Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms

TL;DR

This paper develops a systematic method for analyzing and controlling bifurcations in nonlinear singular systems, especially Hopf-zero singularities, using universal asymptotic unfolding normal forms and Maple implementation.

Contribution

It introduces universal asymptotic unfolding normal forms for nonlinear singular systems and applies them to Hopf-zero singularities, enabling systematic bifurcation analysis and control design.

Findings

Derived novel orbital and parametric normal forms for Hopf-zero singularities.

Proved finite determinacy of steady-state bifurcations for key subfamilies.

Designed a multiple-parametric quadratic state feedback controller for a three-dimensional singular system.

Abstract

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.