Nonlinear dynamical systems with multiple equilibrium points: Theory, applications, and emerging perspectives

Abstract

Multiple equilibrium points are a fundamental feature of many nonlinear dynamical systems and often indicate the presence of competing operating regimes, critical transitions, and basin-dependent behavior. This review presents a focused study of benchmark nonlinear systems in which equilibrium multiplicity plays a central role, including chaotic oscillators, Lur’e-type feedback systems, epidemic models, opinion dynamics in social networks, gradient-flow dynamics in nonconvex optimization, and power electronic converters with constant-power loads. The aim is to show how multiple equilibria arise, what they mean in practical systems, and why their presence makes stability, safety, and performance fundamentally dependent on the global structure of the state space. Across the surveyed examples, equilibrium multiplicity is shown to emerge from recurring mechanisms such as nonlinear feedback, saturation, coupling, bounded interactions, switching effects, and nonconvex energy landscapes. These mechanisms can lead to multistability, basin sensitivity, and unwanted regime shifts under disturbances or parameter variations. The review emphasizes that local stability analysis alone is often insufficient for such systems; instead, a global viewpoint based on basins of attraction, separatrices, safety regions, and regime transitions is required. Finally, emerging tools such as Lyapunov and barrier certificates, contraction theory, dominance analysis, and data-driven verification are discussed as promising directions for analysis and design.