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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Chaos and Its Control in a Fractional Order Glucose-insulin Regulatory System

Journal of Applied Nonlinear Dynamics 11(4) (2022) 875--893 | DOI:10.5890/JAND.2022.12.008

P. Y. Dousseh$^{1}$, L. A. Hinvi$^{1,2}$, C.H. Miwadinou$^{1,3,4}$, A. V. Monwanou$^1$, J. B. Chabi Orou$^1$

$^1$ Laboratoire de M'ecanique des Fluides, de la Dynamique Nonlin'eaire et de la Mod'elisation des Syst`emes ; Institut de Math'ematiques et de Sciences Physiques, Porto-Novo, B'enin

$^2$ D'epartement des Sciences Industrielles et Techniques, INSTI-Lokossa, Universit'e Nationale des Sciences, Technologiques, Ing'enierie et Math'ematiques (UNSTIM) Abomey, B'enin

$^3$ D'epartement de Physique, ENS-Natitingou, Universit'e Nationale des Sciences, Technologies, Ing'enierie et Math'ematiques (UNSTIM) Abomey, B'enin

$^4$ Laboratoire de Physique et Applications de Natitingou, Universit'e Nationale des Sciences, Technologies, Ing'enierie et Math'ematiques (UNSTIM) Abomey, B'enin

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Abstract

In this paper, dynamic behaviors and chaos control of a Fractional Order Glucose-Insulin Regulatory System (FOGIRS) recently introduced in the literature are studied. Some interesting behaviors such as periodic motions, antimonotonicity, and period-doubling route to chaos are observed. Besides, it is found that the minimum effective dimension for the system to generate chaos is 2.904 in the conmensurate order case and 2.882 in the incommensurate order case. To control chaos in this system, two controllers are designed. First, a feedback controller to stabilize the chaotic system to one of its equilibrium points, and a robust adaptive fractional order sliding mode controller is used to globally stabilize the chaotic FOGIRS with uncertain dynamics. Numerical simulations are used to show the effectiveness of those controllers.

References

  1. [1]  Chuedoung, M., Sarika, W., and Lenbury, Y. (2009), Dynamical analysis of a nonlinear model for glucose-insulin system incorporating delays and $\beta$-cells compartment, Nonlinear Analysis, 71(12), e1048-e1058.
  2. [2]  Rother, K.I. (2007), Diabetes treatment-bridging the divide, The New England journal of medecine, 356(15), 1499-1501.
  3. [3]  Herman, W.H. and Zimmet, P. (2012), Type 2 diabetes: an epidemic requiring global attention and urgent action, Diabetes Care, 35(5), 943-944.
  4. [4]  Bolie, V.W. (1961), Coefficients of normal blood glucose regulation, Journal of applied physiology, 16(5), 783-788.
  5. [5]  Ackerman, E., Rosevear, J.W., and McGuckin, W.F. (1964), A mathematical model of the glucose-tolerance test, Physics in medecine and Biology, 9(2), 203-213.
  6. [6]  Bajaj, J.S., Rao, G.S., Rao, J.S., and Khardori, R. (1987), A mathematical model for insulin kinetics and its application to protein-deficient (malnutrition-related) diabetes mellitus (PDDM), Journal of theoretical biology, 126(4), 491-503.
  7. [7]  Geevan, C.P., Rao, J.S., Rao, G.S., and Bajaj, J.S. (1990), A mathematical model for insulin kinetics III. Sensitivity analysis of the model, Journal of theoretical biology, 147(2), 255-263.
  8. [8]  Lenbury, Y., Ruktamatakul, S., and Amornsamarnkul, S. (2001), Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions, Biosystems, 59(1), 15-25.
  9. [9]  Shabestari, P.S., Panahi, S., Hatef, B., Jafari, S., and Sprott, J.C. (2018), A new chaotic model for glucose-insulin regulatory system, Chaos, Solitons and fractals, 112, 44-51.
  10. [10]  Andrievskii, B.R. and Fradkov, A.L. (2003), Control of chaos: methods and applications. I. Methods, Automation and Remote Control, 64(5), 673-713.
  11. [11]  Yin, C., Dadras, S., Zhong, S., and Chen, Y. (2013), Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Applied Mathematical Modelling, 37(4), 2469-2483.
  12. [12]  Chen, D., Liu, Y., Ma, X., and Zhang, R. (2012), Control of a class of fractional-order chaotic systems via sliding mode, Nonlinear Dynamics, 67(1), 893-901.
  13. [13]  Dadras, S. and Momeni, H.R. (2009), Control uncertain Genesio-Tesi chaotic system: Adaptive sliding mode approach, Chaos, Solitons and Fractals, 42(5), 3140-3146.
  14. [14]  Dadras, S. and Momeni, H.R. (2010), Control of fractional-order economical system via sliding mode, Physica A, 389(12), 2434-2442.
  15. [15]  Perruquetti, W. and Barbot, J.P. (2002), Sliding Mode Control in Engineering, Marcel Dekker Inc.: New York.
  16. [16]  Roopaei, M., Sahraei, B.R., and Lin, T.C. (2010), Adaptive sliding mode control in a novel class of chaotic systems, Communications in Nonlinear Science and Numerical Simulation, 15(12), 4158-4170.
  17. [17]  Konishi, K., Hirai, M., and Kokame, H. (1998), Sliding mode control for a class of chaotic systems, Physics Letters A, 245(6), 511-517.
  18. [18]  Nazzal, J.M. and Natsheh, A.N. (2007), Chaos control using sliding mode theory, Chaos, Solitions and Fractals, 33(2), 695-702.
  19. [19]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press: New York.
  20. [20]  Hartley, T.T., Lorenzo, C.F., and Qammer, H.K. (1995), Chaos in a fractional order Chua's system, IEEE Transactions on Circuits and Systems I, 42(8), 485-490.
  21. [21]  Grigorenko, I. and Grigorenko, E. (2003), Chaotic dynamics of the fractional Lorenz system Physical Review Letters, 91(3), 034101.
  22. [22]  Daftardar-Gejji, V. and Bhalekar, S. (2010), Chaos in fractional order Liu system, Computers and mathematics with applications, 59(3), 1117-1127.
  23. [23]  Li, C. and Chen, G. (2004), Chaos and hyperchaos in the fractional-order R\"ossler equations, Physica A, 341, 55-61.
  24. [24]  Liu, X. and Hong, L. (2013), Chaos and adaptive synchronizations in fractional-order systems, International Journal of Bifurcation and Chaos, 23(11), 1350175-1-12.
  25. [25]  Li, C. and Chen, G. (2004), Chaos in the fractional-order Chen system and its control, Chaos, Solitons and Fractals, 22(3), 549-554.
  26. [26]  Faieghi, M.R. and Delavari, H. (2012), Chaos in fractional-order Genesio-Tesi system and its synchronization, Communications in Nonlinear Science and Numerical Simulation, 17(2), 731-741.
  27. [27]  Rajagopal, K., Bayani, A., Jafari, S., Karthikeyan, A., and Hussain, I. (2019), Chaotic dynamics of a fractional order glucose-insulin regulatory system, FITEE, 21(7), 1108-1118.
  28. [28]  Diethelm, K., Ford, N.J., and Freed, A.D. (2002), A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29(1), 3-22.
  29. [29]  Tavazoei, M.S. and Haeri, M. (2008), Chaotic attractors in incommensurate fractional order systems, Physica D, 237(20), 2628-2637.
  30. [30]  Matignon, D. (1996), Stability results for fractional differential equations with applications to control processing, Computational engineering in systems applications, 2(1), 963-968.
  31. [31]  Tavazoei, M.S. and Haeri, M. (2007), A necessary condition for double scroll attractor existence in fractional order systems, Physics Letters A, 367(1-2), 102-113.
  32. [32]  Zeng, C.B., Yang, Q.G., and Wang, J.W. (2011), Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci, Nonlinear Dynamics, 65(4), 457-466.
  33. [33]  Ott, E., Grebogi, C., and Yorke, J.A. (1990), Controlling chaos, Physical Review Letters, 64(11), 1196-9.
  34. [34]  Chen, G. and Yu, X. (2003), Chaos control: theory and applications, Springer-Verlag: Berlin.
  35. [35]  Ahmed, E., El-Sayed, A.M.A., and El-Saka, H.A.A. (2006), On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R\"ossler, Chua and Chen systems, Physics Letters A, 358(1), 1-4.
  36. [36]  Wang, X.Y., He, Y.J., and Wang, M.J. (2009), Chaos control of a fractional-order modified coupled dynamos system, Nonlinear Analysis, 71(12), 6126-6134.
  37. [37]  Chen, G. and Dong, X. (1998), From chaos to order: methodologies, perspectives and applications, World Scientific: Singapore.
  38. [38]  Hajipour, A. and Tavakoli, H. (2017), Dynamic analysis and adaptive sliding mode controller for a chaotic fractional incommensurate order financial system, International Journal of Bifurcation and Chaos, 27(13), 1750198-1-14.

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