Dynamical Analysis and Control of a Carbon Credit Investment System with Chaotic Behavior and Fractional-Order Memory

Abstract

This study investigates the dynamic behavior of a carbon credit investment system using nonlinear analysis, fractional-order modeling, and active control techniques. The analysis focuses on parameters associated with the demand for green investments and the impact of the carbon tax rate. System dynamics are characterized through bifurcation diagrams, phase portraits, time series, and Lyapunov exponents, enabling the identification of distinct dynamic regimes, including periodic and chaotic behaviors. To incorporate memory effects inherent in economic variables, fractional-order formulations based on the Riemann–Liouville operator are employed, revealing that the order of the fractional derivative significantly influences the system's dynamics. To stabilize the system and suppress chaotic behavior, a linear controller of the Linear Quadratic Regulator (LQR) type and a nonlinear controller based on the State-Dependent Riccati Equation (SDRE) are designed. The results demonstrate that both strategies successfully drive the system toward desired equilibrium states while exhibiting low tracking errors. Furthermore, addressing scenarios where not all state variables are directly available for feedback, state observers are utilized to estimate unmeasured variables and supply the necessary information to the controllers. The findings provide a quantitative basis for understanding the complex dynamics associated with carbon credit investments and highlight the applicability of control strategies for system stabilization and the mitigation of undesirable dynamic behaviors.