Suppose where is a vector space of homogeneous

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Unformatted text preview: g which implies . Since the basis vectors are linearly independent this is equivalent to the matrix equation . The simplest case is when the eigenvalues of . Compute the action of . From [3] are real and distinct. Then in Jordan form on the monomial basis vectors of 19 Ch8Lecs . Since is diagonal and component of is nonzero: is proportional to , we have . Only the -th is nonzero, and therefore only the -th row of the Jacobian matrix Then is the dot product of this -th row with the vector term is this dot product is . The -th . Then . Thus we have The vector monomials are eigenfunctions of on with eigenvalues are nonzero we can solve by inverting to obtain . Example. Consider the 1D case with a hyperbolic equilibrium: . If all the : [4] Set so . The homological operator is with Since the nonlinear terms are all proportional to with and all nonlinear terms may be eliminated to obtain , . , there are no resonant terms . This results is consistent with the 20 Ch8Lecs Hartman-Grobman theorem, which tells us that the dynamics of [4] in a neighborhood of the origin are topologically conjugate to those of the linearized system. ■ When one or more of the are zero, For example, consider the case of be Suppose is nontrivial and there may be resonances. for which the eigenvectors of were found to . Then the condition corresponds to . For the case of , and . Then . If then is nontrivial and may have resonant nonlinear terms that cannot be eliminate d. This may not be a surprise since is not hyperbolic. For the case of , and . Then . Notice that this can be zero even if is hyperbolic. In other words, certain nonlinear systems with hyperbolic linear parts may have resonant nonlinear terms that cannot be eliminated by normal form transformations. Compare this to the statement of the Hartman-Grobman theorem, which assures us that the dynamics of a nonlinear system in a neighborhood of a hyperbolic equilibrium point are topologically conjugate to those of the corresponding linearized system. This difference reflects the fact that normal form transformations use diffeomorphisms to eliminate the nonlinear terms. This is a smaller class of transformations than the homeomorphisms used by the Hartman-Grobman theorem. The non-hyperbolic Hartman-Grobman theorem from center manifold theory tells us that the dynamics of a nonlinear system are topologically conjugate to the linearized dynamics on the stable and unstable manifold together with the nonlinear dynamics on the center manifold. Thus we are particularly interested to apply normal form transformations to systems with nonhyperbolic linear parts. Example Double-zero eigenvalue. Consider a 2D system of form [5] The Jacobian for this system evaluated at the origin is 21 Ch8Lecs . is the most typical...
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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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