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Stability and non-standard finite difference method of the generalized Chua’s circuit. (English) Zbl 1228.65120
Summary: We develop a framework to obtain approximate numerical solutions of the fractional-order Chua’s circuit with Memristor using a non-standard finite difference method. Chaotic response is obtained with fractional-order elements as well as integer-order elements. Stability analysis and the condition of oscillation for the integer-order system are discussed. In addition, the stability analyses for different fractional-order cases are investigated showing a great sensitivity to small order changes indicating the poles’ locations inside the physical $s$-plane. The Grünwald-Letnikov method is used to approximate the fractional derivatives. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is an effective and convenient method to solve fractional-order chaotic systems, and to validate their stability.
##### MSC:
 65L12 Finite difference methods for ODE (numerical methods) 34A08 Fractional differential equations 26A33 Fractional derivatives and integrals (real functions) 45J05 Integro-ordinary differential equations 94C05 Analytic circuit theory