Many systems experiencing critical transitions in the real world—such as financial crises, population collapses, and climate change—can be viewed as dynamical systems exhibiting saddle-node bifurcation characteristics. In these systems, the bifurcation point (also known as the critical point) corresponds to a fold catastrophe, and the bifurcation parameters typically change over time, evolving gradually. The rate of these changes can be described by a sweep rate constant. Therefore, the emergence of critical points can be seen as the result of fold catastrophes driven by time-dependent parameter changes.

Such critical transitions pose serious challenges to global sustainable development, particularly concerning extreme climate phenomena, rising sea levels, and environmental changes highlighted by SDG 13 (Climate Action); the resilience of urban infrastructure addressed by SDG 11 (Sustainable Cities and Communities); and the polar glacier changes and ocean dynamics covered by SDG 14 (Life Below Water). If appropriate interventions are not taken before the critical points are reached, the drastic changes triggered by these transitions can have profound impacts on economic stability, environmental sustainability, and public health. Thus, understanding the mechanisms and timing of critical transitions is vital for achieving the Sustainable Development Goals.

This study explores the saddle-node bifurcation problem in discrete systems, aiming to enhance the accuracy of numerical analysis methods to support climate change forecasting and decision-making. We found that if a continuous dynamical system is discretized without matching the time step size to the order of magnitude of the system parameters, the estimation of critical points can be significantly biased. This research precisely describes the relationship between discrete time steps and small parameters, and rigorously proves the asymptotic behaviors of the discrete numerical solutions over different intervals. These findings hold significant value for numerical models in climate science, environmental monitoring, and disaster warning systems.

In fields such as ecology, oceanography, and other sustainable development applications, accurate numerical analysis helps strengthen strategies for climate change adaptation. For example, in the study of polar ice sheet dynamics, predicting the critical points of glacier collapse is crucial for assessing the rate of sea level rise (SDG 13, SDG 14); in the arenas of urban infrastructure and disaster response, dynamical system models can be used to simulate the impacts of climate change on water supply, power grids, and agricultural production (SDG 11).

This study demonstrates that when dealing with real-world problems, if the intervals of discretized data are not adequately fine, the computed critical points may lack reliability, thus affecting the accuracy of the predictions. This finding not only provides a new perspective for numerical analysis of dynamical systems but also offers important theoretical support for addressing climate change, disaster risk management, and sustainable development, aligning with the core spirit of the United Nations Sustainable Development Goals (SDGs).