Systematic study of the Bogdanov–Takens–cusp (codimension-three) bifurcation in state-dependent delay systems

Investigate and develop a systematic theoretical framework for the codimension-three Bogdanov–Takens–cusp bifurcation in delay differential equations with state-dependent delays, including classification, normal-form derivation, and unfolding scenarios consistent with observed proximity of cusp and BT points.

Background

The threshold-delay example exhibits cusp and BT structures in close proximity, suggesting the presence of a Bogdanov–Takens–cusp point. The authors note a lack of systematic paper of this codimension-three bifurcation in the state-dependent delay context.

A systematic theory for BTC in state-dependent delays would unify and explain the observed bifurcation diagrams, complementing existing constant-delay and neuronal-model studies.