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Unformatted text preview: lly parallel to
as
.
Sketch
[C] Saddle:
the origin on
.
Sketch plane, and , several trajectories. . Straight line solutions move away from the origin on
and towards
. Other solution are asymptotic to
as
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
.). Let
, where and are real vectors. The solution is still given by [4], where and
must also be conjugate in order that the solution be real. Let
. Substituting
these expressions into the right hand side of [4] one obtains
[5] 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 nonisolated equilibria, which occur when there is a zero
eigenvalue giving
. 7
[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
along lines
parallel to .
[c] Center:
are obtained for
and
. The analysis is given by [5] 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
eigenvalue:
An eigenvalue has algebraic multiplicity if the characteristic polynomial can be written
where
, i.e. is a fold root of the characteristic polynomial. An eigenvalue
has geometric multiplicity if it has linearly independent eigenvectors ,
i.e.
and
. 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
 MarcEvans

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