Chaotic behavior of a new fractional-order financial system and its predefined-time sliding mode control based on the RBF neural network

Fractional-order financial systems hold significant importance in practical situations. In this work, a novel fractional-order financial system considering the non-constant elasticity of demand was presented, and the system's complex dynamics were studied. The results demonstrated that the system can exhibit diverse chaotic dynamics and periodic oscillations, which are influenced by different fractional orders and system parameters. Then, to stabilize the proposed chaotic system with uncertainties and achieve predefined-time synchronization of master-slave systems, an effective sliding mode control strategy utilizing the RBF neural network was put forward. In real financial markets, uncertainties and perturbations occur suddenly, and excessive control input can lead to resource inefficiency. Therefore, unlike other papers that rely on conservative estimations using upper bounds, this paper used RBF neural network approximation to design a more flexible and robust controller while reducing the control input. Simulation findings reveal that this approach requires less control input than traditional methods without the RBF neural network and converges more rapidly than finite-time, fixed-time, and other predefined-time sliding mode control strategies with the RBF neural network, which validates the feasibility of this approach. Finally, the proposed chaotic system and control method were successfully applied to secure encryption, demonstrating their practical value.