Topological foundation for the steady-state hypothesis in generalized modeling

Abstract

Generalized modeling provides a powerful framework for the local analysis of broad classes of nonlinear dynamical systems without fixing specific functional forms. A foundational step in this approach is the assumption that a steady state exists, followed by the normalization of the state variables and the construction of the Jacobian in terms of generalized parameters. In this paper, we provide a rigorous justification for this assumption across a broad class of finite-dimensional models. Under natural boundary sign conditions on a compact hyperrectangle, the Poincaré–Miranda theorem guarantees the existence of at least one steady state. We demonstrate how these conditions can be routinely verified in compartmental systems via donor-vanishing and upper-face dominance criteria and we derive a Brouwer-degree refinement that establishes topological robustness under admissible perturbations when the boundary inequalities are strict. The general theory is illustrated using representative examples from physics and biology, including a two-patch metapopulation model, an open SEIR system with immigration and a coupled-laser model. Together, these results provide a rigorous topological foundation for the steady-state hypothesis commonly used in generalized modeling, offering a highly practical criterion for verifying it in applications.

Under natural boundary sign conditions on a compact hyper rectangle, the Poincaré–Miranda theorem guarantees the existence of at least one steady state \(x^*\) in generalized gain-loss dynamical systems without requiring specific functional forms. The inward-pointing components of the vector field along opposite boundary faces (\(f_i \ge 0\) on lower faces \(F_i^-\) and \(f_i \le 0\) on upper faces \(F_i^+\)) provide a rigorous, model-independent topological foundation and degree-theoretic robustness for the steady-state hypothesis across physical, ecological, and epidemiological networks