AI 中文总结
本文比较了参数不确定下均衡计算的点方法与区间方法，以生物分子电路模型为测试对象，发现区间方法可提供严格均衡边界，而点方法易遗漏多稳态。
AI 中文摘要
均衡点定义了非线性动力学与控制系统的运行条件，其在参数不确定下的存在性、多重性与稳定性决定了可行运行区间及鲁棒性声明的有效性。当参数被约束在有界集合中时，可采取三类方法：(i)计算采样参数值下的均衡点；(ii)沿参数空间的指定路径追踪均衡点；(iii)识别给定运行域中至少对一个容许参数实现为均衡的状态。本文将标准的点式工作流——直接仿真、数值延拓、残差最小化及多起点牛顿-拉夫逊方法——与经验证的基于区间分析的工作流进行比较。后者(a)可对固定参数提供均衡点的排除、存在性与唯一性的形式化证明，且在参数包含条件下，可对整个参数盒提供均匀的此类证明；(b)在状态空间中构造严格的外包围盒，可证明其包含与全部容许参数集相关的所有均衡点。以非线性常微分方程描述的生物分子电路模型为代表性应用领域，我们在四个参数不确定水平上对三种经典架构进行基准测试，包括接近对称破缺分岔的基因双稳态开关。基于采样与切片的方法可能遗漏或低估多稳态，而基于区间的外包围盒可得到参数不确定诱导的均衡集的数学严格边界。
英文摘要
Equilibrium points define operating conditions for nonlinear dynamical and control systems. Their existence, multiplicity, and stability under parametric uncertainty determine feasible operating regimes and the validity of robustness claims. With parameters constrained to a bounded set, one can (i) compute equilibria at sampled parameter values, (ii) trace equilibria along a prescribed path in parameter space, or (iii) identify states in a given operating domain that are equilibria for at least one admissible parameter realization. We compare standard pointwise workflows-direct simulation, numerical continuation, residual minimization, and a multistart Newton-Raphson method-with validated interval-analysis-based workflows. The latter (a) provide formal certificates of exclusion, existence, and uniqueness of equilibria for fixed parameters and, under parametric inclusion conditions, uniformly over entire parameter boxes, and (b) construct rigorous outer enclosures in state space that provably contain all equilibria associated with the full admissible parameter set. Biomolecular circuit models governed by nonlinear ODEs serve as a representative application domain. We benchmark three canonical architectures across four levels of parameter uncertainty, including a genetic toggle switch near a symmetry-breaking bifurcation. Sampling- and slice-based approaches can miss or underrepresent multistability, whereas interval-based outer enclosures yield mathematically rigorous bounds on the equilibrium set induced by parametric uncertainty.
Comments6 pages, 1 figure, 2 tables
