Thus we are particularly interested to apply normal

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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 (Takens-Bogdanov) bifurcation to third order to obtain , . See Meiss, Chapter 8, problem 18. ■...
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