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Unformatted text preview: hen we obtained the relationship between
Note that the expression for given below  is a diffeomorphism, and that Eq.  exhibits the correct constant of proportionality . The comparison between  and  gives
Thus the family of vector fields is induced by a family of form .  According to the one-dimensional equivalence theorem, theorem 4.10, there is a neighborhood
of the origin for which the dynamics of  is topologically equivalent to those of  because
both systems have two equilibria with the same stability types and arranged in the same order
on the line. □
, an unfolding of
equilibrium point at
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
Ch8Lecs . 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
corresponds to the zero eigenvalue and
are the remaining coordinates.
where and . Suppose that . (nondegeneracy) Then there exists an interval
containing , functions
, and a neighborhood of
such that if there are no equilibria and if
there are two. Suppose that has a -dimensional unstable space
-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
. By assumption,
that there is a neighborhood of
such that is nonsingular; thus the Implicit Function Theorem ensures
where there exists a unique function 13
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-d...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
- Fall '14