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Unformatted text preview: lly parallel to
the origin on
Sketch -plane, and , several trajectories. . Straight line solutions move away from the origin on
. Other solution are asymptotic to
and asymptotic to
as -plane, and , several trajectories. Like Meiss Fig. 2.3 [D] Unstable focus:
. Both the eigenvalues and eigenvectors are
complex conjugate. (To see this just take the complex conjugate of A
, where and are real vectors. The solution is still given by , where and
must also be conjugate in order that the solution be real. Let
these expressions into the right hand side of  one obtains
 6 .
We have written The solution is an exponentially expanding spiral. Example. The following phase portrait was constructed for the linear system with
For this case , , and so that . 0.4 0.2 0 .2 0.1 0.1 0.2 0.2 0.4 0.6 ■
[E] Stable focus:
. The analysis is similar to that for an unstable
focus. The solution is an exponentially contracting spiral.
There are also several cases corresponding to boundaries between the 5 regions on the -plane.
The first two cases correspond to non-isolated equilibria, which occur when there is a zero
[a] Unstable degenerate equilibrium:
. The eigenspace
is a line of
unstable equilibria. Solutions that originate off this line move away exponentially as
along lines parallel to .
[b] Stable degenerate equilibrium:
. The eigenspace
is a line of stable
equilibria. Solutions that originate off this line move away exponentially as
parallel to .
are obtained for
. The analysis is given by  with
. Trajectories are ellipses. This case is particularly important both because it occurs in
bifurcations (chapter 8) and because Hamiltonian systems always have
To analyze the boundary states corresponding to the parabola
we need to recall
some concepts from linear algebra, beginning with two concepts of the multiplicity of an
An eigenvalue has algebraic multiplicity if the characteristic polynomial can be written
, i.e. is a -fold root of the characteristic polynomial. An eigenvalue
has geometric multiplicity if it has linearly independent eigenvectors ,
. A theorem of linear algebra is that the
geometric multiplicity is at most the algebraic multiplicity. When the geometric multiplicity is
less than the algebraic multiplicity, the matrix is defec...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
- Fall '14