New topologies in the unfolding of the Doubly Degenerate Bogdanov-Takens bifurcation

High-codimension bifurcations play a key role in shaping the dynamics of nonlinear models, as their unfoldings establish structured relationships between lower-codimension bifurcations and, ultimately, the attractors they generate. Owing to this unifying and predictive capacity, such bifurcations are attracting growing interest in mathematical biology, particularly in neuroscience. One notable example is the codimension-3 Degenerate Bogdanov-Takens (DBT) bifurcation, proposed as an organizing center for neural dynamics and playing a key role in bursting. The DBT itself arises within the unfolding of an even higher-order bifurcation, the Doubly Degenerate Bogdanov-Takens (DDBT), whose structure remains only partially understood, despite initial indications of its relevance in neural dynamics. In this work, we expand the numerical investigation of the DDBT unfolding through the use of spherical surfaces, corroborating the conjecture that it connects DBT to a highly symmetric codimension-3 bifurcation, but through transitions that partly differ from those proposed in the literature. Notably, one key intermediate passage remains unsolved. We then show that using planes to explore the unfolding allows for new bifurcation topologies and transitions that cannot be found on spheres. We illustrate how transitions to topologies supporting excitability and fold/homoclinic bursting, two biologically relevant behaviors, differ across spheres and planes. Overall, the additional bifurcation structures enrich our understanding of what models with a DDBT bifurcation can do and how these dynamics are related.