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Development and Elaboration of a Compound Structure of Chaotic Attractors with Atangana–Baleanu Operator

  • Emile F. Doungmo GoufoEmail author
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 194)

Abstract

After the finding of the compound structure for standard chaotic attractors, the main concern was related to how to regulate such a fascinating dynamic. Hence, the question about the existence of a compound structure for chaotic attractors generated by fractional systems was raised. In this work, we investigate the existence of compound structure of a chaotic attractor generated from a Atangana–Baleanu fractional system where two cases are studied: the integer case and the fractional one. The model is first solved numerically thanks to the Haar Wavelets scheme whose convergence is proved via error analysis. Numerical simulations are performed and clearly reveal the existence of the desired compound structure in both cases and characterized by the generation of a left-attractor seen as the reflection of a right attractor through the mirror operation. Moreover, those two simple attractors can always be combined together to form the resulting chaotic attractor. The mechanism of forming those simple attractors is shown and leads to a bounded partial attractor. Furthermore, that same mechanism appears to be strongly dependent on two parameters, the model parameter u and the Atangana–Baleanu derivative with order \(\alpha ,\) important in controlling the systems. It is observed that, in the fractional case (\(\alpha =0.9\)), the period-doubling bifurcations start at a higher value of u compared to the integer case (\(\alpha =1\)).

Keywords

Fractional calculus Atangana–Baleanu fractional derivative Chaotic attractors 

References

  1. 1.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)zbMATHCrossRefGoogle Scholar
  2. 2.
    Doungmo Goufo, E.F. Chaotic processes using the two-parameter derivative with non-singular and nonlocal kernel: basic theory and applications. Chaos: Interdiscip. J. Nonlinear Sci. 26(8), 1–21 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Wang, Z., Sun, Y., Van Wyk, B.J., Qi, G., Van Wyk, M.A.: A 3-D four-wing attractor and its analysis. Braz. J. Phys. 39(3), 547–553 (2009)CrossRefGoogle Scholar
  4. 4.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(07), 1465–1466 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Vanecek, A., Celikovsky, S.C.: Control Systems: from Linear Analysis to Synthesis of Chaos. Prentice Hall International (UK) Ltd, Prentice (1996)zbMATHGoogle Scholar
  7. 7.
    Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(03), 659–661 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Özoguz, S., Elwakil, A.S., Kennedy, M.: Experimental verification of the butterfly attractor in a modified lorenz system. Int. J. Bifurc. Chaos 12(07), 1627–1632 (2002)CrossRefGoogle Scholar
  9. 9.
    Lü, J., Zhou, T., Chen, G., Zhang, S.: The compound structure of chen’s attractor. Int. J. Bifurc. Chaos 12(04), 855–858 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Babolian, E., Shahsavaran, A.: Numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets. J. Comput. Appl. Math. 225(1), 87–95 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chen, Y., Yi, M., Yu, C.: Error analysis for numerical solution of fractional differential equation by Haar wavelets method. J. Comput. Sci. 3(5), 367–373 (2012)CrossRefGoogle Scholar
  12. 12.
    Doungmo Goufo, E.F.: Solvability of chaotic fractional systems with 3D four-scroll attractors. Chaos Solitons & Fractals 104, 443–451 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Doungmo Goufo, E.F., Nieto, J.J.: Attractors for fractional differential problems of transition to turbulent flows. J. Comput. Appl. Math. 339, 329–342 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons & Fractals 89, 447–454 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel. Therm. Sci. 20(2), 763–769 (2016)CrossRefGoogle Scholar
  16. 16.
    Gómez-Aguilar, J.F.: Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys. A: Stat. Mech. Appl. 494, 52–75 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)CrossRefGoogle Scholar
  18. 18.
    Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)CrossRefGoogle Scholar
  19. 19.
    Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromag. Waves Appl. 30(15), 1937–1952 (2016)CrossRefGoogle Scholar
  20. 20.
    Ghanbari, B., Gómez-Aguilar, J.F.: Modeling the dynamics of nutrient-phytoplankton-zooplankton system with variable-order fractional derivatives. Chaos, Solitons & Fractals 116, 114–120 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)Google Scholar
  22. 22.
    Gómez-Aguilar, J.F., Torres, L., Yépez-Martínez, H., Baleanu, D., Reyes, J.M., Sosa, I.O.: Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel. Adv. Differ. Equ. 2016(1), 1–17 (2016)zbMATHCrossRefGoogle Scholar
  23. 23.
    Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)CrossRefGoogle Scholar
  24. 24.
    Yépez-Martínez, H., Gómez-Aguilar, J.F., Sosa, I.O., Reyes, J.M., Torres-Jiménez, J.: The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev. Mex. Fís 62(4), 310–316 (2016)MathSciNetGoogle Scholar
  25. 25.
    Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos, Solitons & Fractals 115, 283–299 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: Application to chaos and statistics. Chaos, Solitons & Fractals 114, 516–535 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys. A: Stat. Mech. Appl. 509, 703–716 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Atangana, A.: Non validity of index law in fractional calculus: a fractional differential operator with markovian and non-markovian properties. Phys. A: Stat. Mech. Appl. 505, 688–706 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Doungmo Goufo, E.F., Atangana, A.: Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion. Eur. Phys. J. Plus 131(8), 1–26 (2016)CrossRefGoogle Scholar
  30. 30.
    Lepik, Ü., Hein, H.: Haar Wavelets: With Applications. Springer Science & Business Media, Berlin (2014)Google Scholar
  31. 31.
    Lü, J., Chen, G., Zhang, S.: Dynamical analysis of a new chaotic attractor. Int. J. Bifurc. Chaos 12(05), 1001–1015 (2002)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaFloridaSouth Africa

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