Full Text:   <710>

Summary:  <30>

CLC number: O415.5

On-line Access: 2020-07-10

Received: 2019-02-24

Revision Accepted: 2019-04-14

Crosschecked: 2019-10-18

Cited: 0

Clicked: 668

Citations:  Bibtex RefMan EndNote GB/T7714


Karthikeyan Rajagopal


-   Go to

Article info.
Open peer comments

Frontiers of Information Technology & Electronic Engineering  2020 Vol.21 No.7 P.1108-1118


Chaotic dynamics of a fractional order glucose-insulin regulatory system

Author(s):  Karthikeyan Rajagopal, Atiyeh Bayani, Sajad Jafari, Anitha Karthikeyan, Iqtadar Hussain

Affiliation(s):  Center for Nonlinear Dynamics, Defence University, Bishoftu 1041, Ethiopia; more

Corresponding email(s):   rkarthiekeyan@gmail.com, atiyeh.bayani@yahoo.com, sajad.jafari@tdtu.edu.vn, mrs.anithakarthikeyan@gmail.com, iqtadarqau@gmail.com

Key Words:  Diabetes mellitus, Chaos, Bifurcation, Multistability, Antimonotonicity

Share this article to: More <<< Previous Article|

Karthikeyan Rajagopal, Atiyeh Bayani, Sajad Jafari, Anitha Karthikeyan, Iqtadar Hussain. Chaotic dynamics of a fractional order glucose-insulin regulatory system[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(7): 1108-1118.

@article{title="Chaotic dynamics of a fractional order glucose-insulin regulatory system",
author="Karthikeyan Rajagopal, Atiyeh Bayani, Sajad Jafari, Anitha Karthikeyan, Iqtadar Hussain",
journal="Frontiers of Information Technology & Electronic Engineering",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Chaotic dynamics of a fractional order glucose-insulin regulatory system
%A Karthikeyan Rajagopal
%A Atiyeh Bayani
%A Sajad Jafari
%A Anitha Karthikeyan
%A Iqtadar Hussain
%J Frontiers of Information Technology & Electronic Engineering
%V 21
%N 7
%P 1108-1118
%@ 2095-9184
%D 2020
%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.1900104

T1 - Chaotic dynamics of a fractional order glucose-insulin regulatory system
A1 - Karthikeyan Rajagopal
A1 - Atiyeh Bayani
A1 - Sajad Jafari
A1 - Anitha Karthikeyan
A1 - Iqtadar Hussain
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 21
IS - 7
SP - 1108
EP - 1118
%@ 2095-9184
Y1 - 2020
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/FITEE.1900104

The fractional order model of a glucose-insulin regulatory system is derived and presented. It has been extensively proved in the literature that fractional order analysis of complex systems can reveal interesting and unexplored features of the system. In our investigations we have revealed that the glucose-insulin regulatory system shows multistability and antimonotonicity in its fractional order form. To show the effectiveness of fractional order analysis, all numerical investigations like stability of the equilibrium points, Lyapunov exponents, and bifurcation plots are derived. Various biological disorders caused by an unregulated glucose-insulin system are studied in detail. This may help better understand the regulatory system.


Karthikeyan RAJAGOPAL1,2, Atiyeh BAYANI3, Sajad JAFARI3,4, Anitha KARTHIKEYAN1, Iqtadar HUSSAIN5



Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Ackerman E, Rosevear JW, McGuckin WF, 1964. A mathematical model of the glucose-tolerance test. Phys Med Biol, 9(2):203-213.

[2]Adomian G, 1990. A review of the decomposition method and some recent results for nonlinear equations. Math Comput Model, 13(7):17-43.

[3]Aram Z, Jafari S, Ma J, et al., 2017. Using chaotic artificial neural networks to model memory in the brain. Commun Nonl Sci Numer Simul, 44:449-459.

[4]Baghdadi G, Jafari S, Sprott JS, et al., 2015. A chaotic model of sustaining attention problem in attention deficit disorder. Commun Nonl Sci Numer Simul, 20(1):174-185.

[5]Bajaj JS, Rao GS, Rao JS, et al., 1987. A mathematical model for insulin kinetics and its application to protein-deficient (malnutrition-related) diabetes mellitus (PDDM). J Theor Biol, 126(4):491-503.

[6]Bao BC, Wu PY, Bao H, et al., 2018. Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator. Chaos Sol Fract, 106:161-170.

[7]Bao H, Wang N, Bao BC, et al., 2018. Initial condition- dependent dynamics and transient period in memristor- based hypogenetic jerk system with four line equilibria. Commun Nonl Sci Numer Simul, 57:264-275.

[8]Buscarino A, Fortuna L, Frasca M, et al., 2018. Synchronization of chaotic systems with activity-driven time-varying interactions. J Compl Netw, 6(2):173-186.

[9]Caponetto R, Dongola G, Fortuna L, et al., 2010. New results on the synthesis of FO-PID controllers. Commun Nonl Sci Numer Simul, 15(4):997-1007.

[10]Charef A, Sun HH, Tsao YY, et al., 1992. Fractal system as represented by singularity function. IEEE Trans Autom Contr, 37(9):1465-1470.

[11]Chen DY, Liu YX, Ma XY, et al., 2012. Control of a class of fractional-order chaotic systems via sliding mode. Nonl Dynam, 67(1):893-901.

[12]Chuedoung M, Sarika W, Lenbury Y, 2009. Dynamical analysis of a nonlinear model for glucose-insulin system incorporating delays and β-cells compartment. Nonl Anal Theory Methods Appl, 71(12):e1048-e1058.

[13]Danca MF, 2015. Lyapunov exponents of a class of piecewise continuous systems of fractional order. Nonl Dynam, 81(1-2):227-237.

[14]Diethelm K, 1997. An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal, 5:1-6.

[15]Diethelm K, 2010. The Analysis of Fractional Differential Equations: an Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin, Germany.

[16]Diethelm K, Ford NJ, 2002. Analysis of fractional differential equations. J Math Anal Appl, 265(2):229-248.

[17]Diethelm K, Freed AD, 1999. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Proc Forschung und Wissenschaftliches Rechnen, p.57-71.

[18]Diethelm K, Ford NJ, Freed AD, 2004. Detailed error analysis for a fractional Adams method. Numer Algor, 36(1):31-52.

[19]Dudkowski D, Jafari S, Kapitaniak T, et al., 2016. Hidden attractors in dynamical systems. Phys Rep, 637:1-50.

[20]Elsadany AEA, El-Metwally HA, Elabbasy EM, et al., 2012. Chaos and bifurcation of a nonlinear discrete prey- predator system. Comput Ecol Softw, 2(3):169-180.

[21]Garrappa R, 2011. Predictor-corrector PECE Method for Fractional Differential Equations. MATLAB Central.

[22]Ginoux JM, Ruskeepää H, Perc M, et al., 2018. Is type 1 diabetes a chaotic phenomenon? Chaos Sol Fract, 111:198- 205.

[23]Ginoux JM, Naeck R, Ruhomally YB, et al., 2019. Chaos in a predator–prey-based mathematical model for illicit drug consumption. Appl Math Comput, 347:502-513.

[24]Hadaeghi F, Golpayegani MRH, Jafari S, et al., 2016. Toward a complex system understanding of bipolar disorder: a chaotic model of abnormal circadian activity rhythms in euthymic bipolar disorder. Aust New Zealand J Psych, 50(8):783-792.

[25]Hansen K, 1923. Oscillations in the blood sugar in fasting normal persons. J Int Med, 57(S4):27-32.

[26]Hilborn RC, 2000. Chaos and Nonlinear Dynamics: an Introduction for Scientists and Engineers (2nd). Oxford University Press, Oxford, USA.

[27]Holt TA, 2003. Nonlinear dynamics and diabetes control. Endocrinologist, 13(6):452-456.

[28]Korn H, Faure P, 2003. Is there chaos in the brain? II. Experimental evidence and related models. Comptes Rendus Biol, 326(9):787-840.

[29]Kroll MH, 1999. Biological variation of glucose and insulin includes a deterministic chaotic component. Biosystems, 50(3):189-201.

[30]Kumar PRAV, Sreedevi V, 2017. A review on influence of antidiabetic medications on quality of life. Eur J Pharm Med Res, 4(5):276-288.

[31]Lakshmikantham V, Vatsala AS, 2008. Basic theory of fractional differential equations. Nonl Anal Theory Methods Appl, 69(8):2677-2682.

[32]Lang DA, Matthews DR, Peto J, et al., 1979. Cyclic oscillations of basal plasma glucose and insulin concentrations in human beings. N Engl J Med, 301(19):1023-1027.

[33]Leonov GA, Kuznetsov NV, Vagaitsev VI, 2011. Localization of hidden Chuaʼs attractors. Phys Lett A, 375(23):2230- 2233.

[34]Leonov GA, Kuznetsov NV, Vagaitsev VI, 2012. Hidden attractor in smooth Chua systems. Phys D, 241(18):1482- 1486.

[35]Leonov GA, Kuznetsov NV, Mokaev TN, 2015a. Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commun Nonl Sci Numer Simul, 28(1-3):166-174.

[36]Leonov GA, Kuznetsov NV, Mokaev TN, 2015b. Homoclinic orbits, and self-excited and hidden attractors in a Lorenz- like system describing convective fluid motion. Eur Phys J Spec Top, 224(8):1421-1458.

[37]Liszka-Hackzell JJ, 1999. Prediction of blood glucose levels in diabetic patients using a hybrid AI technique. Comput Biomed Res, 32(2):132-144.

[38]Molnar GD, Taylor WF, Langworthy AL, 1972. Plasma immunoreactive insulin patterns in insulin-treated diabetics. Studies during continuous blood glucose monitoring. Mayo Clin Proc, 47(10):709-719.

[39]Petráš I, 2011. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin, Germany.

[40]Pfeiffer EF, Meyerhoff C, Bischof F, et al., 1993. On line continuous monitoring of subcutaneous tissue glucose is feasible by combining portable glucosensor with microdialysis. Horm Metab Res, 25(2):121-124.

[41]Pham VT, Frasca M, Caponetto R, et al., 2012. Control and synchronization of fractional-order differential equations of phase-locked loop. Chaot Model Simul, 4:623-631.

[42]Preissl H, Lutzenberger W, Pulvermüller F, 1996. Is there chaos in the brain? Behav Brain Sci, 19(2):307-308.

[43]Rajagopal K, Akgul A, Jafari S, et al., 2017a. Chaotic chameleon: dynamic analyses, circuit implementation, FPGA design and fractional-order form with basic analyses. Chaos Sol Fract, 103:476-487.

[44]Rajagopal K, Karthikeyan A, Srinivasan AK, 2017b. FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonl Dynam, 87(4): 2281-2304.

[45]Rajagopal K, Karthikeyan A, Duraisamy P, et al., 2019. Bifurcation, chaos and its control in a fractional order power system model with uncertainties. Asian J Contr, 21(1): 184-193.

[46]Rihan FA, Hashish A, Al-Maskari F, et al., 2016. Dynamics of tumor-immune system with fractional-order. J Tumor Res, 2(1):1000109J.

[47]Rocco A, West BJ, 1999. Fractional calculus and the evolution of fractal phenomena. Phys A, 265(3-4):535-546.

[48]Schiff SJ, Jerger K, Duong DH, et al., 1994. Controlling chaos in the brain. Nature, 370(6491):615-620.

[49]Shabestari PS, Panahi S, Hatef B, et al., 2018. A new chaotic model for glucose-insulin regulatory system. Chaos Sol Fract, 112:44-51.

[50]Sharma PR, Shrimali MD, Prasad A, et al., 2015a. Control of multistability in hidden attractors. Eur Phys J Spec Top, 224(8):1485-1491.

[51]Sharma PR, Shrimali MD, Prasad A, et al., 2015b. Controlling dynamics of hidden attractors. Int J Bifurc Chaos, 25(4): 1550061.

[52]Tavazoei MS, Haeri M, 2007. Unreliability of frequency- domain approximation in recognising chaos in fractional- order systems. IET Signal Process, 1(4):171-181.

[53]Tolba MF, AbdelAty AM, Soliman NS, et al., 2017. FPGA implementation of two fractional order chaotic systems. AEU-Int J Electron Commun, 78:162-172.

[54]Tsirimokou G, Psychalinos C, Elwakil AS, et al., 2018. Electronically tunable fully integrated fractional-order resonator. IEEE Trans Circ Syst II, 65(2):166-170.

[55]Wolf A, Swift JB, Swinney HL, et al., 1985. Determining Lyapunov exponents from a time series. Phys D, 16(3): 285-317.

[56]Yin C, Zhong SM, Chen WF, 2012. Design of sliding mode controller for a class of fractional-order chaotic systems. Commun Nonl Sci Numer Simul, 17(1):356-366.

[57]Zhao CN, Xue DY, Chen YQ, 2005. A fractional order PID tuning algorithm for a class of fractional order plants. Proc IEEE Int Conf on Mechatronics and Automation, p.216-221.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - Journal of Zhejiang University-SCIENCE