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Discrete chaos in fractional delayed logistic maps

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Abstract

Recently the discrete fractional calculus (DFC) started to gain much importance due to its applications to the mathematical modeling of real world phenomena with memory effect. In this paper, the delayed logistic equation is discretized by utilizing the DFC approach and the related discrete chaos is reported. The Lyapunov exponent together with the discrete attractors and the bifurcation diagrams are given.

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References

  1. Wang, X.F., Chen, G.: On feedback anticontrol of discrete chaos. Int. J. Bifurc. Chaos 9, 1435–1441 (1999)

    Article  MATH  Google Scholar 

  2. Yang, X.S., Chen, G.: Some observer-based criteria for discrete-time generalized chaos synchronization. Chaos Solitons Fractals 13, 1303–1308 (2002)

    Google Scholar 

  3. Yan, Z.: QS synchronization in 3D Henon-like map and generalized H\(\acute{e}\)non map via a scalar controller. Phys. Lett. A 342, 309–317 (2005)

    Article  MATH  Google Scholar 

  4. Pareek, N.K., Patidar, V., Sud, K.K.: Image encryption using chaotic logistic map. Image Vis. Comput. 24, 926–934 (2006)

    Article  Google Scholar 

  5. Behnia, S., Akhshani, A., Mahmodi, H., Akhavan, A.: A novel algorithm for image encryption based on mixture of chaotic maps. Chaos Solitons Fractals 35, 408–419 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jalan, S., Amritkar, R.: Self-organized and driven phase synchronization in coupled maps. Phys. Rev. Lett. 90, 014101 (2003)

    Article  Google Scholar 

  7. Atay, F.M., Jost, J., Wende, A.: Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 92, 144101 (2004)

    Article  Google Scholar 

  8. Miller, K.S., Ross, B.: Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, May 1988; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 139–152 (1989)

  9. Bohner, M., Peterson, A.C.: Dynamic Equations on Time Scales: an Introduction with Applications. Birkhauser, Boston (2001)

    Book  Google Scholar 

  10. Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137, 981–989 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Atici, F.M., Senguel, S.: Modeling with fractional difference equations. J. Math. Anal. Appl. 369, 1–9 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Holm, M.T.: The Laplace transform in discrete fractional calculus. Comput. Math. Appl. 62, 1591–1601 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Holm, M.T.: The Theory of discrete fractional calculus: development and application. PhD thesis, University of Nebraska-Lincoln, Lincoln, Nebraska (2011)

  14. Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Abdeljawad, T., Baleanu, D., Jarad, F., Agarwal, R.P.: Fractional sums and differences with binomial coefficients. Discret. Dyn. Nat. Soc. 2013, 104173 (2013)

  16. Anastassiou, G.A.: Principles of delta fractional calculus on time scales and inequalities. Math. Comput. Model. 52, 556–566 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen, F.L., Luo, X.N., Zhou, Y.: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011, 713201 (2011)

  18. Ionescu, C., Machado, J.A.T., Robin, D.K.: Fractional-order impulse response of the respiratory system. Comput. Math. Appl. 62, 845–854 (2011)

    Article  MATH  Google Scholar 

  19. Machado, J.A.T., Galhano, A.: Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn. 56, 401–407 (2009)

    Article  MATH  Google Scholar 

  20. Ortigueira, M.D.: Introduction to fractional linear systems. Part 2: Discrete-time case. IEE Proc. Vis. Image Signal Process. 147, 71–78 (2000)

    Article  Google Scholar 

  21. Jarad, F., Bayram, K., Abdeljawad, T., Baleanu, D.: On the discrete Sumudu transform. Rom. Rep. Phys. 64, 347–356 (2012)

    Google Scholar 

  22. Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    Google Scholar 

  23. Wu, G.C., Baleanu, D., Zeng, S.D: Discrete chaos in fractional sine and standard maps. Phys. Lett. A 378, 484–487 (2014)

    Google Scholar 

  24. Verhulst, P.F.: Recherches math\(\acute{e}\)matiques sur la loi d’accroissement de la population. Nouv. mm. de l’Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1–41 (1845)

    Google Scholar 

  25. Hutchinson, G.E.: Circular casual systems in ecology. Ann. N. Y. Acad. Sci. 50, 221–246 (1948)

    Article  Google Scholar 

  26. Hutchinson, G.E.: An introduction to population ecology. Yale University Press, New Haven (1978)

    MATH  Google Scholar 

  27. Balanov, Z., Krawcewicz, W., Ruan, H.: G. E. Hutchinson’s delay logistic system with symmetries and spatial diffusion. Nonlinear Anal. Real World Appl. 9, 154–182 (2008)

  28. Kolesov, A.Yu., Mishchenko, E.F., Rozov, NKh: A modification of Hutchinson’s equation. Comput. Math. Math. Phys. 50, 1990–2002 (2010)

    Google Scholar 

  29. Maynard, S.J.: Mathematical Ideas in Biology. Cambridge University Press, Cambridge (1968)

    Google Scholar 

  30. May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)

    Article  Google Scholar 

  31. Marotto, F.R.: The dynamics of a discrete population model with threshold. Math. Biosci. 58, 123–128 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  32. Briden, W., Zhang, S.: Stability of solutions of generalized logistic difference equations. Period Math. Hung. 29, 81–87 (1994)

    Google Scholar 

  33. Hernandez-Bermejo, B., Brenig, L.: Some global results on quasipolynomial discrete systems. Nonlinear Anal. Real World Appl. 7, 486–496 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tarasov, V.E., Edelman, M.: Fractional dissipative standard map. Chaos 20, 023127 (2010)

    Google Scholar 

  35. Cheng, J.F.: The theory of fractional difference equations. Xiamen University Press, Xiamen (2011). (in Chinese)

    Google Scholar 

  36. Xiao, H., Ma, Y.T., Li, C.P.: Chaotic vibration in fractional maps (2013). doi:10.1177/1077546312473769

  37. Agarwal, R.P., El-Sayed, A.M.A., Salman, S.M.: Fractional-order Chua’s system: discretization, bifurcation and chaos. Adv. Differ. Equ. 2013, 320 (2013)

  38. Munkhammar, J.: Chaos in a fractional order logistic map. Fract. Calc. Appl. Anal. 16, 511–519 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No. 11301257), the Seed Funds for Major Science and Technology Innovation Projects of Sichuan Provincial Education Department (Grant No. 14CZ0026) and the Innovative Team Program of Sichuan Provincial Universities (Grant No. 13TD0001). The authors also thanks for the referees’ sincere and helpful suggestions to improve the work. One of the authors (G.C. Wu) feels grateful to Prof. Yu Xue for his important comments and remarks during the preliminary stage of writing this manuscript.

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Authors and Affiliations

  1. Key Laboratory of Numerical Simulation of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641112, People’s Republic of China

    Guo-Cheng Wu

  2. College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, People’s Republic of China

    Guo-Cheng Wu

  3. Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

    Dumitru Baleanu

  4. Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Balgat, 06530 , Ankara, Turkey

    Dumitru Baleanu

  5. Institute of Space Sciences, Magurele, Bucharest, Romania

    Dumitru Baleanu

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  1. Guo-Cheng Wu
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  2. Dumitru Baleanu
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Correspondence to Dumitru Baleanu.

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Wu, GC., Baleanu, D. Discrete chaos in fractional delayed logistic maps. Nonlinear Dyn 80, 1697–1703 (2015). https://doi.org/10.1007/s11071-014-1250-3

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  • DOI: https://doi.org/10.1007/s11071-014-1250-3

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