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Unformatted text preview: Jordan form for a system with two zero eigenvalues. The alternative is
identically zero.
We attempt to eliminate the quadratic terms on the right hand side of [5] using a near identity
transformation of form
, where
and
. Denote
. The homological operator is Recalling etc., we have
,
, etc. In this way we construct a matrix representation for
is in the basis . The result . The column space of defines its range,
space is any subspace complementary to
. Since
element of . The second vector may be any linear combination of
independent of
. The simplest choices for are
the second choice leads to
, . The resonant
, it must be an
and that is
and
. Using [6]
. 22
Ch8Lecs This second form has the advantage that it is equivalent to the nonlinear “oscillator”
.
The right hand side of [6] is the singular vector field whose unfolding is known as the Takens Bogdanov bifurcation, which Meiss discusses in section 8.10. ■ Higher Order Normal Forms
Proceed by induction. Suppose that all terms in the range of
order . To this order have been eliminated below ,
where contains the resonant terms through order . Now let and require that the vector field for have only resonant terms through order . Then
.
Substitute these expansions into the relationship between diffeomorphic vector fields that we
reproduce,
,
to obtain
,
,
, [7] where the homological operator is on the left hand side of the equation. Set and equate terms in [7] of order to obtain This equation may be solved since the right hand side is in the range of . 23
Ch8Lecs Example. Continue the normal form transformations for the double zero (TakensBogdanov)
bifurcation to third order to obtain
,
.
See Meiss, Chapter 8, problem 18. ■...
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 Fall '14
 MarcEvans

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