Fractal–fractional modeling and chaos analysis of a financial system with generalized memory kernels

Chaotic behavior in financial systems strongly influences investment strategies, risk management, and policy decisions. Conventional fractional calculus, however, has limitations in capturing the memory and scaling effects that characterize such complexity. To address this gap, the present study employs a novel differential operator that unifies fractal and fractional calculus through the Caputo and Atangana–Baleanu kernels. The objective is to investigate the nonlinear dynamics of a financial chaotic model using fractal–fractional derivative operators. A numerical scheme is implemented to generate system trajectories, and the Lyapunov exponent is applied to assess chaotic transitions. The results show that variations in saving rate, per-investment cost, and demand elasticity significantly affect system stability and regime shifts. Compared with classical fractional formulations, the proposed approach uncovers crossover phenomena in phase portraits and reveals novel attractor structures. These findings provide deeper insight into the mechanisms underlying financial complexity and demonstrate the effectiveness of fractal–fractional calculus as a powerful framework for modeling real-world economic dynamics.