Normal forms for constrained nonlinear differential equations. I. Theory

The theory of normal forms for smooth vector fields is system nonlinear differential-algebraic equations. Such equations are widely encountered in practical circuits and systems when parasitics play an important role in the system´s qualitative behavior. Such parasitics are a called small parameters in the associated singular perturbation problem. The approach taken from here is completely different form the literature on singular perturbation and is based on the general framework described by L.D. Chua and H. Kokuba (see ibid., vol. 35, no. 7, p. 863-880. 1988), namely, the calculation of infinitesimal deformations. A coordinate-free formulation for constrained equations give a local classification according to the extent of the degeneracy of the original constrained equation