The simplest normal form and its application to bifurcation control

This paper is concerned with the computation of the simplest normal forms with perturbation parameters, associated with codimension-one singularities, and applications to control systems. First, an efficient method is presented to compute the normal forms for general semi-simple cases, which combines center manifold theory and normal form theory in one unified procedure. The efficient approach is then applied to find the explicit simplest normal forms of general n-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain either single zero or a purely imaginary pair. In addition to near-identity nonlinear transformation, time and parameter rescalings are used to obtain the simplest normal forms. It is shown that, unlike the classical normal forms, the simplest normal forms for single zero and Hopf singularities are finite up to an arbitrary order, which greatly simplify stability and bifurcation analysis. The new method is applied to consider controlling bifurcations of the Lorenz system and a nonlinear electrical circuit. Symbolic programs have been developed using Maple, which greatly facilitates applications.