Systematic study of Bogdanov–Takens bifurcations in state-dependent delay DDEs

Conduct a systematic bifurcation analysis of Bogdanov–Takens points in delay differential equations with state-dependent delays, including identification, normal forms, and unfolding, extending existing constant-delay results to the state-dependent case.

Background

In the scalar threshold-delay example, the authors observe structures indicative of Bogdanov–Takens dynamics and refer to recent analyses for constant-delay DDEs. They explicitly note the absence of systematic studies for the state-dependent delay setting.

A systematic framework would address how state dependence modifies local and global BT dynamics, including the role of threshold-defined delays and the associated velocity ratio terms.

References

Bogdanov-Takens bifurcations have recently been analyzed for constant delay DDEs in , %(where Figures 5 and S8 resemble the part of Figure~\ref{fig:gamma4ltmultgamma3gt_BT} close to the BT point), but we are not aware of any systematic study of them in state-dependent DDE problems.

— Practicalities of State-Dependent and Threshold Delay Differential Equations (2510.17126 - Humphries et al., 20 Oct 2025) in Section: Examples → Scalar Threshold Delay Example (discussion of BT and cusp points)