To the Derivation of a Mathematical Model of a Chaotic System Using the Example of a Chua Circuit of Fractional Order

Abstract

An analysis of existing mathematical models of a fractional order Chua circuit was carried out, which showed that in all models the fractional order is achieved by replacing the first order derivatives of the system of differential equations that describe the behavior of the classical Chua circuit with fractional order derivatives with some differences in taking into account the parameters and the nature of Chua diode nonlinearity. However, such a replacement has not been justified in any way. The paper considers an approach to form a fractional order Chua circuit, where capacitive elements of a Chua circuit are replaced with specific elements, which component equations represent differential relations of a fractional order (the so-called “elements with fractal impedance”). During preliminary assessment of chaos emergence conditions of the existing fractional order model in MATLAB and its analogue the OrCAD circuit modeling program using element models with fractal impedance, it turned out that the assessment results do not coincide. The hypothesis for the discrepancy between modeling results was the assumption that the mathematical model of a fractional order Chua circuit, obtained by formal replacing the derivative order, does not reflect features of fractal impedance elements and cannot serve as the basis for an analog implementation of a fractional order Chua circuit. A mathematical model of a chaotic system, built on the basis of a fractional-order Chua circuit containing elements with fractal impedance, is obtained. The result comparison of visual observation of chaotic behavior areas of the fractional order Chua system mathematical model described in literature and obtained in this work using the fde_pi12_pc tool in Matlab with a circuit model of the system in OrCAD has been made. It is shown that the percentage of coincidence of the chaos occurrence conditions in the mathematical model in comparison with the circuit model was: with the mathematical model described in the literature - 11.8 %, and with the mathematical model obtained in this work - 97 %. Consequently, the resulting mathematical model corresponds to the circuit model of a fractional-order Chua circuit more accurately.