Abstract
This study develops a two-dimensional nonlinear dynamical model to analyze the interaction between carbon emissions and renewable energy adoption in the context of Sustainable Development Goal 13 (Climate Action). The model is formulated as a system of coupled differential equations in which carbon emissions grow with economic activity but are mitigated through renewable energy deployment, while renewable adoption follows logistic growth constrained by infrastructural limits and is inhibited by high emission levels. The nonlinear interaction terms capture feedback mechanisms and saturation effects that are commonly observed in real-world energy–climate systems but are not adequately represented by linear models. Analytical investigation identifies three equilibrium points: an unstable trivial equilibrium corresponding to an unsustainable baseline, a stable zero-emission equilibrium associated with complete renewable energy adoption, and an intermediate saddle-type equilibrium representing partial stabilization of emissions. Stability analysis shows that long-term sustainability is achievable only when the efficiency of renewable energy in reducing emissions exceeds the intrinsic emission growth rate. Numerical simulations using representative parameter values illustrate how insufficient policy intervention can trap the system in unstable intermediate regimes, whereas sustained support for renewable expansion can steer the dynamics toward a low-emission equilibrium. The results highlight the importance of nonlinear feedbacks, threshold behavior, and policy consistency in emission–energy transitions. The proposed framework provides qualitative insights into climate–energy dynamics and supports the need for coordinated policy measures, technological advancement, and long-term commitment to achieve stable decarbonization pathways.
Keywords: Carbon Emissions, Nonlinear Modelling, Renewable Energy Adoption, SDG 13, Stability Analysis.