In chapter 4 we learned that these changes of

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Unformatted text preview: nlinear terms. 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 multi-index 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 non-resonant 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 .

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