FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks

Fractional-order chaotic oscillators (FOCOs) have been widely studied during the last decade, and some of them have been implemented on embedded hardware like field-programmable gate arrays, which is a good option for fast prototyping and verification of the desired behavior. However, the hardware resources are dependent on the length of the digital word that is used, and this can degrade the desired response due to the finite number of bits to perform computer arithmetic. In this manner, this paper shows the implementation of FOCOs using analog electronics to generate continuous-time chaotic behavior. Charef's method is applied to approximate the fractional-order derivatives as a ratio of two polynomials in the Laplace domain. For instance, two commensurate FOCOs are the cases of study herein, for which we show their dynamical analysis by evaluating their equilibrium points and eigenvalues that are used to estimate the minimum fractional-order that guarantees their chaotic behavior. We propose the use of first-order all-pass and low-pass filters to design the ratio of the polynomials that approximate the fractional-order. The filters are implemented using amplifiers and synthesized on a field-programmable analog array (FPAA) device. Experimental results are in good agreement with simulation results thus demonstrating the usefulness of FPAAs to generate continuous-time chaotic behavior, and to allow reprogramming of the parameters of the FOCOs.