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On a fractionalorder delay MackeyGlass equation
Advances in Difference Equations volume 2016, Article number: 137 (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.
Introduction
Delay differential equations (DDEs) arise in the mathematical description of systems whose time evolution depends explicitly on a past state of the system, as for example in the case of delayed feedback. Neural systems [1], respiration regulation [2], agricultural commodity markets [3], nonlinear optics, and neutrophil populations in the blood [2] are but a few systems in which delayed feedback leads naturally to a description in terms of a delay differential equation. We will restrict our attention to systems modeled by evolutionary delay equations that can be expressed in the form
Here the ‘state’ of the system at time t is \(x(t)\), whose rate of change depends explicitly, via the function f, on the past state \(x(t\tau)\) where τ is a fixed time delay. More general delay equations might be considered: multiple time delays, variable time delays, continuously distributed delays, and higher derivatives all arise in applications and lead to more complicated evolution equations. Nevertheless, equations of the form (1.1) constitute a sufficiently broad class of systems to be of practical importance, and they will provide adequate fodder for the types of problems we wish to consider.
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
The fractional integral of order \(\beta\in\mathbb{R^{+}}\) of the function \(f (t)\), \(t > 0\) is defined by
and the fractional derivative of order \(\alpha\in(n1,n)\) of \(f(t)\), \(t>0\) is defined by
In addition, the following results are the main features in fractional calculus. Let \(\beta, \gamma\in\mathbb{R^{+}}\), \(\alpha\in(0,1)\):

\(I^{\beta}_{a}: L^{1} \rightarrow L^{1}\), 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 \(\lim_{\alpha\rightarrow1} D^{\alpha}_{a} f(x)= \frac{df(x)}{dx}\).
The MackeyGlass equation is a nonlinear time delay differential equation, which was proposed as a model of hematopoiesis, given by
where γ, c, ρ, τ are real parameters, and \(x_{\tau}\) represents the value of the variable x at time \((t\tau)\). Depending on the values of the parameters, this equation displays a range of periodic and chaotic dynamics.
In this work, we will show that considering a fractionalorder derivative with delay in equation (1.2) will exhibit more complex and richer dynamics.
Consider the fractionalorder delay MackeyGlass equation given in the form
with the initial condition
where \(\alpha \in (0, 1]\), \(\rho \in R^{+}\), and \(c > 0\). In equation (1.3), we consider delay in the last term.
Discretization process
In this part, we apply the discretization process represented in [22, 23], and [24] for discretizing the delay fractionalorder Mackey Glass equation with piecewise constant arguments given by
with initial condition (1.4).
The steps of the discretization process are as follows.
Let \(t \in [0,r)\), then \(\frac{t}{r} \in [0,1)\). That is,
and the solution of (2.2) is given by
Let \(t \in [r,2r)\), then \(\frac{t}{r} \in [1,2)\). That is,
and the solution of (2.3) is given by
Repeating the process we can easily get
Let \(t \rightarrow (n+1)r \), we obtain the discretization
It is worth to pay attention here that Euler’s discretization method is an approximation for the derivative while the predictorcorrector method is an approximation for the integral. However, our proposed discretization method here is an approximation for the righthand side of the system under consideration as is pretty clear from (2.4). Moreover, we have noticed that when \(\alpha \rightarrow1\), the discretization will be Euler’s discretization.
In the following, we will discuss two cases of the delay: Case I: \(\tau=r\), and Case II: \(\tau=2r\).
Case I: \(\tau=r\)
In this case we have a secondorder difference equation given by
Existence and stability of fixed points
To find the fixed points of system (3.1), we first split it into two firstorder difference equations as follows:

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 \(\mathrm{fix}_{2}=(\sqrt[c]{\rho1},\sqrt[c]{\rho1})\).
In order to study the local stability of these fixed points, we need the moduli of the eigenvalues of the Jacobian matrix evaluated at each of the fixed points [25]. The Jacobian matrix of system (3.2) evaluated at any fixed point \((x,y)\) is obtained by
The eigenvalues associated to the Jacobian matrix are
where
The fixed points \(\mathrm{fix}_{1}\), \(\mathrm{fix}_{2}\) of the system equation (3.1) are stable if \(\lambda_{i} < 1\), \(i=1,2\). In order to study the qualitative behavior of the solution of system (3.2) we rely on the Jury criteria given generally by

1.
\(F:= 1+ T +D > 0\),

2.
\(TC:= 1 T +D > 0\),

3.
\(H:= 1D > 0\),
where the trace and determinant of the Jacobian matrix are given, respectively, by
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
The Jacobian matrix at the first fixed point \(\mathrm{fix}_{1}\) is obtained by
which has two eigenvalues,
According to the Jury criteria [26, 27], where \(T=1\), \(D= \frac{r^{\alpha}(1\rho)}{\Gamma(1+\alpha)}\), the first condition is always satisfied, while the second and third may be violated. That is, the fixed point \(\mathrm{fix}_{1}\) loses stability via a flip bifurcation when \(\rho>1\), and via a NeimarkSacker bifurcation when \(\rho> \frac {r^{\alpha}\Gamma(1+\alpha)}{r^{\alpha}}\). □
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
Calculating the Jacobian matrix at the second fixed point \(\mathrm{fix}_{2}\) of system (3.2) we obtain
which has two eigenvalues,
where the trace and determinant of \(J(\mathrm{fix}_{2})\) are given, respectively, by
According to the Jury criteria, the three conditions may be all violated. That is, \(\mathrm{fix}_{2}\) 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 \(\rho< \frac{cR}{cR1}\). □
Case II: \(\tau=2r\)
In this section, we take the delay to be \(\tau=2r\) in equation (1.3). Applying the discretization process we end up with a system of thirdorder difference equations given by
To study the fixed points of system (4.1) we first split it into three firstorder difference equations as follows:
In the following, we study the local stability of the fixed points of the system (4.2).
Existence and stability of fixed points
System (4.2) has the following 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 \(\mathrm{fix}\, x_{2}=(\sqrt[c]{\rho1},\sqrt[c]{\rho1},\sqrt[c]{\rho1})\).
By considering a Jacobian matrix for one of these fixed points and calculating their eigenvalues, we can investigate the stability of each fixed point based on the roots of the system characteristic equation. The Jacobian matrix is given by
where \(R=\frac{r^{\alpha}}{\Gamma(1+\alpha)}\), \(S=\frac{\rho+\rho z^{c}(1c)}{(1+z^{c})^{2}}\).
Linearizing the system (4.2) about \(\mathrm{fix}\, x_{1}\) yields the following characteristic equation:
Let
From the Jury test, if \(P(1)>0\), \(P(1)<0\), and \(a_{3}<1\), \(b_{3}>b_{1}\), \(c_{3}>c_{2}\), where \(b_{3}=1a^{2}_{3}\), \(b_{2}=a_{1}a_{3}a_{2}\), \(b_{1}=a_{2}a_{3}a_{1}\), \(c_{3}=b^{2}_{3}b^{2}_{1}\), and \(c_{2}=b_{3}b_{2}b_{1}b_{2}\), then the roots of \(P(\lambda)\) satisfy \(\lambda <1\) and thus \(\mathrm{fix}_{1}\) is asymptotically stable.
The first condition gives \(\rho<1\), while the second condition gives
The remaining conditions give the following inequalities:

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

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

\(c_{3}> c_{2}\Rightarrow(1(R\rho R)^{2})^{2}(R\rho R)^{2}>(R\rho R)^{2}1+(R\rho R)\).
Linearizing the system (4.2) about \(\mathrm{fix}\,x_{2}\) yields the following characteristic equation:
Let
From the Jury test, if \(P(1)>0\), \(P(1)<0\), and \(a_{3}<1\), \(b_{3}>b_{1}\), \(c_{3}>c_{2}\), where \(b_{3}=1a^{2}_{3}\), \(b_{2}=a_{1}a_{3}a_{2}\), \(b_{1}=a_{2}a_{3}a_{1}\), \(c_{3}=b^{2}_{3}b^{2}_{1}\), and \(c_{2}=b_{3}b_{2}b_{1}b_{2}\), then the roots of \(P(\lambda)\) satisfy \(\lambda <1\) and thus \(\mathrm{fix}_{1}\) is asymptotically stable.
We are going to check these conditions at \(\mathrm{fix}\, x_{2}\):

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

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

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

\( b_{3}  > b_{1} \Rightarrow (1+\frac{c^{2}R^{2}}{\rho ^{2}}(1\rho)^{2}) > \frac{cR(\rho1)}{\rho}\).
Thus, any condition may be violated resulting in instability of \(\mathrm{fix}\, x_{2}\).
Numerical simulation
In this section, a numerical simulation is carried out with the aid of Matlab to illustrate our theoretical results and to reveal the more complex dynamics of equation (1.2) in the two cases \(\tau =r\) and \(\tau=2r\). In all numerical simulations, we take \(c=6\), and \(r=0.5\). First of all, let us consider system (3.2). Indeed, if one is interested in determining whether a dynamical system is chaotic or not, often just a few of the largest Lyapunov characteristic exponents (LCEs) may provide the answer. This actually is so because a positive LCE is a good indicator for chaos. Since for nonchaotic systems all LCEs are nonpositive, the presence of a positive LCE has often been used to help determine if a system is chaotic or not. In this paper, we compute the LCEs via the Householder QRbased methods described in [28]. For system (3.2), we get when \(\alpha=0.95\), \(c=6\), and \(r=0.5\), \(\mathrm{LCE}1=0.3397\), and \(\mathrm{LCE}2=0.185\) as shown in Figure 1. We vary the parameter ρ and fix the other parameters, c, r, and α. Bifurcation diagrams of system (3.2) are also shown in Figure 1 for different values of the fractionalorder parameter α. If we consider \(\alpha=0.95\), it is shown that the fixed point \(\mathrm{fix}_{1}(0,0)\) is stable if \(\rho<1\), and at \(\rho>1\) it losses stability via a flip bifurcation. Afterwards, a stable periodic solution of period 2 appears, then the periodic solution of period 4 becomes unstable, and a periodic solution of period 8 appears and chaos happens. Figure 2 shows the different phase plane for system (3.2) for \(\alpha=0.95\). For \(\rho=1.55\), Figure 2(a) shows an invariant closed curve bifurcating from \(\mathrm{fix}_{1}(0,0)\), while for \(\rho =3.6\), Figure 2(b) shows a chaotic attractor. Now we vary the parameter α from 0 to 1 and fix ρ to plot the bifurcation diagram for system (3.2) as a function of α as shown in Figure 3.
Next, we turn to the second case, when \(\tau=2r\). Figure 4 shows the bifurcation diagram for system (4.2) as a function of ρ. If \(\rho=0.95\), the figure shows that the fixed point \(\mathrm{fix}\,x_{1}\) becomes unstable when \(\rho>1\) as it loses stability via a flip bifurcation. Then the stable period2 orbit appears at \(\rho =1.2\), which in turn loses stability; then chaos appears. Finally, Figure 5 shows the phase plane for system (4.2) for \(\alpha=0.95\). Figure 5(a) shows a stable fixed point \(\mathrm{fix}\, x_{1}\) for \(\rho=0.6\), Figure 5(b) shows a double scroll for \(\rho =1.2\), and Figure 5(c), (d) shows chaotic attractors for \(\rho =2\), and \(\rho=2.7\), respectively.
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.
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ElSayed, A.M., Salman, S.M. & Elabd, N.A. On a fractionalorder delay MackeyGlass equation. Adv Differ Equ 2016, 137 (2016). https://doi.org/10.1186/s136620160863x
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Keywords
 fractionalorder delay MackeyGlass equations
 fixed points
 local stability
 discretization
 Lyapunov exponent
 bifurcation
 chaos