Comparing Point and Interval Methods for Equilibrium Computation under Parametric Uncertainty

Abstract

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.