Top

Published in:

2018 | OriginalPaper | Chapter

Bifurcation Analysis and Chaotic Behaviors of Fractional-Order Singular Biological Systems

Authors : Komeil Nosrati, Christos Volos

Published in: Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this chapter, singular system theory and fractional calculus are utilized to model the biological systems in the real world, some fractional-order singular (FOS) biological systems are established, and some qualitative analyses of proposed models are performed. Through the fractional calculus and economic theory, a new and more realistic model of biological systems predator-prey, logistic map and SEIR epidemic system have been extended, and besides some mathematical analysis, the numerical simulations are considered to illustrate the effectiveness of the numerical method to explore the impacts of fractional-order and economic interest on the presented systems in biological contexts. It will be demonstrated that the presence of fractional-order changes the stability of the solutions and enrich the dynamics of system. In addition, singular models exhibit more complicated dynamics rather than standard models, especially the bifurcation phenomena and chaotic behaviors, which can reveal the instability mechanism of systems. Toward this aim, some materials including several definitions and existence theorems of uniqueness of solution, stability conditions and bifurcation phenomena in FOS systems and detailed introductions to fundamental tools for discussing complex dynamical behavior, such as chaotic behavior have been added.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

next chapter Chaos and Bifurcation in Controllable Jerk-Based Self-Excited Attractors

For the singular system (2), if \(\det (sE - A) \ne 0\) for some complex number \(s\), then the pair \((E,A)\) is said to be regular (Yang et al. 2012).

Abdelouahab MS, Hamri NE, Wang J (2012) Hopf bifurcation and chaos in fractional-order modified hybrid optical system. Nonlin Dyn 69(1):275–284MathSciNetCrossRefMATH

Alligood K, Sauer T, Yorke J (1997) An introduction to dynamical systems. Springer, New YorkMATH

Area I, Batarfi H, Losada J, Nieto JJ et al (2015) On a fractional order Ebola epidemic model. Adv Differ Equ 278(1):1–12MathSciNetMATH

Atanackovic TM, Stankovic B (2004) An expansion formula for fractional derivatives and its application. Fract Calculus Appl Anal 7(3):365–378MathSciNetMATH

Ayasun S, Nwankpa CO, Kwatny HG (2004) Computation of singular and singularity induced bifurcation points of differential-algebraic power system model. IEEE Trans Cir Syst I 51(8):1525–1537MathSciNetCrossRefMATH

Baker GL, Gollub JP (1990) Chaotic dynamics; an introduction. Cambridge University Press, CambridgeMATH

Campbell SL (1980) Singular systems of differential equations. Priman, LondonMATH

Caputo M (1966) Linear models of dissipation whose Q is almost frequency independent. Ann Geophys 19(4):383–393

Chakraborty K, Das S, Kar TK (2011) Optimal control of effort of a stage structured prey–predator fishery model with harvesting. Nonlin Anal Real World Appl 12(6):3452–3467MathSciNetCrossRefMATH

Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resource. Wiley, New YorkMATH

Dai L (1989) Singular control system. Springer, New YorkCrossRef

Diethelm K (2010) The analysis of fractional differential equations. Springer, BerlinCrossRefMATH

Doungmo Goufo EF, Maritz R, Munganga J (2014) Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence. Adv Differ Equ 278(1):1–9MathSciNetMATH

Duan GR (2010) Analysis and design of descriptor linear systems. Springer, New YorkCrossRefMATH

Freedman HI (1980) Deterministic mathematical models in population ecology. Marcel Dekker, New YorkMATH

Gakkhar S, Naji RK (2003) Existence of chaos in two-prey, one-predator system. Chaos Soli Frac 17(4):639–649MathSciNetCrossRefMATH

Gakkhar S, Singh B (2007) The dynamics of a food web consisting of two preys and a harvesting predator. Chaos Soli Frac 34(4):1346–1356MathSciNetCrossRefMATH

Giannakopoulos K, Deliyannis T, Hadjidemetriou J (2002) Means for detecting chaos and hyperchaos in nonlinear electronic circuits. In: 14th international conference on digital signal processing, Santorini, Greece, 1–3 July 2002

Glendinning P, Perry LP (1997) Melnikov analysis of chaos in a simple epidemiological model. J Math Biol 35(3):359–373MathSciNetCrossRefMATH

Gordon H (1954) The economic theory of a common property resource: the fishery. J Polit Econ 62(2):124–142CrossRef

Greenhalgh D, Khan QJA, Lewis FI (2004) Hopf bifurcation in two SIRS density dependent epidemic models. Math Comp Model 39(11):1261–1283MathSciNetCrossRefMATH

Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New YorkCrossRefMATH

Hartman P (2002) Ordinary differential equations. Cambridge University Press, CambridgeCrossRefMATH

Kaczorek T (2011) Selected problems of fractional systems theory. Springer, LondonCrossRefMATH

Kaczorek T, Rogowski K (2015) Fractional linear systems and electrical circuits. Springer, BialystokCrossRefMATH

Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. In: Proceedings of the royal society, London

Kielhoefer H (2004) Bifurcation theory: an introduction with applications to PDEs. Springer, New YorkCrossRef

Kot M (2001) Elements of mathematical biology. Cambridge University Press, Cambridge

Kumar A, Daoutidis P (1999) Control of nonlinear differential-algebraic equation systems with applications to chemical process. CRC Press, LondonMATH

Kuznetsov YA, Piccardi C (1994) Bifurcation analysis of periodic SEIR and SIR epidemic models. Math Bio 32(2):109–121MathSciNetCrossRefMATH

Lewis FL (1986) A survey of linear singular systems. Circuits Syst Signal Proc 5(1):3–36MathSciNetCrossRefMATH

Li XZ, Gupur G, Zhu GT (2001) Threshold and stability results for an age-structured SEIR epidemic model. Comp Math Appl 42(6):883–907MathSciNetCrossRefMATH

Liu Z, Lu P (2014) Stability analysis for HIV infection of CD4 + T-cells by a fractional differential time-delay model with cure rate. Adv Differ Equ 1:1–20MathSciNet

Luenberger DG (1977) Dynamic Equations in Descriptor Form. IEEE Trans Automat Control 22(3):312–321MathSciNetCrossRefMATH

Luenberger DG, Arbel A (1997) Singular dynamic Leontief systems. Econometrica 45:991–995CrossRefMATH

Marszalek W, Trzaska ZW (2005) Singularity-induced bifurcations in electrical power system. IEEE Trans Pow Syst 20(1):302–310CrossRef

Masoud M, Masoud S, Caro L et al (2006) Introducing a new learning method for fuzzy descriptor systems with the aid of spectral analysis to forecast solar activity. J Atmo Sol-Terr Phy 68(18):2061–2074CrossRef

May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467CrossRefMATH

May RM, Oster GF (1976) Bifurcation and dynamic complexity in simple ecological models. Amer Nat 110(974):573–599CrossRef

Munkhammar J (2013) Chaos in a fractional order logistic map. Fract Calc Appl Anal 16(3):511–519MathSciNetCrossRefMATH

N’Doye I, Darouach M, Zasadzinski M et al (2013) Robust stabilization of uncertain descriptor fractional-order systems. Automatica 49(6):1907–1913MathSciNetCrossRefMATH

Nosrati K, Shafiee M (2017) Dynamic analysis of fractional-order singular Holling type-II predator–prey system. Appl Math Comput 313:159–179MathSciNetCrossRef

Olsen LF, Schaffer WM (1990) Chaos versus periodicity: alternative hypotheses for childhood epidemics. Science 249:499–504

Ozalp N, Demirci E (2011) A fractional order SEIR model with vertical transmission. Math Comput Model 54(1):1–6MathSciNetCrossRefMATH

Petras I (2011) Fractional-order nonlinear systems: Modeling, analysis and simulation. Springer, New YorkCrossRefMATH

Podlubny I (1998) Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, CaliforniaMATH

Rivero M, Trujillo JJ, Vazquez L et al (2011) Fractional dynamics of population. Appl Math Comput 218(3):1089–1095MathSciNetCrossRefMATH

Rostamy D, Mottaghi E (2016) Stability analysis of a fractional-order epidemics model with multiple equilibriums. Adv Differ Equ. https://doi.org/10.1186/s13662-016-0905-4 MathSciNet

Sotomayor J (1973) Generic bifurcations of dynamical systems. Dynamical Systems. Academic Press, New YorkMATH

Sun CJ, Lin YP, Tang SP (2007) Global stability for a special SEIR epidemic model with nonlinear incidence rates. Chaos Soli Frac 33(1):290–297MathSciNetCrossRefMATH

Tavazoei MS, Haeri M, Attari M, Bolouki S et al (2009a) More details on analysis of fractional-order Van der Pol oscillator. J Vib Control 15(6):803–819

Tavazoei MS, Haeri M, Attari M (2009b) A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8):1886–1890

Tavazoei MS (2010) A note on fractional-order derivatives of periodic functions. Automatica 46(5):945–948MathSciNetCrossRefMATH

Venkatasubramanian V, Schaettler H, Zaborszky J (1995) Local bifurcations and feasibility regions in differential-algebraic systems. IEEE Trans Auto Contr 40(12):1992–2013MathSciNetCrossRefMATH

Wu GC, Baleanu D (2014) Discrete fractional logistic map and its chaos. Nonlin Dyn 75(1):283–287MathSciNetCrossRefMATH

Xu WB, Liu HL, Yu JY et al (2005) Stability results for an age-structured SEIR epidemic model. J Sys Sci Inf 3(3):635–642

Yao YU, Zhuang JIAO, Chang-Yin SUN (2013) Sufficient and necessary condition of admissibility for fractional-order singular system. Acta Autom Sin 39(12):2160–2164MathSciNet

Yang C, Zhang Q, Zhou L (2012) Stability analysis and design for nonlinear singular systems. Springer, Berlin

Yude, J, Qiu J (2015) Stabilization of fractional-order singular uncertain systems. ISA Trans 56:53-64

Yue M, Schlueter R (2004) Bifurcation subsystem and its application in power system analysis. IEEE Trans Pow Syst 19(4):1885–1893CrossRef

Zhang JS (1990) Singular system economy control theory. Tsinghua Press, Beijing

Zhang Y, Zhang QL, Zhao LC et al (2007) Tracking control of chaos in singular biological economy systems. J Nor Uni 28(2):157–164MathSciNetMATH

Zhang Y, Zhang QL (2007) Chaotic control based on descriptor bioeconomic systems. Cont Dec 22(4):445–452MathSciNet

Zhang G, Zhu L, Chen B (2010) Hopf bifurcation and stability for a differential-algebraic biological economic system. Appl Math Comput 217(1):330–338MathSciNetMATH

Zhang Q, Liu C, Zhang X (2012) Complexity, analysis and control of singular biological systems. Springer, LondonCrossRefMATH

Zhang Y, Zhang Q, Yan XG (2014) Complex dynamics in a singular Leslie-Gower predator–prey bioeconomic model with time delay and stochastic fluctuations. Phys A 404:180–191MathSciNetCrossRef

Zhang X, Chen Y (2017) Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: the 0 < α < 1 case. ISA Trans. https://doi.org/10.1016/j.isatra.2017.03.008

Title: Bifurcation Analysis and Chaotic Behaviors of Fractional-Order Singular Biological Systems
Authors: Komeil Nosrati
Christos Volos
Publisher: Springer International Publishing
Book: Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors
Print ISBN: 978-3-319-71242-0

Electronic ISBN: 978-3-319-71243-7

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-71243-7_1

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner