Modulating Chaotic Oscillations in Autocatalytic Reaction Networks Using Atangana–Baleanu Operator

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


Many mathematical models describing dynamics that occur in autocatalytic reaction networks have been proved to be chaotic, exhibiting orbits with unpredictable outcomes. Is it always possible to modulate that chaos? We use Haar wavelet numerical method to investigate a fractional system modeling autocatalytic reaction networks, where particular attention is made on biochemical systems of four-component networks. The convergence of the method is detailed through error analysis. Graphical representations reveal that the dynamic of the whole system is characterized by limit-cycles followed by period-doubling bifurcations that culminate with chaos, depending on the change of the total concentration of cofactor. The behavior of the system becomes more unpredictable as the concentration of cofactor increases, but the phenomenon is shown to be regulated by an additional parameter, the order of the fractional derivative \(\gamma ,\) which plays an important role in triggering and controlling the appearance of chaos. Moreover, the chaotic behavior observed in the cascade diagram of the pure fractional case is proven to appear earlier, showing that the parameter \(\gamma \) is a valuable tool to regulate the chaos observed in some biochemical systems.


Fractional calculus Atangana–Baleanu fractional derivative Autocatalytic reaction networks 



The work of EF Doungmo Goufo was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa and a grant from the Simons Foundation. EF. Doungmo Goufo would also like to thank his post-doctoral fellows for their valuable comments.


  1. 1.
    Stadler, P.F., Schuster, P.: Mutation in autocatalytic reaction networks. J. Math. Biol. 30(6), 597–631 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Gerdts, C.J., Sharoyan, D.E., Ismagilov, R.F.: A synthetic reaction network: chemical amplification using nonequilibrium autocatalytic reactions coupled in time. J. Am. Chem. Soc. 126(20), 6327–6331 (2004)CrossRefGoogle Scholar
  3. 3.
    Blackmond, D.G.: An examination of the role of autocatalytic cycles in the chemistry of proposed primordial reactions. Angewandte Chemie 121(2), 392–396 (2009)CrossRefGoogle Scholar
  4. 4.
    Di Cera, E., Phillipson, P.E., Wyman, J.: Limit-cycle oscillations and chaos in reaction networks subject to conservation of mass. Proc. Natl. Acad. Sci. 86(1), 142–146 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Filisetti, A., Graudenzi, A., Serra, R., Villani, M., Füchslin, R.M., Packard, N., Kauffman, S.A., Poli, I.: A stochastic model of autocatalytic reaction networks. Theory Biosci. 131(2), 85–93 (2012)CrossRefGoogle Scholar
  6. 6.
    Bazhlekova, E.G.: Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal. 3(3), 213–230 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)CrossRefGoogle Scholar
  8. 8.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 1–13 (2015)Google Scholar
  9. 9.
    Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015)Google Scholar
  10. 10.
    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
  11. 11.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel. Therm. Sci. 20(2), 763–769 (2016)CrossRefGoogle Scholar
  12. 12.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Reyes-Reyes, J.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Modell. 40(21–22), 9079–9094 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)Google Scholar
  14. 14.
    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
  15. 15.
    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
  16. 16.
    Atangana, A., Gómez-Aguilar, J.F.: A new derivative with normal distribution kernel: theory, methods and applications. Phys. A: Stat. Mech. Appl. 476, 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)CrossRefGoogle Scholar
  19. 19.
    Saad, K., Gómez, F.: Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense. Rev. Mex. Fís 64(5), 539–547 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Gómez-Aguilar JF. Irving-Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel. Chaos Soliton. Fract. 2017; (95) 35: 1-7MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    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
  24. 24.
    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
  25. 25.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A: Stat. Mech. Appl. 491, 406–424 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F., Valtierra-Rodríguez, M.: Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order. Phys. A: Stat. Mech. Appl. 487, 1–21 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    Doungmo Goufo EF. Chaotic processes using the two-parameter derivative with non-singular and nonlocal kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016; 26(8), 1-21MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  30. 30.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam (2006)zbMATHGoogle Scholar
  31. 31.
    Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–13 (2017)CrossRefGoogle Scholar
  32. 32.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary order. Academic Press Inc, Cambridge (1974)zbMATHGoogle Scholar
  33. 33.
    Podlubny, I.: The Laplace transform method for Linear Differential Equations of the Fractional order (1997).
  34. 34.
    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
  35. 35.
    Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (2013)zbMATHGoogle Scholar
  36. 36.
    Brockmann, D., Hufnagel, L.: Front propagation in reaction-superdiffusion dynamics: taming Lévy flights with fluctuations. Phys. Rev. Lett. 98(17), 178–301 (2007)CrossRefGoogle Scholar
  37. 37.
    Doungmo Goufo, E.F.: Stability and convergence analysis of a variable-order replicator-mutator process in a moving medium. J. Theor. Biol. 403, 178–187 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Doungmo Goufo, E.F.: Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Math. Modell. Anal. 21(2), 188–198 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pooseh, S., Rodrigues, H.S., Torres, D.F.: Fractional derivatives in dengue epidemics. AIP Conf. Proc. 1389(1), 739–742 (2011)CrossRefGoogle Scholar
  40. 40.
    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
  41. 41.
    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
  42. 42.
    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
  43. 43.
    Lepik Ü, Hein H. Haar Wavelets: With Applications. Springer Science & Business Media, Berlin (2014)Google Scholar
  44. 44.
    Tonelli, L.: Sullintegrazione per parti. Rend. Acc. Naz. Lincei 5(18), 246–253 (1909)zbMATHGoogle Scholar
  45. 45.
    Fubini, G.: Opere scelte II. Cremonese, Roma (1958)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaFloridaSouth Africa
  2. 2.Institute of Groundwater Studies, Faculty of Natural and Agricultural SciencesUniversity of Free StateBloemfonteinSouth Africa

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