关于Shilnikov-Hopf分岔的Bosch-Simó猜想
The Bosch and Simó conjecture on the Shilnikov-Hopf bifurcation
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中文总结 AI 辅助
本文对Bosch-Simó猜想提供严格数学证明，构建耗散返回映射证明Shilnikov-Hopf分岔中存在持续稳健的大奇异吸引子，刻画了有序与混沌运动的过渡。
中文摘要 AI 辅助
我们对Bosch-Simó猜想（M. Bosch、C. Simó，1993年，《Physica D》，第217-229页）提供了严格的数学证明，该猜想涉及Shilnikov-Hopf分岔展开中奇异吸引子的丰富性。通过构建一类通用的耗散返回映射，这类映射能捕捉这个余维二场景的基本全局几何，我们证明了在Broer-Simó-Tatjer意义下的“大”奇异吸引子是动力学中持续存在且统计稳健的特征。我们证明，对于具有正勒贝格测度的参数集，该系统存在支撑唯一Sinai-Ruelle-Bowen（SRB）测度的吸引子。该证明依赖于通过强横向收缩和奇异角膨胀实现的秩一模型的降维。此外，我们还表明，这种混沌状态中穿插着产生超稳定周期轨道的参数序列，从而刻画了分岔奇异极限下有序运动与混沌运动之间的过渡。
英文摘要
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.
