Callahan’s conjecture on the number of stable steady patterns for ℓ ≤ 6

Establish whether only ten topologically distinct stable steady patterns can arise for wavenumbers ℓ ≤ 6 in O(3)-equivariant Turing bifurcation normal forms, as conjectured by Callahan, and assess implications for the amplitude equations of bulk-surface reaction-diffusion systems in a ball.

Background

The paper connects the derived amplitude equations to previously studied O(3)-equivariant normal forms for patterns on the sphere. In this context, Callahan formulated a conjecture limiting the number of distinct stable steady patterns for ℓ ≤ 6.

Verifying or refuting this conjecture in the present bulk-surface setting would clarify the landscape of stable patterns and guide expectations for which symmetry types can emerge and persist near Turing bifurcations.