Singular transcritical Bogdanov–Takens organising centre

Establish whether the spatial and parametric collision of the singular transcritical bifurcation with the singular Bogdanov–Takens/singular Andronov–Hopf structure in the five-compartment Kuznetsov tumour–immune model constitutes a singular transcritical Bogdanov–Takens bifurcation, and derive its complete topological unfolding under variation of the effector-cell supply rate α₁, effector-cell death rate β₁, and singular perturbation parameter ε.

Background

The paper identifies a singular transcritical bifurcation and a nilpotent pseudo-singularity associated with a singular Andronov–Hopf bifurcation near the origin of the blown-up tumour–immune system. As the parameters approach the singular limit, these structures collide spatially and parametrically. The authors conjecture that this collision forms a new singular transcritical Bogdanov–Takens organising centre, which would organise the nearby bifurcation geometry involving tumour-free and tumour-present states.

The paper provides a local modified Bogdanov–Takens normal form and describes the relevant singular transcritical and singular Andronov–Hopf phenomena, but it does not establish the full topological classification. A complete resolution requires a simultaneous geometric parameter blow-up in ε, α₁, and β₁ and a rigorous analysis of the resulting unfolding.