Lecture8

# Thus we are particularly interested to apply normal

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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. ■...
View Full Document

Ask a homework question - tutors are online