Analytical location of Shilnikov–Hopf bifurcations near triple-zero singularities

Determine analytically the location of Shilnikov–Hopf bifurcations and their associated large strange attractors within the unfolding of singularities with triple zero eigenvalues.

Background

The paper studies a codimension-two Shilnikov–Hopf configuration and establishes large strange attractors for a specific return-map framework. It then identifies a higher-codimension setting involving singularities with triple zero eigenvalues, where the analytical placement of the bifurcations and their associated attractors has not been resolved. The authors explicitly describe this as an open challenge and defer it to future work.

References

This study provides a systematic methodology for characterising observable chaos in dissipative systems near a homoclinic network. Building upon the foundational observations in , a prominent open challenge remains the analytical location of Shilnikov--Hopf bifurcations and their associated ``large'' strange attractors within the unfolding of singularities with triple zero eigenvalues (cf. Eq. (28)).

— The Bosch and Simó conjecture on the Shilnikov-Hopf bifurcation  (2608.13021 - Rodrigues, 13 Aug 2026) in Section “Discussion and concluding remarks,” subsection “Concluding remarks”