 Research
 Open Access
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
 chaos
1 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
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