Normal forms of double Hopf oscillators with radial nonlinearities

Abstract

We are concerned with the simplest normal forms of non-resonant double Hopf oscillators with radial nonlinearities. Due to the required Lie algebraic structure for preserving the radial structure in the normal form process, we appeal to the Lie algebra of radial and rotational transformation generators. The structure constants of the Lie algebra are presented, where we notice a Lie ideal structure for the rotational vector fields. This gives rise to an effective reduction technique through the use of its corresponding quotient Lie algebra. Yet, computations of kernel spaces of the generalized homological maps seem a formidable project and thus, we employ the multiple Lie brackets method to skip explicit kernel space derivations. The simplest normal forms and orbital normal forms of the nonlinear family are presented. We also address their parametric normal forms as multiple parametric perturbations of the corresponding vector fields. Transformation symmetry groups associated with truncated normal forms are also derived. An efficient algorithm is provided for the normal form computations using computer algebra systems. Finally, we derive state-feedback controllers for a two-arm vibrator using normal forms of radial double Hopf singularity. This is to illustrate how our normal forms can be applied on classical control engineering problems as a new alternative nonlinear controller technique to the existing methods in nonlinear control theory.