AI 中文总结
本研究提出一种基于区间分析的经验证区间工作流，可认证不确定非线性系统中鞍结、霍普夫分岔候选点的存在性或不存在性，为鲁棒分析设计提供严格保证。
AI 中文摘要
非线性动力系统中的定性转变（如稳定性丧失、振荡起始、多稳态出现）划定了工作区间，且会作为参数不确定性下鲁棒分析与设计的隐式约束出现。当参数由数据推断时，可容许值自然表示为不确定性集合，这催生了对工作区间转变候选点存在与否的认证测试。我们提出一种经验证的区间工作流，其将鞍结和霍普夫候选点条件编码为增广代数方程组，并应用Krawczyk算子，在规定的状态-参数盒上认证：要么（i）候选解的存在性与局部唯一性，要么（ii）经认证的不存在性。对不确定合成基因网络常微分方程模型的数值实验，在双参数切片上为双稳态电路生成了经局部认证的鞍结候选点包围域，还借助劳斯-赫尔维茨特化方法为三状态振荡器生成了经认证的霍普夫候选点包围域。所得证书旨在通过补充非验证的逐点基线（如离散参数值处的牛顿法求解）和基于采样的工作流，为用户指定的参数切片提供严格的存在/不存在保证，从而支持有界不确定性下的工作区间感知分析与设计。
英文摘要
Qualitative transitions in nonlinear dynamical systems (e.g., loss of stability, onset of oscillations, emergence of multistability) delimit operating regimes and can arise as implicit constraints in robust analysis and design under parametric uncertainty. When parameters are inferred from data, admissible values are naturally represented as uncertainty sets, motivating certified tests for the presence or absence of regime-transition candidates. We propose a validated interval workflow that encodes saddle-node and Hopf candidate conditions as square augmented algebraic systems and applies the Krawczyk operator to certify, over a prescribed state-parameter box, either (i) existence and local uniqueness of a candidate solution or (ii) certified absence. Numerical experiments on uncertain synthetic gene-network ODE models yield locally certified saddle-node candidate enclosures on a two-parameter slice for a bistable circuit and certified Hopf candidate enclosures for a three-state oscillator using a Routh-Hurwitz specialization. The resulting certificates are intended to support regime-aware analysis and design under bounded uncertainty by complementing non-validated, pointwise baselines (e.g., Newton method solves at discrete parameter values) and sampling-based workflows with rigorous presence/absence guarantees on user-specified parameter slices.
Comments6 pages, 3 figures, 3 tables
