Equilibrium and stability of coupled nonlinear energy-storing components.

Abstract

Coupled systems of nonlinear components occur across physics and engineering and can display rich behaviors such as multistability, snap-through, and memory. These phenomena play a key role in physical intelligence, where functional behavior emerges from structure rather than algorithmic control. Yet, predicting the equilibrium states and stability of these systems is challenging due to intricate energy landscapes and the presence of multiple, often disconnected, equilibria. Here, we introduce a general static framework based on a parametric space in which each point represents an element-level equilibrium. In this space, system-level equilibria appear as a one-dimensional manifold, enabling their visualization without the ambiguities that appear in conventional displacement maps. By combining this formulation with a continuation strategy and a local stability assessment, the method traces both stable and unstable equilibria, including isolated isolas inaccessible by standard loading paths. We validate the method experimentally in the mechanical domain using coupled nonlinear-spring and inflatable systems, achieving quantitative agreement between predictions and measurements. This unifying approach offers a powerful tool for the design and analysis of nonlinear coupled systems, enabling systematic exploration of their equilibrium landscapes across disciplines.