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Non-standard finite difference schemes for solving fractional-order Rössler chaotic and hyperchaotic systems. (English) Zbl 1228.65119
Summary: The non-standard finite difference method (for short NSFD) is implemented to study the dynamic behaviors in the fractional-order Rössler chaotic and hyperchaotic systems. The Grünwald-Letnikov method is used to approximate the fractional derivatives. We found that the lowest value to have chaos in this system is $$2.1$$ and hyperchaos exists in the fractional-order Rössler system of order as low as $$3.8$$. Numerical results show that the NSFD approach is easy to implement and accurate when applied to differential equations of fractional order.

##### MSC:
 65L12 Finite difference and finite volume methods for ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34A08 Fractional ordinary differential equations and fractional differential inclusions 26A33 Fractional derivatives and integrals 34C28 Complex behavior and chaotic systems of ordinary differential equations 45J05 Integro-ordinary differential equations
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##### References:
 [1] G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, vol. 371, 2002, pp. 461-580. · Zbl 0999.82053 [2] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J., Fractional calculus models and numerical methods, (2009), World Scientific Singapore [3] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with riemann – liouville fractional derivatives, Rheol. acta, 45, 765-771, (2006) [4] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [5] Magin, R.L., Fractional calculus models of complex dynamics in biological tissues, Comput. math. appl., 59, 1586-1593, (2010) · Zbl 1189.92007 [6] Momani, S.; Odibat, Z., Analytical solution of a time-fractional navier – stokes equation by Adomian decomposition method, Appl. math. comput., 177, 2, 488-494, (2006) · Zbl 1096.65131 [7] Odibat, Z.; Momani, S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. math. appl., 58, 11-12, (2009) [8] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003 [9] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictorcorrector approach for the numerical solution of fractional differential equations, Nonlinear dyn., 29, 3-22, (2002) · Zbl 1009.65049 [10] Li, C.P.; Peng, G.J., Chaos in chen’s system with a fractional order, Chaos solitons fractals, 22, 443-450, (2004) · Zbl 1060.37026 [11] Li, C.P.; Dao, X.H.; Guo, P., Fractional derivatives in complex plane, Nonlinear anal. TMA, 71, 1857-1869, (2009) · Zbl 1173.26305 [12] Li, C.P.; Wang, Y.H., Numerical algorithm based on Adomian decomposition for fractional differential equations, Comput. math. appl., 57, 1672-1681, (2009) · Zbl 1186.65110 [13] Zhou, T.S.; Li, C.P., Synchronization in fractional-order differential systems, Physica D, 212, 111-125, (2005) · Zbl 1094.34034 [14] Cang, J.; Tan, Y.; Xu, H.; Liao, S.J., Series solutions of non-linear Riccati differential equations with fractional order, Chaos solitons fractals, 40, 1-9, (2009) · Zbl 1197.34006 [15] Erjaee, G.H., Numerical bifurcation of predator – prey fractional differential equations with a constant rate harvesting, J. phys. conf. ser., 96, 12-45, (2009) [16] G. Hussian, M. Alnaser, S. Momani, Non-standard discretization of fractional differential equations, in: Proceeding of 8th Seminar of Differential Equations and Dynamical Systems in Isfahan, Iran, 2008. [17] Li, C.P.; Chen, G., Chaos and hyperchaos in the fractional-order Rössler equations, Physica A, 341, 55-61, (2004) [18] Mickens, R.E., Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: implications for numerical analysis, Numer. methods partial differential equations, 5, 313-325, (1989) · Zbl 0693.65059 [19] Mickens, R.E.; Smith, A., Finite-difference models of ODE’s: influence of denominator functions, J. franklin inst., 327, 143-149, (1990) · Zbl 0695.93063 [20] Mickens, R.E., Nonstandard finite difference models of differential equations, (1994), World Scientific Singapore · Zbl 0925.70016 [21] Mickens, R.E., Nonstandard finite difference schemes for reaction – diffusion equations, Numer. methods partial differential equations, 15, 201-214, (1999) · Zbl 0926.65085 [22] Mickens, R.E., Advances in the applications of nonstandard finite difference schemes, (2005), World Scientific Singapore · Zbl 1079.65005 [23] R.E. Mickens, Applications of Nonstandard Finite Difference Schemes, Singapore, 2000. · Zbl 0989.65101 [24] Rössler, O.E., An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062 [25] Rössler, O.E., An equation for hybercaos chaos, Phys. lett. A, 71, 155-157, (1979) · Zbl 0996.37502 [26] Mickens, R.E., Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer. methods partial differential equations, 23, 672-691, (2007) · Zbl 1114.65094 [27] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 [28] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons Press New York · Zbl 0789.26002 [29] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [30] Li, C.P.; Deng, W.H., Remarks on fractional derivatives, Appl. math. comput., 187, 777-784, (2007) · Zbl 1125.26009
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