Bifurcation Control and Universal Unfolding for Hopf-Zero Singularities with Leading Solenoidal Terms

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 the 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 primary bifurcations of equilibria and 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.

Bifurcation Controller Designs for the Generalized Cusp Plants of Bogdanov--Takens Singularity with an Application to Ship Control

“…We remark that a different unfolding leads to slightly different dynamics than the case presented here; for example see the transition set in figure 9 and the bifurcation varieties given in figure 13(a). Yet our bifurcation control analysis is sufficient for a comprehensive study of all these cases; also see [20,Proposition 6.2].…”

“…The condition r = s = 2 and a simple Maple programming imply that the parameters (µ 1 , µ 3 ) and (µ 3 , µ 4 ) can play the role of the distinguished parameters (asymptotic unfolding parameters), i.e., ν 2 and ν 3 are diffeomorphic polynomials in terms of µ 1 and µ 3 ; see [20]. We choose d 2 := 1, d 3 := d 6 := b, and set…”

“…Each change in the qualitative dynamics of a singular system in the vicinity of an equilibrium is called a local bifurcation. The most common local bifurcations in nonlinear planar systems are the changes in the number of equilibria, limit cycles, homoclinic and heteroclinic cycles, and changes in their stabilities and/or the existence of bi-stabilities; e.g., see [12,13,20,23,28,36]. An important challenging problem in engineering control is to predict and locate possible trajectories of these types in an engineering singular plant.…”

“…This analysis provides a prediction of possibilities in the system's qualitative dynamics, but it does not represent an actual qualitative and quantitative dynamics in terms of the original input parameters of the system. Hence, the only useful normal form approach for their possible engineering applications is through parametric normal forms and this has been recently addressed for a few cases: Hopf, single zero, the generalized saddle-node case of Bogdanov-Takens, and a case of Hopf-zero singularity; see [10,17,20,21,45]. In this paper we treat the generalized cusp case of Bogdanov-Takens singular systems.…”

“…This is called control bifurcations; see [1,24,25,27,29]. Thus, this paper investigates the use of a recently introduced notion of truncated universal asymptotic unfolding normal form [20] for bifurcation control of a family of Bogdanov-Takens singularity. (A truncated universal asymptotic unfolding normal form here refers to a suitable truncation of the simplest parametric normal form.)…”