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Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model

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Abstract

In this paper, we report on the dynamics of a discrete-time mathematical model, which is obtained by the forward Euler method from the continuous-time mathematical model of the glycolysis process. More specifically, here we investigate the parameter-space of a two-dimensional map resulting from this discretization process. Different places where period-doubling and Naimark–Sacker bifurcations occur are determined. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region which is a result of a Naimark–Sacker bifurcation. We identify period-adding, Farey, and Fibonacci sequences of periodic structures embedded in this quasiperiodic region.

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References

  1. S.H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, New York, 1994)

    Google Scholar 

  2. J.M. Berg, J.L. Tymoczko, L. Stryer, Biochemistry (W H Freeman, New York, 2002)

    Google Scholar 

  3. F.A. Davidson, B.P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel’kov model. Proc. R. Soc. Edinb. Sect. A 130, 507516 (2000)

    Article  Google Scholar 

  4. M.X. Wang, Non-constant positive steady-states of the Sel’kov model. J. Differ. Equ. 190, 600–620 (2003)

    Article  Google Scholar 

  5. R. Peng, Qualitative analysis of steady states to the Sel’kov model. J. Differ. Equ. 241, 386–398 (2007)

    Article  Google Scholar 

  6. M. Wei, J. Wu, G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction. Nonlinear Anal. RWA 22, 155–175 (2015)

    Article  Google Scholar 

  7. Q. Din, Bifurcation analysis and chaos control in discrete-time glycolysis models. J. Math. Chem. 56, 904–931 (2018)

    Article  CAS  Google Scholar 

  8. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 2002)

    Google Scholar 

  9. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990)

    Book  Google Scholar 

  10. M. Sekikawa, N. Inaba, Doubly twisted Neimark–Sacker bifurcation and two coexisting two-dimensional tori. Phys. Lett. A 380, 171–176 (2016)

    Article  CAS  Google Scholar 

  11. Q. Din, A novel chaos control strategy for discrete-time Brusselator models. J. Math. Chem. 56, 3045–3075 (2018)

    Article  CAS  Google Scholar 

  12. Q. Din, T. Donchev, D. Kolev, Stability, bifurcation analysis and chaos control in chlorine dioxide-iodine-malonic acid reaction. MATCH Commun. Math. Comput. Chem. 79, 577–606 (2018)

    Google Scholar 

  13. J.A.C. Gallas, Structure of the parameter-space of the Hénon map. Phys. Rev. Lett. 70, 2714–2717 (1993)

    Article  CAS  PubMed  Google Scholar 

  14. H.G. Schuster, W. Just, Deterministic Chaos, An Introduction (Wiley-VCH, Weinheim, 2005)

    Book  Google Scholar 

  15. H.A. Albuquerque, R.M. Rubinger, P.C. Rech, Self-similar structures in a 2D parameter-space of an inductorless Chua’s circuit. Phys. Lett. A 372, 4793–4798 (2008)

    Article  CAS  Google Scholar 

  16. C. Bonatto, J.A.C. Gallas, Accumulation horizons and period adding in optically injected semiconductor lasers. Phys. Rev. E 75, 055204 (2007)

    Article  CAS  Google Scholar 

  17. T.S. Krüger, P.C. Rech, Dynamics of an erbium-doped fiber dual-ring laser. Eur. Phys. J. D 66, 12 (2012)

    Article  CAS  Google Scholar 

  18. F.G. Prants, P.C. Rech, Organization of periodic structures in a damped-forced oscillator. Eur. Phys. J. B 87, 196 (2014)

    Article  CAS  Google Scholar 

  19. P.C. Rech, Period-adding structures in the parameter-space of a driven Josephson junction. Int. J. Bifurcat. Chaos 25, 1530035 (2015)

    Article  Google Scholar 

  20. P.C. Rech, Spiral organization of periodic structures in the Lorenz–Stenflo system. Phys. Scr. 91, 075201 (2016)

    Article  CAS  Google Scholar 

  21. S.L.T. de Souza, A.M. Batista, M.S. Baptista, I.L. Caldas, J.M. Balthazar, Characterization in bi-parameter space of a non-ideal oscillator. Physica A 466, 224–231 (2017)

    Article  Google Scholar 

  22. P.C. Rech, How to embed shrimps in parameter planes of the Lorenz system. Phys. Scr. 92, 045201 (2017)

    Article  CAS  Google Scholar 

  23. P.C. Rech, Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system. Eur. Phys. J. B 90, 251 (2017)

    Article  CAS  Google Scholar 

  24. M. Borghezan, P.C. Rech, Chaos and periodicity in Valli’s model for El Niño. Chaos Solitons Fractals 97, 15–18 (2017)

    Article  Google Scholar 

  25. V. Wiggers, P.C. Rech, Chaos, periodicity, and quasiperiodicity in a radio-physical oscillator. Int. J. Bifurcat. Chaos 27, 1730023 (2017)

    Article  Google Scholar 

  26. A. da Silva, P.C. Rech, Numerical investigation concerning the dynamics in parameter planes of the Ehrhard–Müller system. Chaos Solitons Fractals 110, 152–157 (2018)

    Article  Google Scholar 

  27. R.A. Dunlap, The Golden Ratio and Fibonacci Numbers (World Scientific, Singapore, 2003)

    Google Scholar 

Download references

Acknowledgements

The author thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, and Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina-FAPESC, Brazilian Agencies, for financial support.

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  1. Departamento de Física, Universidade do Estado de Santa Catarina, Joinville, 89219-710, Brazil

    Paulo C. Rech

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  1. Paulo C. Rech
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Correspondence to Paulo C. Rech.

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Rech, P.C. Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model. J Math Chem 57, 632–637 (2019). https://doi.org/10.1007/s10910-018-0976-4

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