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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps

Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 391--402 | DOI:10.5890/DNC.2015.11.003

M. Edelman

Dept. of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA

Courant Institute of Mathematical Sciences, New York University,251 Mercer St., New York, NY 10012, USA

Department of Mathematics, BCC, CUNY, 2155 University Avenue, Bronx, New York 10453, USA

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In this paper we extend the notion of an α-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic α- Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.


The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University. The author expresses his gratitude to E. Hameiri, H. Weitzner, and G. Ben Arous for the opportunity to complete this work at the Courant Institute and to V. Donnelly for technical help.


  1. [1]  Edelman, M. (2014), Fractional Maps as Maps with Power-Law Memory, in: Nonlinear Dynamics and Complexity, Eds.: A. Afraimovich, A. C. J. Luo, and X. Fu, pp. 79-120, New York, Springer.
  2. [2]  Carla, M.A. Pinto, Mendes Lopes, A., and Tenreiro Machado, J.A. (2012), A review of power laws in real life phenomena, Commun. Nonlin. Sci. Numer. Simul., 17, 3558-3578.
  3. [3]  Tarasov, V.E. and Zaslavsky, G.M. (2008), Fractional equations of kicked systems and discrete maps, J. Phys. A, 41, 435101.
  4. [4]  Stanislavsky, A.A. (2006), Long-term memory contribution as applied to the motion of discrete dynamical system, Chaos, 16, 046105.
  5. [5]  Miller, K.S. and Ross, B. (1989), Fractional Difference Calculus, in: Univalent Functions, Fractional Calculus, and Their Applications, Eds. H. M. Srivastava and S. Owa, pp. 139-151, Chichester, Ellis Howard.
  6. [6]  Gray, H.L. and Zhang, N.-F. (1988), On a new definition of the fractional difference, Math. Comput., 50, 513-529.
  7. [7]  Agarwal, R.P. (2000), Difference equations and inequalities, Marcel Dekker, New York.
  8. [8]  Atici, F.M. and Eloe, P.W. (2009), Initial value problems in discrete fractional calculus, Proc. Am. Math. Soc., 137, 981-989.
  9. [9]  Anastassiou, G.A., Discrete Fractional Calculus and Inequalities,
  10. [10]  Chen, F., Luo, X., and Zhou, Y. (2011), Existence Results for Nonlinear Fractional Difference Equation, Adv. Differ. Eq., 2011, 713201.
  11. [11]  Wu, G.-C., Baleanu, D., and Zeng, S.-D. (2014), Discrete chaos in fractional sine and standard maps, Phys. Lett. A, 378, 484-487.
  12. [12]  Wu, G.-C. and Baleanu, D. (2014), Discrete fractional logistic map and its chaos, Nonlin. Dyn., 75, 283-287.
  13. [13]  Edelman, M. and Tarasov, V. E. (2009), Fractional standard map, Phys. Lett. A, 374, 279-285.
  14. [14]  Tarasov, V.E. (2009), Differential equations with fractional derivative and universal map with memory, J. Phys. A: Math. Theor., 42, 465102.
  15. [15]  Tarasov, V.E. (2009), Discrete map with memory from fractional differential equation of arbitrary positive order, J. Math. Phys., 50, 122703.
  16. [16]  Tarasov, V.E. (2011), Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields, and Media, HEP & Springer, Beijing & New York.
  17. [17]  Edelman, M. (2011), Fractional Standard Map: Riemann-Liouville vs. Caputo, Commun. Nonlin. Sci. Numer. Simul., 16, 4573-4580.
  18. [18]  Edelman, M. and Taieb, L.A. (2013), New Types of Solutions of Non-Linear Fractional Differential Equations, in: Advances in Harmonic Analysis and Operator Theory; Series: Operator Theory: Advances and Applications, Eds.: A. Almeida, L. Castro, and F.-O. Speck, 229, pp. 139-155, Basel, Springer.
  19. [19]  Edelman, M. (2013), Fractional Maps and Fractional Attractors. Part I: α-Families of Maps, Discontinuity, Nonlinearity, and Complexity, 1, 305-324.
  20. [20]  Edelman, M. (2013), Universal Fractional Map and Cascade of Bifurcations Type Attractors, Chaos, 23, 033127.
  21. [21]  Chirikov, B.V. (1979), A universal instability of many dimensional oscillator systems, Phys. Rep., 52, 263-379.
  22. [22]  Lichtenberg, A.J. and Lieberman, M.A. (1992), Regular and Chaotic Dynamics, Springer, Berlin. 1992.
  23. [23]  Zaslavsky, G.M. (2008), Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
  24. [24]  May, R.M. (1976), Simple mathematical models with very complicated dynamics, Nature, 261, 459-467.
  25. [25]  Samko, S.G., Kilbas, A.A., and Marichev, O.I.(1993), Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York.
  26. [26]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam.
  27. [27]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.
  28. [28]  Lalescu, C.C. (2010), Patterns in the sine map bifurcation diagram, arXiv:1011.6552.
  29. [29]  Edelman, M. (2014), Caputo standard α-family of maps: Fractional difference vs. fractional, Chaos, 24, 023137.
  30. [30]  Edelman, M. (2014), Fractional maps and fractional attractors. Part II: fractional difference α-families of maps, arXiv:1404.4906v2.