Full investigation of the dynamics of low-wavenumber normal forms

Conduct a comprehensive investigation of the dynamics of the O(3)-equivariant normal-form amplitude equations for small wavenumbers (e.g., ℓ = 1, 2, 3) arising from bulk-surface reaction-diffusion systems in a ball, including systematic analysis of invariant subspaces, steady-state branches, stability, and global phase portraits.

Background

Section 3 presents explicit normal forms for ℓ = 1, 2, and 3, discusses invariant manifolds, and derives some steady-state conditions, but emphasizes that a full dynamical analysis is not available—even for these low-dimensional cases.

Completing this analysis would clarify the range of possible behaviors (steady, periodic, or more complex) inherent to these normal forms and strengthen the connection between weakly nonlinear predictions and observed bulk-surface pattern dynamics.