IDENTIFYING UNSTABLE EQUILIBRIA IN MULTI-STABLE DYNAMICAL SYSTEMS USING A LOCAL REGRESSION METHOD
Abstract
Multistability and unstable equilibria play a crucial role in nonlinear dynamical systems, from discrete and continuous mechanical structures to non-mechanical systems such as immune and ecosystems. However, locating unstable equilibria or identifying multistability in an unknown dynamical system remains challenging. Here, we propose a novel, iterative, and self-validated numerical approach to identify unstable equilibria and construct a bifurcation diagram using only transient responses. No prior knowledge or global mathematical model (e.g., PDEs or FEA) is required. The approach constructs local polynomial models in the vicinity of each equilibrium, accurately predicts their locations, delineates the valid region for each local model, and enables stability assessment via eigenvalue analysis. Both linear and quadratic regression algorithms are presented. The approach has been validated on a multi-DoF discrete mass-spring system, a continuous bistable structure, and a Lotka-Volterra type ecosystem. It demonstrates strong effectiveness and generalizability, successfully locating multiple equilibria with one or more negative eigenvalues and ultimately constructing a local bifurcation diagram in the state space that reveals which stable state the system will approach for given initial conditions.
Publication Title
ASME International Mechanical Engineering Congress and Exposition Proceedings Imece
Recommended Citation
Guan, Y. (2025). IDENTIFYING UNSTABLE EQUILIBRIA IN MULTI-STABLE DYNAMICAL SYSTEMS USING A LOCAL REGRESSION METHOD. ASME International Mechanical Engineering Congress and Exposition Proceedings Imece, 5-A https://doi.org/10.1115/IMECE2025-164467
