Integrated Circuit Design of Fractional-Order Chaotic Systems Optimized by Metaheuristics

Nowadays, a huge amount of research is done on introducing and implementing new fractional-order chaotic systems. In the majority of cases, the implementation is done using embedded hardware, and very seldom does it use integrated circuit (IC) technology. This is due to the lack of design automation tools ranging from the system level down to layout design. At the system level, the challenge is guaranteeing chaotic behavior by varying all parameters while optimizing dynamical characteristics, such as the Lyapunov spectrum and the Kaplan-Yorke dimension. Using embedded hardware, the implementation is straightforward, but one must perform a scaling process for IC design, in which the biases may be lower than 1 volt but the amplitudes of the state variables of the chaotic systems can have values higher than one. In this manner, this paper describes three levels of abstraction to design fractional-order chaotic systems: The first one shows the optimization of a case study, the mathematical model of the fractional-order Lorenz system to find the fractional-orders of the derivatives, and the coefficients that generate better chaotic behavior. The second level is the block description of a solution of the mathematical model, in which the fractional-order derivatives are approximated in the Laplace domain by several approximation methods. The third level shows the IC design using complementary metal-oxide-semiconductor (CMOS) technology. The transfer functions approximating the fractional-order derivatives are synthesized by active filters that are designed using operational transconductance amplifiers (OTAs). The OTAs are also used to design adders and subtractors, and the multiplication of variables is done by designing a CMOS four-quadrant multiplier. The paper shows that the simulation results scaling the mathematical model to have amplitudes lower than +/- 1 are in good agreement with the results using CMOS IC technology of 180 nm.