Bifurcation, Tipping, and Regime Shifts in Nonlinear Energy-Balance Models

This study develops a mathematical framework based on the potential-function approach for interpreting tipping points and regime shifts in nonlinear dynamical systems through the Energy-Balance Model (EBM) and its nonlinear extensions. Building on classical bifurcation theory, the paper defines a tipping process as a coupled evolution involving (1) a bifurcation — the merging or loss of equilibria — and (2) a transition from one stable state to another. Using potential-function analysis, bifurcations are identified where both the slope and curvature of the potential vanish, providing a clear geometric picture of the merging of local extrema at an inflection point where stability is lost. Three representative EBMs are analyzed. The linear EBM [Type (I)] shows a continuous, trend-following regime shift. The quadratic EBM [Type (II)] exhibits an abrupt, globally unstable tipping. The cubic EBM [Type (III)] reveals global bistability, hysteresis, and critical slowing down. After tipping, the type (II) and (III) systems transit to an unstable and stable state, respectively. The analysis connects geometric curvature, dynamic dissipation, and irreversible transitions within a unified potential-landscape perspective. Comparisons with the nondissipative Lorenz model reveal that forced, dissipative regime shifts and conservative turning-point dynamics share analogous barrier-crossing structures. Overall, the proposed classification clarifies the mechanisms underlying tipping phenomena and provides a rigorous foundation for interpreting nonlinear transitions and predictability in climate and other complex systems.