This paper investigates a three-species reaction-diffusion model with fractional-order derivatives and proposes an innovative numerical method for its simulation. The method integrates an optimized Grünwald-Letnikov discretization scheme, enhanced by a short-memory principle, with a high-order accurate nine-point compact difference scheme, enabling an efficient and stable solution of fractional operators. Rigorous convergence and stability analyses confirm the theoretical reliability of the algorithm. Through stability and Turing bifurcation analyses, the study systematically reveals the regulatory mechanism of the fractional-order exponent on the dynamic behavior of the system. The numerical results demonstrate that the present method accurately captures the effect of fractional derivatives on the formation process of spatial patterns.

This paper investigates the chaotic dynamics and control of a fractional-order energy-saving and emission-reduction (ESER) system using a high-precision method. By constructing a four-dimensional fractional-order system, the complex interactions between new energy, carbon emissions, economic growth, and carbon trading volume are studied. A novel high-precision numerical method, based on generating functions, is introduced to analyze the dynamic behaviors of the system at various fractional derivatives and parameters. Numerical simulations find all kinds of dynamic behaviors, including stable, rapid divergence, and chaotic attractors. In this study, we design feedback and adaptive control strategies to achieve chaos synchronization and system stability. The results highlight the effectiveness of the proposed control methods in stabilizing the system and synchronizing drive-response systems, underscoring the pivotal role of fractional-order derivatives in regulating system dynamics.