  In this paper, we study jumps of nonlinear DAEs caused by inconsistent
initial values. First, we propose a simple normal form called the index-1
nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the
notion of consistency projector for linear DAEs to the nonlinear case. By an
example, we compare our proposed nonlinear consistency projectors with two
existing consistent initialization methods to show that the two existing
methods are not coordinate-free, i.e., the consistent points calculated by the
two methods are not invariant under nonlinear coordinates transformations. Next
we propose a singular perturbed system approximation for nonlinear DAEs, which
is an ordinary differential equation (ODE) with a small perturbation parameter,
we show that the solutions of the proposed perturbation system approximate both
the jumps resulting from the nonlinear consistency projectors and the $\mathcal
C^1$-solutions of the DAE. At last, we use a numerical simulation of a
nonlinear DAE model arising from an electric circuit to illustrate the
effectiveness of the proposed singular perturbed system approximation of DAEs.
