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Unformatted text preview: act, 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 . ■
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 7
where and . Then the flows are related by
.  If is a diffeomorphism we can also find a relationship between the corresponding vector
fields. Differentiate  with respect to : Set to obtain the desired relationship.
, where and Example. Let 
. Compare with Eq. 4.34 in Meiss.
and . Then , where
and  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
Example. The singularity associated with the saddle-node bifurcation is
degenerate because it has a zero eigenvalue. Two unfolding of are
.■ . is 8
Ch8Lecs 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
. 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.
. 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
, . , 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 9
form , the higher order term satisfies . A general unfolding of ,
where will h...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
- Fall '14