Chaotic dynamics analysis and control of fractional-order ESER systems using a high-precision Caputo derivative approach

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.