FPGA realization of fractional-order chaotic oscillators using ABM and EFORK methods.

Despite the growing theoretical interest in fractional-order chaotic systems, there is a lack of systematic, hardware-oriented methodology for their real-time digital realization that preserves chaotic dynamical properties and optimize computational resource utilization, particularly in the context of FPGA implementations using different fractional-order numerical integration schemes. This work presents the development of two reconfigurable VHDL modules for the digital emulation of fractional-order chaotic oscillators based on the Adams-Bashforth-Moulton (ABM) and explicit fractional-order Runge-Kutta (EFORK) methods. Fractional-order implementations of the Lorenz, Rabinovich-Fabrikant, Li, Chen, and Yu-Wang oscillators are realized on an Artix-7 FPGA platform. To validate the proposed architectures, Matlab simulations of each oscillator using both numerical methods are first conducted, followed by a bifurcation analysis of the Lyapunov exponents (LEs) as a function of the fractional-order. It is shown that, in general, ABM yields positive LEs with larger magnitudes and over wider intervals than EFORK, except for the Chen oscillator. Finally, the logic resources required to emulate each oscillator on an FPGA device are analyzed, and the results show that Runge-Kutta systematically reduces LUT usage due to a more hardware-efficient computational structure, particularly by avoiding division operations, which are more resource-intensive in FPGA implementations. As a reference, Adams-Bashforth-Moulton implementation requires the fewest logic resources for the Lorenz system, with 47,780 LUTs, whereas for Runge-Kutta, Li system requires 38,860 LUTs. Overall, the results highlight a clear trade-off between dynamical complexity and resource efficiency in the digital emulations of fractional-order chaotic oscillators on FPGA devices.