
Alternately, we can study the equilibria algebraically. Let
of [1] correspond to
. When
or
there are none.
. Then equilibrium points
. The equilibria are
,
. add
These exist only for
, arrows to graph We can write [1] in the form
,
[2]
where the dot now refers to the derivative with respect to dimensionless time. The
eigenvalues of the equilibrium points are
,
which depends on through . Since
, its eigenvalue is negative and
is a stable
equilibrium point. Since
, its eigenvalue is positive and
is an unstable equilibrium
point. As increases toward the equilibria converge and their eigenvalues approach 0. At
the equilibria merge as a single non-hyperbolic equilibrium point and for
they
disappear.
Draw a bifurcation diagram for the logistic model with constant harvesting by plotting the
equilibria as a function of .
sketch
-,
-, the two branches of the saddle-node
A saddle-node bifurcation occurs at a critical value of the parameter,
.
Simplify the logistic model with constant harvesting by centering the bifurcation at the origin.
Let
and
. Then [2] becomes
.
[3]
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This is the normal form of the saddle-node bifurcation. For
there are two hyperbolic
equilibrium points, for
there is a single nonhyperbolic equilibrium, and for
there
are no equilibria. ■
The saddle-node bifurcation requires 3 conditions on the vector field:



Singularity condition (the equilibrium point [3] is nonhyperbolic at
).
Non-degeneracy condition (the coefficient of
in [3] is nonzero).
Transversality condition (that guarantees that the parameter perturbs the
nonhyperbolic equilibrium point in a transverse way:
in [3]).
Local Bifurcations of Limit Cycles
To further illustrate the meaning of “local bifurcation” let’s briefly describe local bifurcations of
limit cycles.
sketch limit cycle , section , intersection
Consider a Poincaré map
.

corresponds to a fixed point of :
.
The eigenvalues of
are the Floquet multipliers (not including the trivial unit multiplier).
Real and imaginary axes, region of evls within the unit circle for
, arrows showing that
evls can traverse the circle though
,
, or a complex conjugate pair can pass out of
the unit circle off the real axis.
Example A saddle-node bifurcation of limit cycles.
Suppose that for
there is a semistable limit cycle and an associated Poincaré map:
sketch semistable limit cycle in 2D and nearby trajectories
sketch
-,
-, the line
, the Poincaré map is concave down and
tangent to the line at the origin, cobweb paths inside and outside the limit cycle
sketch pair of limit cycles. The outer one is stable and the inner one unstable.
Nearby trajectories.
 sketch
-,
-, the line
, the Poincaré map has shifted so there are
now two intersections with the line. Three cobweb paths.
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sketch spiral flow without limit cycle no intersection between the Poincare map with
the line
.
The saddle node bifurcation corresponds to a single eigenvalue reaching the unit circle at
and then disappearing.■
Global Bifurcations of Limit Cycles
Global bifurcations cannot be described by a local analysis.
Example Homoclinic bifurcation in
sketch like Meiss Fig. 8.18.
the limit cycle for
 ■
.
and
without the point
and axis. Notice
8.2 Preservation of Equilibria
Consider
. An equilibrium point
satisfies
. When we linearize
about the equilibrium point, the Jacobian matrix is
. If the Jacobian matrix has a
zero eigenvalue then it is singular and we say the equilibrium point is degenerate. Otherwise
the Jacobian matrix is nonsingular and the equilibrium point is nondegenerate.
Contrast this with the definition of a nonhyperbolic equilibrium point, where only the real part
of an eigenvalue need be zero. For example, if a Jacobian matrix has a pair of complex
conjugate imaginary eigenvalues and all of the other eigenvalues are nonzero, it is not singular.
Example Fishery model with constant harvesting. One way that we have written the equation
of motion is
.
Let
be an equilibrium point.
.
When
and
, the equilibrium point is degenerate;
corresponds to the saddle-node bifurcation. For
the equilibrium point disappears. ■
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If an equilibrium point is nondegenerate, it cannot be removed by sufficiently small
perturbations (such as a small change in the value of
Theorem 8.1 (Implicit Function). Let be an open set in
. Suppose there is a point
such that
nonsingular
matrix. Then there are open sets
function
for which
and
and
and
and
.
with
is a
and a unique
The proof of this famous theorem probably appears in your favorite analysis book.
To gain a rough understanding of why the condition on the Jacobian is necessary , expand
about
:
.
Neglect the higher-order terms and solve for
to obtain
.
This can be done for arbitrary
only if
is nonsingular.
When discussing the system
, the application of the implicit function theorem to
preservation of equilibria corresponds to
and
.
Corollary 8.2 (Preservation of a Nondegenerate Equilibrium) Suppose the vector field
is
in both and and that is a nondegenerate equilibrium point for parameter value
Then there exists a unique
curve of equilibria
passing through at .
Proof. The matrix
governs the stability of the equilibrium, and since
of its eigenvalues nonzero, is nonsingular. Then theorem 8.1 implies that there is a
neighborhood of for which there is a curve of equilibria
. □
8.3 Unfolding Vector Fields
Change of Variables and Topological Conjugacy
Recall topological conjugacy between flows corresponds to the diagram:
,
.
has all
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where
is a homeomorphism. The diagram implies
.
Bifurcation theory studies systems that depend on parameters. Let
be the flow of
and let
be the flow of
. If these flows are conjugate for each
value of then there is the diagram:
.
where
is a homeomorphism. The diagram implies
.
Example. Consider the system
. Change variables using
to obtain
. Note that
is a homeomorphism (in
fact, a diffeomorphism). The corresponding flows are topologically conjugate. Meiss extends
this terminology to the vector fields; he also calls and conjugate. ■
For a fixed value of ,
family of vector fields.
gives a vector field. For all possible values of ,
A family of vector fields
is induced by a family
such that
. If the family
the same dynamics as or is simpler than .
Example. The family
map
values of
if there is a continuous map
is induced by the family , then
is induced by the family
. The family
gives a
has
using the
has the same dynamics as . (In fact, for corresponding
and , the vector fields are the same.) ■
Example. Consider the family
and the constant map
family of vector fields induced by using the map is
is clearly simpler than the family . ■
Let
be the flow of
and be the flow of
vector fields and are conjugate for corresponding values of
diagram
. The
. The family
where the
and . Then we have the
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,
where
and
. Then the flows are related by
.
[1]
If is a diffeomorphism we can also find a relationship between the corresponding vector
fields. Differentiate [1] with respect to :
Set
to obtain the desired relationship.
,
where
and
Example. Let
[2]
. Compare with Eq. 4.34 in Meiss.
and
. Then
, where
and [2] gives
verified by making the indicated substitutions for
and
. This can be
and . ■
Unfolding Vector Fields
Consider a vector field that has a degenerate orbit. This could be a degenerate equilibrium
point or, for example, the homoclinic orbit in the homoclinic bifurcation. Then we say fulfills
a singularity condition. A family of vector fields
is an unfolding of
if
.
Example. The singularity associated with the saddle-node bifurcation is
degenerate because it has a zero eigenvalue. Two unfolding of are
and
. ■
.
is
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An unfolding
of is versal if it contains all possible qualitative dynamics that can occur
near to . This means that every other unfolding in some neighborhood of will have the
same dynamics as some family induced by
.
Example. Let
and
. We will show below that is a versal
unfolding of the saddle node bifurcation. However,
is not versal. For example, there is
no value of for which the system
has two equilibrium points.
An unfolding is miniversal if it is a versal unfolding with the minimum number of parameters.
Example. Let
and
. The flows of
are diffeomorphic. To see this complete the square in :
and
.
The change of variables
and
converts
into .
Note that is a diffeomorphism. Since
may be any real number, the families of vector
fields and have the same dynamics. However, has two parameters and has only one..
Then is a versal unfolding, but not a miniversal unfolding of the saddle node bifurcation. ■
8.4 Saddle-Node Bifurcation in One Dimension
Consider the ODE
on
Theorem 8.3 Suppose that
origin,
,
.
, and that
with a nonhyperbolic equilibrium at the
satisfies the nondegeneracy condition
.
Then there is a
such that when
that there is a unique extremal value
, there is an open interval
, containing
.
There are two equilibria in when
.
Proof. Let
, one when
and zero when
. The singularity and nondegeneracy conditions imply
.
such
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Since
form
, the higher order term satisfies
. A general unfolding of
,
where
will have
[3]
and
. Consider the function
.
Since
, the implicit function theorem guarantees neighborhoods and of
the origin such that when
there is a unique
such that
and
. Since
there must also be an open interval
, containing
, for
which
is monotone. On
,
is either concave up (when
) or concave
down (when
). Therefore
is the unique extremal value for when
, and
.
determines whether the critical point is a minimum or a maximum. Since
when
, this remains true by continuity for small enough ;
for
. Therefore, when
, has a minimum at
and is
positive on
. This implies that, if
there are no zeros of , if
there is a
single zero, and if
there are two zeros. Similar considerations apply when
. □
Picture for

-,
:
-,
,

For

,

is a stable equilibrium and
is an unstable equilibrium.
The bifurcation set is the set of all points in parameter space at which the bifurcation takes
place.
A bifurcation is codimension
on parameters.
if the bifurcation set is determined by
independent conditions
According to theorem 8.3, the saddle-node bifurcation depends on a single quantity:
.
The bifurcation takes place when
. The saddle-node is a codimension 1 bifurcation.
Example. Let
. Find the saddle-node bifurcation set, that is, the locus
of saddle-node bifurcations in
space.
is defined by
. From
we have
. Then
. The
bifurcation set corresponds to the single condition
. ■
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Corollary 8.4 If f satisfies the hypotheses of theorem 8.3 and there is a single parameter
such that the transversality condition
holds, then a saddle-node bifurcation
occurs as crosses zero.
Proof. Recall that
defined
was defined to satisfy
. Thus the chain rule implies
. We also
.
Since
, the sign of
changes as
crosses zero. □
Remark. When there is a vector of parameters , application of the corollary requires that
for
as crosses .
Example. Let
bifurcation takes place as
Example. Let
and
system
.
If
, a saddle-node
crosses . Otherwise the bifurcation takes place when
, an unfolding of
. Then
. By the corollary, a bifurcation occurs as crosses zero. For this
, and a saddle-node bifurcation also takes place when
.
Example. Let
, an unfolding of
. Note that
does not satisfy the transversality condition. In fact
is an equilibrium of
. A saddle-node bifurcation clearly does not occur in this unfolding of . ■
, which
for all values of
Theorem 8.5 Under the hypotheses of theorem 8.3, the saddle-node bifurcation has the
miniversal unfolding
.
Proof. Expand
[1]
in Taylor series about
:
.
Identifying
and
gives
.
Let
and
[2]
. Then
.
[3]
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Compare [2] and [3] with the form of section 8.3, Eq. [2], which is reproduced below. To obtain
that equation, we assumed that the flows of
and
were related by the
diffeomorphism
where
. Then we obtained the relationship between
vector fields
.
Note that the expression for
given below [2] is a diffeomorphism, and that Eq. [3]
exhibits the correct constant of proportionality
. The comparison between [2] and
[3] gives
.
Thus the family of vector fields
is induced by a family of form
.
[4]
According to the one-dimensional equivalence theorem, theorem 4.10, there is a neighborhood
of the origin for which the dynamics of [4] is topologically equivalent to those of [1] because
both systems have two equilibria with the same stability types and arranged in the same order
on the line. □
Transcritical Bifurcation
Consider
, an unfolding of
equilibrium point at
never disappears.
bifurcation. Sketch the bifurcation diagram.

-,
. This is not a versal unfolding; the
is a normal form of the transcritical
-, two lines of equilibria with stability exchanged at the origin

-,
-, with parabolas intersecting the abscissa at
stability of equilibrium points
and
to determine
This bifurcation is sometimes referred to as an exchange of stability between the two
equilibrium points, and is encountered frequently in applications. To see the relationship with
the saddle-node bifurcation, begin by completing squares:
.
Let
and
to see that that flow of
is homeomorphic to that of
. The family of vector fields is induced by
using
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. Notice that
cannot be positive. As increases through zero, values of
take the following path through the saddle-node bifurcation diagram:

-,

-, curves of equilibria corresponding to SN bifurcation, path of
As a bug crawls along this path, it sees the transcritical bifurcation diagram shown above
(modulo some distortion due to the coordinate transformations).
8.6 Saddle-Node Bifurcation in
Theorem 8.6 (saddle node) Let
, and suppose that
satisfies
. (singularity)
Choose coordinates so that
is diagonal in the zero eigenvalue and set
where
corresponds to the zero eigenvalue and
are the remaining coordinates.
Then
,
where
and
. Suppose that
.
(nondegeneracy)
Then there exists an interval
containing , functions
and
, and a neighborhood of
such that if
there are no
equilibria and if
there are two. Suppose that has a -dimensional unstable space
and an
-dimensional stable space. Then, when there are two equilibria, one has a
-dimensional unstable and an
-dimensional stable manifold and the other has a
-dimensional unstable manifold and an
-dimensional stable manifold.
Proof The equilibria are solutions of
,
[1]
.
By assumption,
that there is a neighborhood of
such that
is nonsingular; thus the Implicit Function Theorem ensures
where there exists a unique function
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[2]
and
. Substitute this into
to obtain
.
Consequently, the problem has been reduced to the one-dimensional case; we need only check
that satisfies the same criteria as Theorem 8.3, the one-dimensional case. It is easy to see
that
. Since is , so is , and differentiation of [2] with respect to gives
.
Since
, this implies that
. This relationship helps
compute the required derivatives of :
.
Thus the needed hypotheses for Theorem 8.3 are satisfied and there exists an extremal value
such that when crosses zero the number of equilibria changes from zero to two. The
stability of the equilibria follows by considering the stability of equilibria in the one dimensional
case in conjunction with the nonhyperbolic Hartman-Grobman theorem from our earlier
studies of the center manifold. □
Now we can see why this is called a saddle-node bifurcation. Consider theorem 8.6 for
sketch in 1D case; two equilibria for
equilibria for
.
sketch for
for
sketch for
node; for
, degenerate equilibrium for
: for
a stable node and saddle; for
a converging flow without equilibrium
, no
a degenerate node;
: for
a saddle and an unstable node; for
a diverging flow without equilibrium
a degenerate
Tranversality
The following theorem gives a condition that guarantees that
parameter is varied.
.
changes sign as a
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Corollary 8.7 (transversality) Assume that
any single parameter such that
satisfies the hypotheses of theorem 8.6. If
,
is
(transversality)
then a saddle-node bifurcation takes place when
Proof. Show that
. Use
to denote the critical point of
as a function of . Then
The first derivatives
crosses zero.
and
and
both vanish by hypothesis, then the transversality
. □
assumption gives
Example. Let’s apply the systematic procedure suggested in the statement of theorem 8.6 to
the system
[3]
,
which has an equilibrium at the origin. First, rewrite this in matrix form
.
The Jacobian matrix
and
has eigenvalues
and
. Corresponding eigenvectors are
. The matrix equation has the general form
,
where
. Let the transformation matrix
. Then
.
Set
and
.
or
. We have
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.
This has the form
,
,
given in the statement of theorem 8.6 (see [1]). The system satisfies the singularity conditions
stated in the theorem and the nondegeneracy condition
.
Furthermore,
, so the transversality condition is satisfied. Since
, has a
minimum and since
, the minimum decreases through as increases.
Going back to the original system [3], we can easily solve for the equilibria to get
, confirming our result. ■
and
8.5 Normal Forms
In chapter 2 we transformed linear systems in order to put them in a simple form (diagonalizing
the matrices in the semisimple case). In chapter 4 we learned that these changes of
coordinates were diffeomorphisms. In this section we continue the program of applying
diffeomorphisms to transform
to simpler forms, only now we seek to simplify the
nonlinear terms.
Homological Operator
Let
and have an equilibrium point at the origin. Further assume has as many
derivatives as necessary for the manipulations below. Expand in power series
,
where
[1]
is a vector of homogeneous polynomials of degree . Let
.
A basis for
is the set of monomials
,
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where
and
This compact notation is known as multi-index notation.
For example,
is three dimensional. Let
( factors)
be the space of vectors of homogeneous polynomials on
.
Example.
has dimension 6 and the basis
■
Denoting the standard basis vectors of
provide a basis for
Example. let
by
, the vector monomials
.
and
. The first degree terms in the power series [1] may be written
. ■
We will construct the “simplest” vector field
that is diffeomorphic with
transformation. Let represent the new variable so that
by a near identity
and the diffeomorphism is
.
In a small neighborhood of the equilibrium point of
transformation and is invertible.
Recall from chapter 4 that if
, then
,
.
at the origin,
generates the flow
and if
is close to the identity
generates the flow
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If there is a diffeomorphism
then
between the two flows so that
,
or
Set
to obtain a relationship between the vector fields
.
[2]
We will choose to eliminate as many of the nonlinear terms in
the quadratic terms, setting
. Write
attempt to choose
so as to eliminate so that
expansions for and into [2], we find
as possible. First consider
. We will
. Substituting the
or
.
The linear terms on the two sides of the equation match. Equating the quadratic terms gives
,
[3]
which is an equation for the unknown function .
is called the homological operator.
is
a linear operator on the space of degree vector fields:
(see problem 6). Eq. [3]
can be solved if and only if
, where
is the range of . Consequently, we
introduce the following direct sum decomposition of
:
,
where
is a complement to
, and we split
.
into two parts:
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The function
is the resonant part of . We now reconsider the derivation of Eq [3], only
aiming to eliminate the non-resonant terms. Let
. Then insert the
new expansions for and into [2] to obtain
,
,
,
,
which is guaranteed to have a solution for
.
Matrix Representation
Prior to applying the near identity transformations, we assume that coordinates have been
transformed to bring the
matrix
into Jordan form. Let
, be the
standard unit basis vectors in this coordinate system.
Any linear operator on a finite dimensional space has a matrix representation. Suppose
, where is a vector space of homogeneous polynomials and
and let
represent a basis for . Then
and is given by a linear combination
of basis vectors:
.
This defines the
and
matrix as a representation of the action of
we have
,
on
. Writing
which implies
.
Since the basis vectors are linearly independent this is equivalent to the matrix equation
.
The simplest case is when the eigenvalues of
. Compute the action of
. From [3]
are real and distinct. Then in Jordan form
on the monomial basis vectors
of
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.
Since is diagonal and
component of
is nonzero:
is proportional to , we have
. Only the -th
is nonzero, and therefore only the -th row of the Jacobian matrix
Then
is the dot product of this -th row with the vector
term is this dot product is
. The -th
.
Then
.
Thus we have
The vector monomials
are eigenfunctions of on
with eigenvalues
are nonzero we can solve
by inverting to obtain
.
Example. Consider the 1D case with a hyperbolic equilibrium:
. If all the
:
[4]
Set
so
. The homological operator is
with
Since the nonlinear terms are all proportional to with
and all nonlinear terms may be eliminated to obtain
,
.
, there are no resonant terms
. This results is consistent with the
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Hartman-Grobman theorem, which tells us that the dynamics of [4] in a neighborhood of the
origin are topologically conjugate to those of the linearized system. ■
When one or more of the
are zero,
For example, consider the case of
be
Suppose
is nontrivial and there may be resonances.
for which the eigenvectors of
were found to
. Then the condition
corresponds to
.
For the case of ,
and
. Then
. If
then
is nontrivial and may have resonant nonlinear terms that cannot be eliminated. This may not be a
surprise since is not hyperbolic.
For the case of ,
and
. Then
. Notice that this can be zero
even if is hyperbolic. In other words, certain nonlinear systems with hyperbolic linear parts
may have resonant nonlinear terms that cannot be eliminated by normal form transformations.
Compare this to the statement of the Hartman-Grobman theorem, which assures us that the
dynamics of a nonlinear system in a neighborhood of a hyperbolic equilibrium point are
topologically conjugate to those of the corresponding linearized system. This difference
reflects the fact that normal form transformations use diffeomorphisms to eliminate the
nonlinear terms. This is a smaller class of transformations than the homeomorphisms used by
the Hartman-Grobman theorem.
The non-hyperbolic Hartman-Grobman theorem from center manifold theory tells us that the
dynamics of a nonlinear system are topologically conjugate to the linearized dynamics on the
stable and unstable manifold together with the nonlinear dynamics on the center manifold.
Thus we are particularly interested to apply normal form transformations to systems with
nonhyperbolic linear parts.
Example Double-zero eigenvalue. Consider a 2D system of form
[5]
The Jacobian for this system evaluated at the origin is
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.
is the most typical Jordan form for a system with two zero eigenvalues. The alternative is
identically zero.
We attempt to eliminate the quadratic terms on the right hand side of [5] using a near identity
transformation of form
, where
and
. Denote
. The homological operator is
Recalling
etc., we have
,
,
etc. In this way we construct a matrix representation for
is
in the basis
. The result
.
The column space of defines its range,
space is any subspace complementary to
. Since
element of . The second vector may be any linear combination of
independent of
. The simplest choices for are
the second choice leads to
,
. The resonant
, it must be an
and that is
and
. Using
[6]
.
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This second form has the advantage that it is equivalent to the nonlinear “oscillator”
.
The right hand side of [6] is the singular vector field whose unfolding is known as the TakensBogdanov bifurcation, which Meiss discusses in section 8.10. ■
Higher Order Normal Forms
Proceed by induction. Suppose that all terms in the range of
order . To this order
have been eliminated below
,
where
contains the resonant terms through order
. Now let
and require that the vector field for have only resonant terms through order . Then
.
Substitute these expansions into the relationship between diffeomorphic vector fields that we
reproduce,
,
to obtain
,
,
,
[7]
where the homological operator is on the left hand side of the equation. Set
and equate terms in [7] of order
to obtain
This equation may be solved since the right hand side is in the range of
.
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Example. Continue the normal form transformations for the double zero (Takens-Bogdanov)
bifurcation to third order to obtain
,
.
See Meiss, Chapter 8, problem 18. ■