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detection of turning points; Newton method; Newton flow; basins of attraction; qualitative analysis; normal forms of the flow; global convergence; singularity theory; bifurcation singularity; imperfect bifurcation
The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcation singularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship between the scenario and the actual performance of Newton method is studied. Both theoretical and experimental arguments are presented in order to quaetion the claim that a particular bifurcation singularity $organiyes$ the Newton method assuming small parameter perturbations.
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