[Paper Review] Parametric normal form classification for Eulerian and rotational non-resonant double Hopf singularities

Majid Gazor, Ahmad Shoghi|arXiv (Cornell University)|Dec 30, 2018

Quantum chaos and dynamical systems39 references4 citations

TL;DR

This paper presents a novel parametric normal form classification for Eulerian and rotational vector fields with non-resonant double Hopf singularities, using Lie bracket methods, time rescaling, and block matrix analysis. The key contribution is a symmetry-preserving infinite-level normal form that retains structural Eulerian properties and enables robust bifurcation analysis and control design for multiple-input systems with arbitrary symbolic coefficients.

ABSTRACT

In this paper we provide novel results on the infinite level normal form and orbital normal form classifications of nonlinear Eulerian and rotational vector fields with two pairs of non-resonant imaginary modes. We use the method of multiple Lie brackets and its extension along with time rescaling for orbital normal form classification. Furthermore, we apply two reduction techniques. The first is to use the radical Lie ideal of rotational vector fields and its corresponding quotient Lie algebra. The second technique is to employ a Schur complement block matrix type in Gaussian elimination and analysis of block matrices. The infinite level parametric normal form classification are also presented. The latter is also viewed as a normal form result for multiple-input controlled systems with non-resonant double Hopf singularity. We also discuss nonlinear symmetry transformations associated with the nonlinear symmetry group of the simplest normal forms. Symbolic normal form transformation generators are derived for computer algebra implementation. Further, the results are efficiently implemented and verified using Maple for all three types of normal form computations up to arbitrary degree, where they can also include both small bifurcation parameters and arbitrary symbolic constant coefficients.

Motivation & Objective

• To develop a symmetry-preserving normal form classification for Eulerian and rotational vector fields with non-resonant double Hopf singularities.
• To address the limitations of truncated classical normal forms that destroy Eulerian structural symmetry and misrepresent dynamics.
• To provide a parametric normal form framework that serves as a universal unfolding for multiple-input controlled systems with non-resonant double Hopf singularities.
• To enable symbolic computer algebra implementation of normal forms up to arbitrary degree, including small bifurcation parameters and symbolic coefficients.
• To derive symbolic generators for normal form transformations for use in computational algebra systems like Maple.

Proposed method

• Application of the method of multiple Lie brackets and its extension to classify orbital and infinite-level normal forms.
• Use of time rescaling to facilitate orbital normal form classification in the presence of rotational symmetry.
• Employment of the radical Lie ideal of rotational vector fields and its quotient Lie algebra to simplify classification.
• Utilization of Schur complement block matrix techniques and Gaussian elimination for block matrix analysis in normal form computation.
• Derivation of symbolic normal form transformation generators for implementation in computer algebra systems such as Maple.
• Development of a parametric normal form framework that preserves structural symmetry through all normalization steps.

Experimental results

Research questions

• RQ1 How can the infinite-level normal form of a nonlinear Eulerian and rotational system with non-resonant double Hopf singularity be classified while preserving its structural symmetry?
• RQ2 What is the parametric normal form of a multiple-input system with non-resonant double Hopf singularity, and how does it serve as a universal unfolding?
• RQ3 How can the normal form classification be made robust against small perturbations and errors in real-world control applications?
• RQ4 What symbolic transformation generators can be derived to enable efficient computer algebra implementation of the normal form up to arbitrary degree?
• RQ5 How do nonlinear symmetry transformations of the simplest normal forms relate to the structure of the parametric normal form?

Key findings

• The infinite-level normal form of the system is uniquely determined and takes the form $\dot{z_i} = \sum_{j+k=0}^{2} c^i_{j,k}|z_1|^{2j}|z_2|^{2k}z_i + \sum_{k\geq 3} c^i_{0,k}|z_2|^{2k}z_i$, with $c^i_{j,k} = b_{j,k} + I a^i_{j,k}$, preserving the Eulerian structure.
• The coefficients $b_{2,1}$ and $b_{1,2}$ are explicitly derived in terms of system parameters, including $\omega_1, \omega_2$, and nonlinear coupling terms $a_i$, with complex rational dependence on frequency ratios and symmetry parameters.
• The parametric normal form is shown to be a universal unfolding of the original system, enabling robust bifurcation analysis and control design under small perturbations.
• The method successfully preserves the Eulerian and rotational symmetries throughout normalization, avoiding the structural distortion common in truncated classical normal forms.
• The symbolic normal form generators are implemented in Maple, enabling computation of normal forms up to arbitrary degree with both symbolic coefficients and small bifurcation parameters.
• The results are verified through symbolic computation, demonstrating the feasibility and accuracy of the proposed framework for high-degree normal form analysis.