Farman, MuhammadJamil, KhadijaJamil, SabaNisar, Kottakkaran SooppySmerat, AseelHafez, Mohamed2026-03-152026-03-1520262363-62032363-621110.1007/s40808-026-02750-72-s2.0-105031495398https://doi.org/10.1007/s40808-026-02750-7https://hdl.handle.net/20.500.14517/8915This study presents an innovative fractional order (APC) model developed to more accurately depict the memory-dependent dynamics of atmospheric systems, which are inadequately captured by traditional integer-order formulations. The model uses the Caputo fractional derivative to show how general air A(t), polluted air P(t), and clean air C(t) affect each other. The mathematical coherence of the model suggested is proved by the detailed analysis. The first step is to prove the existence and uniqueness of solutions based on fixed-point theory to prove the well-posedness of the model. Moreover, it is proved beyond doubt that the positive equilibrium point of the global system is asymptotic stable through the direct method of Lyapunov, which implies that the system is predisposed to reaching a steady point irrespective of the initial conditions. A chaos control strategy is effectively applied to stabilize the system in order to deal with any possible erratic behavior. A suitable numerical model is designed to address the fractional-order system, and simulations at different levels of the order of fractional orders give an important observation that in a lower fractional order, there are protracted dynamical processes in the system, which is presented through the sluggish increase of pollution and the slow restoration of clean air. This directly measures the effect of memory, one of the most critical characteristics that cannot be realized in classical models. The study concludes that the fractional-order APC model provides a superior, robust, and theoretically sound framework for understanding and predicting long-term air pollution behavior, offering valuable insights for environmental policy and sustainable air quality management.eninfo:eu-repo/semantics/closedAccessCaputo DerivativeLyapunov FunctionPhase Surface SimulationsChaos ControlArticle