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Homological Operator
Let
and have an equilibrium point at the origin. Further assume has as many
derivatives as necessary for the manipulations below. Expand in power series
,
where [1] is a vector of homogeneous polynomials of degree . Let
. A basis for is the set of monomials
, 16
Ch8Lecs where
and
This compact notation is known as multiindex notation.
For example,
is three dimensional. Let
( factors)
be the space of vectors of homogeneous polynomials on
.
Example. has dimension 6 and the basis
■ Denoting the standard basis vectors of provide a basis for
Example. let by , the vector monomials .
and . The first degree terms in the power series [1] may be written .■
We will construct the “simplest” vector field that is diffeomorphic with transformation. Let represent the new variable so that by a near identity and the diffeomorphism is .
In a small neighborhood of the equilibrium point of
transformation and is invertible.
Recall from chapter 4 that if
, then
,
. at the origin, generates the flow and if is close to the identity generates the flow 17
Ch8Lecs If there is a diffeomorphism
then between the two flows so that ,
or Set to obtain a relationship between the vector fields
. [2] We will choose to eliminate as many of the nonlinear terms in
the quadratic terms, setting
. Write
attempt to choose
so as to eliminate so that
expansions for and into [2], we find as possible. First consider
. We will
. Substituting the or
.
The linear terms on the two sides of the equation match. Equating the quadratic terms gives
, [3] which is an equation for the unknown function .
is called the homological operator.
is
a linear operator on the space of degree vector fields:
(see problem 6). Eq. [3]
can be solved if and only if
, where
is the range of . Consequently, we
introduce the following direct sum decomposition of
:
,
where is a complement to , and we split
. into two parts: 18
Ch8Lecs The function
is the resonant part of . We now reconsider the derivation of Eq [3], only
aiming to eliminate the nonresonant terms. Let
. Then insert the
new expansions for and into [2] to obtain
,
,
,
,
which is guaranteed to have a solution for . Matrix Representation
Prior to applying the near identity transformations, we assume that coordinates have been
transformed to bring the
matrix
into Jordan form. Let
, be the
standard unit basis vectors in this coordinate system.
Any linear operator on a finite dimensional space has a matrix representation. Suppose
, where is a vector space of homogeneous polynomials and
and let
represent a basis for . Then
and is given by a linear combination
of basis vectors:
.
This defines the
and matrix as a representation of the action of
we have
, on . Writin...
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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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