The Bosch and Simó conjecture on the Shilnikov-Hopf bifurcation

Abstract: We provide a rigorous mathematical proof of the Bosch-Simó conjecture ( M. Bosch, C. Simó (1993), Physica D, 217--229) regarding the abundance of strange attractors in the unfolding of a Shilnikov-Hopf bifurcation. By constructing a general class of dissipative return maps that capture the essential global geometry of this codimension-two scenario, we demonstrate that "large" strange attractors -- in the sense of Broer-Simó-Tatjer -- are a persistent and statistically robust feature of the dynamics. We prove that, for a parameter set of positive Lebesgue measure, the system admits attractors supporting a unique Sinai-Ruelle-Bowen (SRB) measure. The proof relies on a dimensional reduction to a rank-one model, enabled by strong transverse contraction and singular angular expansion. Furthermore, we show that this chaotic regime is interspersed with sequences of parameters yielding superstable periodic orbits, providing a characterisation of the transition between ordered and chaotic motion in the singular limit of the bifurcation.