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Numerical Computation Of Normal Forms For Nonlinear Systems
AuthorPandey, Hari Datt
AdvisorPinsky, Mark A.
Mathematics and Statistics
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Poincare's normal forms method has become a prevailing approach to qualitative analysisof local bifurcation phenomena undergoing in nonlinear dynamical systems. However itsnumerical accuracy is frequently depreciated in extended neighborhoods of a coresolution due to poor convergence or divergence of the underlying successiveapproximations. This limits the practical value of the method.In past decades substantial efforts were devoted to improve convergence of thenormalizing approximations. This resulted in the celebrated KAM - theory.This thesis develops a numerical approach to deriving the normal forms andcorresponding nonlinear change of variables which eliminates poorly convergent ordivergent successive approximations. We adopt the functional structure of the normalforms method but determine the corresponding coefficients by minimizing the leastsquare error of the system of solutions which emanate from a given neighborhood of acore solution. We show on a few examples that this approach provides a higher accuracythan the standard technique.