Abstract
ODEs with a small parameter ε multiplying derivatives, which have been studied as a singular perturbation problem, are formulated in a coordinate-free manner, as a pair of a vector field and a tensor field, in order to treat them as a bifurcation problem. This formulation is an extension of the classical interpretation of autonomous ODEs as vector fields, and it is called a constrained system. A method to obtain normal forms for constrained systems is given as well as several results of computing them. Unfoldings of these normal forms include a large part of typical behaviors observed in such ODEs with ε. The local classification of characteristic surfaces is also discussed.
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Oka, H. Constrained systems, characteristic surfaces, and normal forms. Japan J. Appl. Math. 4, 393–431 (1987). https://doi.org/10.1007/BF03167813
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DOI: https://doi.org/10.1007/BF03167813