Speaker
Description
High-codimension bifurcations critically contribute to 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.
One notable example is the codimension-3 Degenerate Bogdanov-Takens (DBT) singularity, proposed in the literature as an organizing center for neural dynamics, and playing a key role in fast-slow bursting. The DBT itself arises within the unfolding of an even higher-order singularity, the Doubly Degenerate Bogdanov-Takens (DDBT), whose structure remains only partially understood \cite{khibnik1998global,krauskopf2016codimension}.
We numerically explore the DDBT four-dimensional unfolding using spheres in parameter space, highlighting similarities and differences with existing conjectures. We show that using planes to explore the unfolding allows for additional bifurcation topologies. We reference the literature for examples of how these topologies emerge in neural dynamics.
Even though one key transition remains to be solved to complete the puzzle, the intermediate configurations and additional bifurcation structures reveal a rich repertoire of dynamical behaviors. These results improve our understanding of what models with a DDBT point can do and how these behaviors are related \cite{saggio2025new}.
Bibliography
@article{khibnik1998global,
title={Global study of a family of cubic Li{\'e}nard equations},
author={Khibnik, Alexander I and Krauskopf, Bernd and Rousseau, Christiane},
journal={Nonlinearity},
volume={11},
number={6},
pages={1505--1519},
year={1998}
}
@incollection{krauskopf2016codimension,
title={A codimension-four singularity with potential for action},
author={Krauskopf, Bernd and Osinga, Hinke M},
booktitle={Mathematical Sciences with Multidisciplinary Applications: In Honor of Professor Christiane Rousseau. And In Recognition of the Mathematics for Planet Earth Initiative},
pages={253--268},
year={2016},
publisher={Springer}
}
@article{saggio2025new,
title={New topologies in the unfolding of the Doubly Degenerate Takens-Bogdanov singularity},
author={Saggio, Marisa},
journal={bioRxiv},
pages={2025--11},
year={2025},
publisher={Cold Spring Harbor Laboratory}
}
