On a fractionalorder delay MackeyGlass equation
 Ahmed MA ElSayed^{1}Email author,
 Sanaa M Salman^{2} and
 Naemaa A Elabd^{3}
https://doi.org/10.1186/s136620160863x
© ElSayed et al. 2016
Received: 10 February 2016
Accepted: 11 May 2016
Published: 18 May 2016
Abstract
In this paper, a fractionalorder MackeyGlass equation with constant delay is considered. The local stability of the fixed points is analyzed. Moreover, a discretization process is applied to convert the fractionalorder delay equation to its discrete analog. A numerical simulation including Lyapunov exponent, phase diagrams, bifurcation, and chaos is carried out using Matlab to ensure theoretical results and to reveal more complex dynamics of the equation after discretization.
Keywords
fractionalorder delay MackeyGlass equations fixed points local stability discretization Lyapunov exponent bifurcation chaos1 Introduction
DDEs arise in many areas of mathematical modeling: for example, population dynamics (taking into account the gestation times), infectious diseases (accounting for the incubation periods), physiological and pharmaceutical kinetics (modeling, for example, the body’s reaction to CO_{2}, etc. in circulating blood), chemical kinetics (such as mixing reactants), the navigational control of ships and aircraft (with, respectively, large and short lags), and more general control problems (see for example [4–6]).
On the other hand, fractional calculus is a generalization of classical differentiation and integration to arbitrary (noninteger) order [7–9]. Many mathematicians and applied researchers have tried to model real processes using fractional calculus [10–16]. In recent years differential equations with fractionalorder have attracted many researchers because of their applications in many areas of science and engineering. Analytical and numerical techniques have been implemented to study such equations. The fractional calculus has allowed the operations of integration and differentiation to be applied for any fractional order [17–21].
We recall the basic definitions (Caputo) and properties of fractionalorder differentiation and integration.
Definition 1

\(I^{\beta}_{a}: L^{1} \rightarrow L^{1}\), and if \(f(x)\in L^{1}\), then \(I^{\gamma}_{a} I^{\beta}_{a} f(x)=I^{\gamma+\beta}_{a} f(x)\).

\(\lim_{\beta\rightarrow n} I^{\beta}_{a} f(x) = I^{n}_{a} f(x)\) uniformly on \([a,b ]\), \(n=1,2,3,\ldots \) , where \(I^{1}_{a} f(x)=\int_{a}^{x} f(s) \,ds\).

\(\lim_{\beta\rightarrow0} I^{\beta}_{a} f(x)=f(x)\) weakly.

If \(f(x)\) is absolutely continuous on \([a,b]\), then \(\lim_{\alpha\rightarrow1} D^{\alpha}_{a} f(x)= \frac{df(x)}{dx}\).
In this work, we will show that considering a fractionalorder derivative with delay in equation (1.2) will exhibit more complex and richer dynamics.
2 Discretization process
The steps of the discretization process are as follows.
In the following, we will discuss two cases of the delay: Case I: \(\tau=r\), and Case II: \(\tau=2r\).
3 Case I: \(\tau=r\)

For all values of the parameter ρ, system (3.2) has one fixed point, namely, \(\mathrm{fix}_{1}=(0,0)\).

For \(\rho>1\), we have an additional fixed point, which is \(\mathrm{fix}_{2}=(\sqrt[c]{\rho1},\sqrt[c]{\rho1})\).
 1.
\(F:= 1+ T +D > 0\),
 2.
\(TC:= 1 T +D > 0\),
 3.
\(H:= 1D > 0\),
Proposition 1
The fixed point \(\mathrm{fix}_{1}\) is locally asymptotically stable if \(\rho <(1+2/R)\), and losses stability via a flip bifurcation when \(\rho>1\) and via a NeimarkSacker bifurcation when \(\rho> \frac{r^{\alpha }\Gamma(1+\alpha)}{r^{\alpha}}\).
Proof
Proposition 2
The fixed point \(\mathrm{fix}_{2}\) of system (3.2) is stable if \(\rho< \frac{cR}{2+cR}\), and it loses stability via a pitchfork bifurcation if \(\rho> \frac{cR}{2+cR}\), via a flip bifurcation if \(\rho>1\), and via a NeimarkSacker bifurcation if \(\rho< \frac{cR}{cR1}\).
Proof
4 Case II: \(\tau=2r\)
Existence and stability of fixed points

For all parameter values, there is only one fixed point \(\mathrm{fix}\, x_{1}=(0,0,0)\).

For \(\rho>1\), there is an additional fixed point \(\mathrm{fix}\, x_{2}=(\sqrt[c]{\rho1},\sqrt[c]{\rho1},\sqrt[c]{\rho1})\).

\(a_{3}<1 \Rightarrow\rho>1\frac{1}{R}\),

\( b_{3}>b_{1} \Rightarrow(1(R\rho R)^{2})>(R\rho R)\),

\(c_{3}> c_{2}\Rightarrow(1(R\rho R)^{2})^{2}(R\rho R)^{2}>(R\rho R)^{2}1+(R\rho R)\).

\(p(1)>0 \Rightarrow \rho > 1\),

\(p(1)<0 \Rightarrow \rho < \frac{cr^{\alpha}}{cr^{\alpha }2\Gamma(\alpha+1)}\),

\(a_{3} < 1 \Rightarrow \rho < \frac{cr^{\alpha}}{cr^{\alpha }\Gamma(\alpha+1)}\),

\( b_{3}  > b_{1} \Rightarrow (1+\frac{c^{2}R^{2}}{\rho ^{2}}(1\rho)^{2}) > \frac{cR(\rho1)}{\rho}\).
5 Numerical simulation
6 Conclusion
In this paper, the dynamic behavior of a fractionalorder delay MackeyGlass equation is investigated after applying a discretization process to it. We have considered two different cases for the delay τ, the first is when \(\tau=r\), and the second is when \(\tau =2r\), where r is the discretization parameter. Stability of the fixed points and local bifurcations of fixed points of the discretized systems in the two cases was are analyzed. A numerical simulation was carried out to ensure our theoretical analysis and to reveal the more complex dynamics of the system.
Declarations
Acknowledgements
The authors would like to thank the referees of this manuscript for their valuable comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 an der Heiden, U: Delays in physiological systems. J. Math. Biol. 8, 345364 (1979) MathSciNetView ArticleMATHGoogle Scholar
 Glass, L, Mackey, MC: Pathological conditions resulting from instabilities in physiological control systems. Ann. N.Y. Acad. Sci. 316, 214235 (1979) View ArticleMATHGoogle Scholar
 Mackey, MC: Commodity price fluctuations: price dependent delays and non linearities as explanatory factors. J. Econ. Theory 48(2), 497509 (1989) MathSciNetView ArticleMATHGoogle Scholar
 Baleanu, D, Magin, RL, Bhalekar, S, DaftardarGejji, V: Chaos in the fractional order nonlinear Bloch equation with delay. Commun. Nonlinear Sci. Numer. Simul. 25(13), 4149 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Wu, GC, Baleanu, D: Discrete chaos in fractional delayed logistic maps. Nonlinear Dyn. 80(4), 16971703 (2015) MathSciNetView ArticleGoogle Scholar
 Jarad, F, Abdeljawad, T, Baleanu, D: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218(18), 92349240 (2012) MathSciNetMATHGoogle Scholar
 Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
 Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) MATHGoogle Scholar
 Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993) MATHGoogle Scholar
 Yang, XJ, Srivastava, HM, Cattani, C: Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics. Rom. Rep. Phys. 67(3), 752761 (2015) Google Scholar
 Yang, XJ, Srivastava, HM: An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 29(1), 499504 (2015) MathSciNetView ArticleGoogle Scholar
 Yang, XJ, Baleanu, D, Srivastava, HM: Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 47, 5460 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Wang, J, Ye, Y, Pan, X, Gao, X, Zhuang, C: Fractional zerophase filtering based on the RiemannLiouville integral. Signal Process. 98(5), 150157 (2014) View ArticleGoogle Scholar
 Wang, J, Ye, Y, Gao, Y, Gao, X, Qian, S: Fractional compound integral with application to ECG signal denoising. Circuits Syst. Signal Process. 34, 19151930 (2015) MathSciNetView ArticleGoogle Scholar
 Wang, J, Ye, Y, Pan, X, Gao, X: Paralleltype fractional zerophase filtering for ECG signal denoising. Biomed. Signal Process. Control 18, 3641 (2015) View ArticleGoogle Scholar
 Wang, J, Ye, Y, Gao, X: Fractional 90^{∘} phaseshift filtering based on the doublesided GrunwaldLetnikov differintegrator. IET Signal Process. 9(4), 328334 (2015) View ArticleGoogle Scholar
 Bhrawy, AH, Alhamed, YA, Baleanu, D, AlZahrani, AA: New spectral techniques for systems of fractional differential equations using fractionalorder generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17, 11371157 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Hafez, RM, EzzEldien, SS, Bhrawy, AH, Ahmed, EA, Baleanu, D: A Jacobi GaussLobatto and GaussRadau collocation algorithm for solving fractional FokkerPlanck equations. Nonlinear Dyn. 82(3), 14311440 (2015) MathSciNetView ArticleGoogle Scholar
 Bhrawy, AH, Hafez, RM, Alzahrani, E, Baleanu, D, Alzahrani, AA: Generalized LaguerreGaussRadau scheme for the first order hyperbolic equations in a semiinfinite domain. Rom. J. Phys. 60, 918934 (2015) Google Scholar
 Bhrawy, AH, Abdelkawy, MA, Alzahrani, AA, Baleanu, D, Alzahrani, E: A ChebyshevLaguerre GaussRadau collocation scheme for solving time fractional subdiffusion equation on a semiinfinite domain. Proc. Rom. Acad., Ser. A 16, 490498 (2015) MathSciNetGoogle Scholar
 Doha, EH, Bhrawy, AH, Baleanu, D, EzzEldien, SS, Hafez, RM: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 2015, 15 (2015) MathSciNetView ArticleGoogle Scholar
 El Raheem, ZF, Salman, SM: On a discretization process of fractionalorder logistic differential equation. J. Egypt. Math. Soc. 22, 407412 (2014) MathSciNetView ArticleMATHGoogle Scholar
 ElSayed, AMA, ElRaheem, ZF, Salman, SM: Discretization of forced Duffing system with fractionalorder damping. Adv. Differ. Equ. 2014, 66 (2014) MathSciNetView ArticleGoogle Scholar
 ElSayed, AMA, Salman, SM: On a discretization process of fractionalorder Riccati differential equation. J. Fract. Calc. Appl. 4(2), 251259 (2013) Google Scholar
 Elaidy, SN: An Introduction to Difference Equations, 3rd edn. Undergraduate Texts in Mathematics. Springer, New York (2005) Google Scholar
 Palis, J, Takens, F: Hyperbolicity and Sensitive Chaotic Dynamics and Homoclinic Bifurcation. Cambridge University Press, Cambridge (1993) MATHGoogle Scholar
 Puu, T: Attractors, Bifurcation and Chaos: Nonlinear Phenomena in Economics. Springer, Berlin (2000) View ArticleMATHGoogle Scholar
 Udwadia, FE, von Bremen, H: A note on the computation of the largest pLyapunov characteristic exponents for nonlinear dynamical systems. J. Appl. Math. Comput. 114, 205214 (2000) View ArticleMATHGoogle Scholar