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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
##### References:
 [1] Chua, L.: Memristor–the missing circuit element, IEEE transactions on circuit theory 18, No. 8, 507-519 (1971) [2] Strukov, D. B.; Snider, G. S.; Stewart, D. R.: The missing memristor found, Nature 435, 80-83 (2008) [3] Sabatier, J.; Agrawal, O. P.; Machado, J. A. Tenreiro: Advances in fractional calculus; theoretical developments and applications in physics and engineering, (2007) [4] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) [5] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) [6] Podlubny, I.: Fractional differential equations, (1999) [7] Li, C. P.; Deng, W. H.: Remarks on fractional derivatives, Applied mathematics and computation 187, 777-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163 [8] Zaslavsky, G. M.: Chaos, fractional kinetics, and anomalous transport, Physics reports 371, 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9 [9] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J.: Fractional calculus models and numerical methods, (2009) [10] Haba, T. C.; Loum, G. L.; Ablart, G.: An analytical expression for the input impedance of a fractal tree obtained by a microelectronical process and experimental measurements of its non-integral dimension, Chaos, solitons and fractals 33, 364-373 (2007) [11] Radwan, A. G.; Elwakil, A. S.; Soliman, A. M.: Fractional-order sinusoidal oscillators: design procedure and practical examples, IEEE transactions on circuits systems I 55, 2051-2063 (2008) [12] Sugi, M.; Hirano, Y.; Miura, Y. F.; Saito, K.: Simulation of fractal immittance by analog circuits: an approach to the optimized circuits, IEICE transactions on fundamentals electronics communications computer science 82, No. 8, 1627-1634 (1999) [13] G.W. Bohannan, S.K. Hurst, L. Springler, Electrical component with fractional order impedance, Utility Patent Application, 11/30/2006, US20060267595. [14] A.G. Radwan, M.A. Zidan, K.N. Salama, HP Memristor mathematical model for periodic signals and DC, in: IEEE International Midwest Symposium on Circuits and Systems, MWSCAS 2010, pp. 861–864. [15] Momani, S.: Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Applied mathematics and computation 165, No. 2, 459-472 (2005) · Zbl 1070.65105 · doi:10.1016/j.amc.2004.06.025 [16] Momani, S.: An explicit and numerical solutions of the fractional KdV equation, Mathematics and computer simulation 70, No. 2, 110-118 (2005) · Zbl 1119.65394 · doi:10.1016/j.matcom.2005.05.001 [17] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, Journal mathematical analysis and applications 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194 [18] G. Hussian, S. Momani, Non-standard discretization of fractional differential equations, in: Proceeding of 8th Seminar of Differential Equations and Dynamical Systems in, Isfahan, Iran, 2008. [19] Erjaee, G. H.: Numerical bifurcation of predator–prey fractional differential equations with a constant rate harvesting, Journal of physics: conference series 96, 012045 (2007) [20] Mickens, R. E.: Nonstandard finite difference models of differential equations, (1994) · Zbl 0810.65083 [21] Mickens, R. E.; Smith, A.: Finite difference models of ordinary differential equations: influence of denominator models, Journal of the franklin institution fractals. 327, 143-145 (1990) · Zbl 0695.93063 · doi:10.1016/0016-0032(90)90062-N [22] Mickens, R. E.: Applications of nonstandard finite difference schemes, (2000) [23] Mickens, R. E.: Advances in the applications of nonstandard finite difference schemes, (2005) [24] Itoh, M.; Chua, L. O.: Memristor oscillators, International journal of bifurcation and chaos 18, No. 11, 3183-3206 (2008) · Zbl 1165.94300 · doi:10.1142/S0218127408022354 [25] Radwan, A. G.; Soliman, A. M.; Elwakil, A. S.; Sedeek, A.: On the stability of linear systems with fractional order elements, Chaos, solitons and fractals 40, No. 5, 2317-2328 (2009) · Zbl 1198.93151 · doi:10.1016/j.chaos.2007.10.033 [26] Moaddy, K.; Hashim, I.; Momani, S.: Non-standard finite difference schemes for solving fractional-order Rössler chaotic and hyperchaotic systems, Computers and mathematics with applications 62, No. 3, 1068-1074 (2011) · Zbl 1228.65119 · doi:10.1016/j.camwa.2011.03.059 [27] M. Perc, User friendly programs for nonlinear time series analysis, Retrieved 7 August, 2010. http://www.matjazperc.com/ejp/time.html.