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Unformatted text preview: imensional 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 HartmanGrobman theorem from our earlier
studies of the center manifold. □
Now we can see why this is called a saddlenode 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 14
Ch8Lecs Corollary 8.7 (transversality) Assume that
any single parameter such that satisfies the hypotheses of theorem 8.6. If , is (transversality) then a saddlenode 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 15
Ch8Lecs .
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
no...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans

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