arXiv subjects

Rudra Prakash

Publications and source records attributed to Rudra Prakash.

3 recordsLinked to original sources

arXiv2026

Comparing Point and Interval Methods for Equilibrium Computation under Parametric Uncertainty

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.

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arXiv2026

Certified Detection of Bifurcation Candidates in Uncertain Nonlinear Systems using Interval Analysis

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.

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arXiv2026

Computational Complexity Analysis of Interval Methods in Solving Uncertain Nonlinear Systems

This paper analyzes the computational complexity of validated interval methods for uncertain nonlinear systems and steady-state enclosure. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes explicit the dependence on the problem dimension $n$, the initial search region size $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivision$+$filter, interval constraint propagation, interval Newton, and interval Krawczyk, and identify dominant cost drivers. We also show that the computation of the determinant and inverse of interval matrices via naive Laplace expansion exhibits factorial growth with increasing matrix dimension, motivating specialized interval linear algebra. We complement the worst-case bounds with computational results on two application-motivated biochemical steady-state models (a Hill-type regulatory network and an enzyme-saturation-based winner-take-all circuit) in dimensions $n\in\{2,5,10\}$, including instances that process millions of boxes. The resulting analysis and experiments support the practical design of validated solvers for uncertainty-aware steady-state screening tasks such as robust operating-point certification and multistability assessment.

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