Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Several Fractional Differences and Their Applications to Discrete Maps

Journal of Applied Nonlinear Dynamics 4(4) (2015) 339--348 | DOI:10.5890/JAND.2015.11.001

Guo-Cheng Wu$^{1}$, Dumitru Baleanu$^{2}$,$^{3}$, Sheng-Da Zeng$^{3}$

$^{1}$ Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, China

$^{2}$ Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530, Balgat, Ankara, Turkey

$^{3}$ Institute of Space Sciences, Magurele-Bucharest, Romania

Download Full Text PDF



Several definitions of fractional differences are discussed. Their applications to fractional maps are compared. As an example, the logistic equation of integer order is discretized by these fractional difference methods. The comparative results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.


This work was financially supported by the National Natural Science Foundation of China (Grant No.11301257), the Innovative Team Program of Sichuan Provincial Universities (Grant No. 13TD0001) and the Seed Funds for Major Science and Technology Innovation Projects of Sichuan Provincial Education Department (Grant No.14CZ0026).


  1. [1]  Machado, J.A.T. and Galhano, A. (2009), Approximating fractional derivatives in the perspective of system control, Nonlinear Dyn. 56, 401-407.
  2. [2]  Ortigueira, M., Coito, F.J. and Trujillo, J.J. (2013), A new look into the discrete-time fractional calculus: derivatives and exponentials, In Procsseding of Fractional Differentiation and Its Applications, 6, 629-634.
  3. [3]  Edelman, M. and Tarasov, V.E. (2009), Fractional standard map, Phys. Let. A 374, 279-285.
  4. [4]  Edelman, M. (2011), Fractional standard map: Riemann-Liouville vs. Caputo,Commun. Nonlinear Sci. Numer. Simulat.16, 4573-4580.
  5. [5]  Cheng, J.F. (2011), The theory of fractional difference equations, Xiamen University Press: Xiamen(in Chinese).
  6. [6]  Mumkhamar, J. (2013), Chaos in a fractional order logisitic map, Frac. Calc. Appl. Anal. 16, 511-519.
  7. [7]  Agarwal, R.P., El-Sayed, A.M.A. and Salman, S.M. (2013), Fractional-order Chua's system: discretization, bifurcation and chaos, Adv. Diff. Equ. 2013, 320.
  8. [8]  Wu, G.C. and Baleanu, D. (2014), Discrete fractional logistic map and its chaos, Nonlinear Dyn. 75, 283-287.
  9. [9]  Wu, G.C., Baleanu, D. and Zeng, S.D. (2014), Discrete chaos in fractional sine and standard maps, Phys. Lett. A 378, 484-487.
  10. [10]  Wu, G.C., Baleanu, D. (2015), Discrete chaos in the fractional delayed logistic maps, Nonlinear Dyn 80, 1697-1703.
  11. [11]  Wu, G.C., Baleanu, D. (2014), Chaos synchronization of the discrete fractional logistic map, Sign. Proc. 102, 96-99.
  12. [12]  Abdeljawad, T., Baleanu, D., Jarad, F., Agarwal, R.P. (2013), Fractional sums and differences with binomial coefficients, Discret. Dyn. Nat. Soc. 2013, Article ID 104173, 6 pages.
  13. [13]  Jarad, F., Bayram, K., Abdeljawad, T., Baleanu, D. (2012), On the discrete Sumudu transform, Rom. Rep. Phys. 64, 347-356.
  14. [14]  Miller, K.S. and Ross, B. (1989), Fractional difference calculus, in: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, May 1988, in: Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 139-152.
  15. [15]  Atici, F.M. and Eloe, P.W. (2007), A transform method in discrete fractional calculus, Int. J. Diff. Equ. 2, 165-176.
  16. [16]  Atici, F.M. and Eloe, P.W. (2009), Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137, 981-989.
  17. [17]  Atici, F.M. and Sengül, S. (2010), Modeling with fractional difference equations, J. Math. Anal. Appl. 369, 1-9.
  18. [18]  Anastassiou, G.A. (2011), About Discrete Fractional Calculus with Inequalities, Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library Volume 5, 575-585.
  19. [19]  Abdeljawad, T. (2011), On Riemann and Caputo fractional differences, Comput. Math. Appl. 62, 1602-1611.
  20. [20]  Podlubny, I. (1999), Fractional differential equations, Academic Press: San Diego.
  21. [21]  Bohner, M. and Peterson, A.C. (2001), Dynamic equations on time scales: An introduction with applications, Birkhauser.
  22. [22]  Xiao, H., Ma, Y.T. and Li, C.P. (2014), Chaotic vibration in fractional maps, J. Vib. Contr. 20, 964-972.
  23. [23]  Oldham, K.B. and Spanier, J. (1974), The fractional calculus, Academic Press: NewYork.
  24. [24]  Meerschaert, M.M. and Tadjeran, C. (2004), Finite difference approximations for fractional advection- dispersion flow equations, J. Comput. Appl. Math. 172, 65-77.
  25. [25]  Tadjeran, C., Meerschaert, M.M. and Scheffer, H.P. (2006), A second-order accuratenumerical approximation for the fractional diffusion equation, J. Comput. Phys. 213, 205-213.
  26. [26]  Scjerer, R., Kalla, S.L., Tang, Y.F. and Huang, J.F. (2011), The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl. 62, 902-917.
  27. [27]  Machado, J.A.T. (1997), Analysis and design of fractional-order digital control systems, Systems Analysis Modelling Simulation, 27, 107-122.
  28. [28]  Machado, J.A.T. (2001), Discrete-time fractional-order controllers, Frac. Calc. Appl. Anal. 4, 47-66.
  29. [29]  Ortigueira, M.D. and Trujillo, J.J. (2011), Generalized Grunwald-Letnikov Fractional Derivative and Its Laplace and Fourier Transforms, J. Comput. Nonlin. Dyn. 6, 034501.
  30. [30]  Pu, Y.F., Zhou, J.L. and Yuan, X. (2010), Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Tran. Image Proc. 19 , 491-511.
  31. [31]  Pu, Y.F. and Zhou, J.L. (2011), A novel approach for multi-scale texture segmentation based on fractional differential, Int. J. Comput. Math. 88, 58-78.
  32. [32]  Diethelm, K. (2011), Analysis of fractional differential equations, J. Math. Anal. Appl. 265, 229-248.
  33. [33]  Li, C.P. and Peng, G.J. (2004), Chaos in Chen's system with a fractional order, Chaos, Soliton. Fract. 22, 443-450.
  34. [34]  Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012), Fractional calculus models and numerical methods, World Scientific: Singapore.
  35. [35]  Chen, F.L., Luo, X.N. and Zhou, Y. (2011), Existence results for nonlinear fractional difference equation, Adv. Diff. Equa. 2011, Article ID 713201, 12 pages.
  36. [36]  Wu, G.C., Baleanu, D. (2015), Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps, Commun. Nonlinear Sci. Numer. Simulat. 22, 95-100.
  37. [37]  Edelman, M. (2014), Fractional Maps and Fractional Attractors. Part II: Fractional Difference α-Families of Maps, arXiv:1404.4906v2.
  38. [38]  Altintan, D. (2006), Extension of the lgoisitic equation with piecewise constant arguments and population dynamcis, Master dissertation: Turkey.
  39. [39]  Akhmet, M.U. (2008), Stability of differential equations with picewise constant arguments of generalized type, Nonlinear Anal. 68, 794-803.
  40. [40]  El-Sayed, A.M.A. and Salman, S. (2013), Chaos and bifurcation of the Logistic discontinuous dynamical systems with piecewise constant arguments, Malaya Journal of Matematik, 3, 14-20.
  41. [41]  El-Sayed, A.M.A. and Nasr, M.E. (2013), Discontinuous dynamical systems and fractional-orders difference equations, J. Fract. Calc. Appl. 4, 130-138.