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Complete rigorous theory for codimension-two bifurcations in state-dependent delay equations

Develop a complete bifurcation theory for codimension-two bifurcations in retarded functional differential equations with state-dependent delays, including existence of sufficiently smooth center manifolds and normal-form reductions necessary to classify and analyze these bifurcations.

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Background

The paper notes that Hopf bifurcation results for state-dependent delays have only recently been established, whereas numerical packages such as DDE-Biftool already detect more complex (codimension-two) bifurcations. However, a rigorous theoretical framework for these higher-codimension phenomena remains incomplete, primarily due to unresolved smoothness of the center manifold.

This gap limits the formal justification for numerical continuation and normal-form analyses of codimension-two phenomena in systems with state-dependent delays.

References

On the other hand the numerical bifurcation and continuation package DDE-Biftool has had the ability to detect Hopf bifurcations since before those results were derived, and can also find more complicated co-dimension two bifurcations for which there is still not a complete theory because of the lack of smoothness of the centre manifold.

— Practicalities of State-Dependent and Threshold Delay Differential Equations (2510.17126 - Humphries et al., 20 Oct 2025) in Section: Delay Differential Equations as Dynamical Systems → Invariant Sets and Bifurcations