Chaos management and synchronization in the Liu financial system via sliding mode and passive control

The Liu financial dynamical system is a chaotic model that has been extensively studied for its applicability to financial and economic activities. This study examines chaos management and synchronization in this system. Two popular control techniques in non-linear system stabilization—sliding mode control (SMC) and passive control (PC)—are compared. While PC makes use of the system's passivity to accomplish control goals, SMC is made to guarantee strong synchronization in the face of external disruptions and parametric uncertainties. Lyapunov theory is used for stability analysis of both approaches, guaranteeing convergence of synchronization faults. In terms of accuracy, resilience, and computational efficiency, numerical simulations show each approach's advantages and disadvantages. The findings offer important insights into how well SMCs and PCs work to manage unpredictability in financial systems and provide valuable guidance. The results show that SMC achieves synchronization within t ≤ 1, while PC takes t ≤ 13. Maximum synchronization error in SMC reduces from 5.67 to 0.02, whereas PC reduces from 6.21 to 0.03. SMC, although faster, requires two controllers, whereas PC, using one controller, adjusts the interest rate for synchronization. These findings highlight the trade-offs in speed, complexity, and practical implementation of both methods for financial stability.