  In this paper we introduce universal asymptotic unfolding normal forms for
nonlinear singular systems. Next, we propose an approach to find the parameters
of a parametric singular system that they play the role of universal unfolding
parameters. These parameters effectively influence the local dynamics of the
system. We propose a systematic approach to locate local bifurcations in terms
of these parameters. Here, we apply the proposed approach on Hopf-zero
singularities whose the first few low degree terms are incompressible. In this
direction, we obtain novel orbital and parametric normal form results for such
families by assuming a nonzero quadratic condition. Moreover, we give a
truncated universal asymptotic unfolding normal form and prove the finite
determinacy of the steady-state bifurcations for two most generic subfamilies
of the associated amplitude systems. We analyze the local bifurcations of
equilibria, limit cycles and the secondary Hopf bifurcation of invariant tori.
The results are successfully implemented and verified using Maple. By employing
the proposed approach, we design an effective multiple-parametric quadratic
state feedback controller for a singular system on a three dimensional central
manifold with two imaginary uncontrollable modes. We illustrate how our program
systematically identifies the distinguished (universal unfolding) parameters,
derives the estimated transition varieties in terms of these parameters, and
locates the local primary and secondary bifurcations of equilibria, limit
cycles and invariant tori. This approach is useful in designing efficient
nonlinear feedback controllers (single or multiple inputs) for local
bifurcation control in engineering problems.
