Orbital and parametric normal forms for families of Hopf-zero singularity

Abstract

This paper explores the simplest truncated orbital and parametric normal forms of controlled Hopf zero singularities. We assume a quadratic generic condition and complete the remaining results on their simplest truncated orbital and parametric normal forms of Hopf-zero singularities. Different normal form styles are explored for their potential applications in bifurcation control. We obtain their associated universal asymptotic unfolding normal forms. We derive coefficient normal form formulas of the most generic cases and present the relations between the controller coefficients and asymptotic universal unfolding parameters. These play an important role in their potential applications in bifurcation control. Finally, the results are implemented on a controlled Chua circuit system to illustrate the applicability of our results.