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Unformatted text preview: g which implies .
Since the basis vectors are linearly independent this is equivalent to the matrix equation
.
The simplest case is when the eigenvalues of
. Compute the action of
. From [3] are real and distinct. Then in Jordan form
on the monomial basis vectors of 19
Ch8Lecs .
Since is diagonal and
component of
is nonzero: is proportional to , we have
. Only the th
is nonzero, and therefore only the th row of the Jacobian matrix Then
is the dot product of this th row with the vector
term is this dot product is . The th .
Then .
Thus we have The vector monomials
are eigenfunctions of on
with eigenvalues
are nonzero we can solve
by inverting to obtain
.
Example. Consider the 1D case with a hyperbolic equilibrium: . If all the :
[4] Set
so . The homological operator is with
Since the nonlinear terms are all proportional to with
and all nonlinear terms may be eliminated to obtain , .
, there are no resonant terms
. This results is consistent with the 20
Ch8Lecs HartmanGrobman theorem, which tells us that the dynamics of [4] in a neighborhood of the
origin are topologically conjugate to those of the linearized system. ■
When one or more of the
are zero,
For example, consider the case of
be Suppose is nontrivial and there may be resonances.
for which the eigenvectors of
were found to . Then the condition corresponds to .
For the case of ,
and
. Then
. If
then
is nontrivial and may have resonant nonlinear terms that cannot be eliminate d. This may not be a
surprise since is not hyperbolic.
For the case of ,
and
. Then
. Notice that this can be zero
even if is hyperbolic. In other words, certain nonlinear systems with hyperbolic linear parts
may have resonant nonlinear terms that cannot be eliminated by normal form transformations.
Compare this to the statement of the HartmanGrobman theorem, which assures us that the
dynamics of a nonlinear system in a neighborhood of a hyperbolic equilibrium point are
topologically conjugate to those of the corresponding linearized system. This difference
reflects the fact that normal form transformations use diffeomorphisms to eliminate the
nonlinear terms. This is a smaller class of transformations than the homeomorphisms used by
the HartmanGrobman theorem.
The nonhyperbolic HartmanGrobman theorem from center manifold theory tells us that the
dynamics of a nonlinear system are topologically conjugate to the linearized dynamics on the
stable and unstable manifold together with the nonlinear dynamics on the center manifold.
Thus we are particularly interested to apply normal form transformations to systems with
nonhyperbolic linear parts.
Example Doublezero eigenvalue. Consider a 2D system of form
[5] The Jacobian for this system evaluated at the origin is 21
Ch8Lecs .
is the most typical...
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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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