Recently, we presented three low-order Energy Balance Models (EBMs) to propose a definition of climate tipping processes consisting of a bifurcation followed by a subsequent transition. These models were formulated as first-order ordinary differential equations comprising three key components: a time-varying forcing term, a linear feedback term, and a nonlinear quadratic or cubic term. Among these systems, the cubic EBM, which admits a double-well potential structure, represents a minimal framework for capturing regime coexistence, bifurcations, and tipping behavior. In this study, we further establish explicit mathematical connections between the cubic EBM and a class of higher-order dynamical systems, including the cubic oscillator and the nondissipative Lorenz model, both of which support oscillatory solutions. These connections highlight a shared potential-based structure underlying seemingly disparate systems. We also demonstrate the physical relevance of the cubic nonlinear term by linking the cubic EBM to the ice–albedo feedback model. Through a systematic reduction from a smooth hyperbolic-tangent albedo formulation, we show that the cubic nonlinearity represents the leading-order saturation mechanism that introduces an effective nonlinear negative feedback. In addition, we examine how bistability — a defining feature of the cubic EBM — also emerges in more comprehensive climate models, such as ice-sheet models, despite their higher dimensionality and complexity. This comparison underscores the value of the cubic EBM as a normal-form representation for understanding tipping points, regime shifts, and path dependence across a broad hierarchy of climate and geophysical models. Beyond the forward reduction pathway emphasized in this study, we outline backward embedding (upscaling) strategies for extending the cubic EBM to more sophisticated yet dynamically traceable systems, thereby providing a systematic route for linking idealized EBMs with higher-order climate models.
Bo-Wen Shen (Sat,) studied this question.