Data-driven neural modeling and chaos control in fractional-order financial dynamical systems

Being universal approximators, neural network based methods form the backbone of modern scientific computing by providing techniques to approximate solutions of mathematical problems where analytical solutions are either difficult or impossible to derive. These methods transform continuous models, such as differential or integral equations, into a machine learning model, which can be trained up to the desired accuracy level. In the current study, we investigated the dynamics of a nonlinear fractional financial chaotic model. Chaotic characteristics of the underlying system are explored through a feedforward neural network trained in an autoregressive manner. Chaos is captured successfully for various fractional orders and for multiple settings of the system’s parameters. It is also validated through the benchmark properties of chaos, such as Lyapunov exponents and the Kaplan–Yorke dimension. The trained model is also tested qualitatively through the 2D and 3D attractors for each combination of fractional and financial parameters. Existence and uniqueness of the solutions are also proved. The memory-dependent behavior of the model is successfully emulated by the proposed feedforward neural network framework. Furthermore, a detailed stability analysis is performed for selected fractional orders. To regulate chaotic oscillations, a feedback gain matrix control strategy is implemented. This approach effectively suppresses chaos and guides the trajectories toward desired equilibria. Finally, a stability analysis of the controlled system is conducted to confirm convergence and robustness of the proposed control strategy.