stabiliteedist

Stabiliteedist is a proposed quantitative measure used in the study of dynamical systems to quantify the robustness of a system’s stability to parameter perturbations. The term blends stability with a notion of distance to a destabilizing change and is intended to provide a single scalar margin for robustness.

Definition and interpretation

For a continuous-time system dx/dt = f(x, p) with an equilibrium x* that is locally stable for nominal parameters p0, the stabiliteedist S(x*, p0) is defined as the infimum of the norm of perturbations Δp such that the perturbed system f(x, p0+Δp) loses stability. Equivalently, if A(p) denotes the Jacobian ∂f/∂x evaluated at x* for parameter p, then S = inf{ ||Δp|| : A(p0+Δp) has an eigenvalue with Re λ ≥ 0 } (or in discrete time, spectral radius ≥ 1). The value depends on the chosen norm and the admissible perturbations considered.

Computation

Calculations typically rely on linearized analysis or robust control methods, including eigenvalue sensitivity, structured singular value (mu) analysis, and numerically optimized perturbations subject to stability constraints. The measure is largely local, reflecting stability properties near the specified equilibrium and under the assumed model.

Applications and limitations

Stabiliteedist can be used to compare robustness across designs, inform controller synthesis, or assess resilience in models from engineering, economics, or ecology. It provides a concise robustness margin that complements Lyapunov-based stability criteria. However, its value depends on the perturbation class and norm chosen, and it may not capture nonlinear global stability or dynamics beyond the neighborhood of the equilibrium. See also distance to instability, stability margin, and robustness analysis.