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Hopf bifurcation analysis in a fractionalorder survival red blood cells model and \(\mathit{PD}^{\alpha} \) control
Advances in Difference Equations volume 2018, Article number: 10 (2018)
Abstract
In this paper, we put forward a fractionalorder survival red blood cells model and study the dynamics through the Hopf bifurcation. When the delay transcends the threshold, a series of Hopf bifurcations occur at the positive equilibrium. Then, a fractionalorder Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) controller is applied to the proposed model for the Hopf bifurcation control. It is discovered that by setting proper parameters, the \(\mathit{PD}^{\alpha} \) controller can delay or advance the onset of Hopf bifurcations. Therefore the Hopf bifurcation of the fractionalorder survival red blood cells model becomes controllable to achieve desirable behaviors. Finally, numerical examples are presented to demonstrate the theoretical analysis.
1 Introduction
Fractional calculus was born in 1695 as an important branch of mathematics, almost simultaneously with classical calculus. Compared with integerorder derivatives, it has been found that fractional derivatives have the superiority of accuracy and flexibility when used to describe some nonclassical phenomena in natural science and engineering applications such as neurons [1], finance systems [2], biological systems [3], and so on. Especially in biological systems, fractional calculus has more advantages than traditional integerorder calculus in describing molecular dynamics with memory characteristics and historical dependence [4, 5]. Fractional calculus accumulates the global information of the function in a weighted form, which is also called memory. A large number of examples show that the fractional calculus has a more universal meaning than the integer calculus.
In recent years, there have been many papers about the stability and the Hopf bifurcation analysis of integerorder survival red blood cells models [6–9]. With the rapid development of biomedical and molecular biology, researchers have proposed some new research topics [10–12]. Many biological systems have shown the feature of fractal geometry, the characteristics of memory and diversity discharge activities which cannot be described accurately by using the classical calculus theory. It is known that the integerorder calculus is only determined by the local character of the function, while the fractionalorder one can gather the global information of the function in the weighted form [13]. In the biological field, Magin argued that the fractionalorder derivative can describe the activities of the organism more accurately [10, 11]. In [12], the dynamics of a red blood cells model is fully described by linear fractionalorder differential equations, and the theory of fractional calculus provides a concise way to describe and quantify the biomechanical behaviors of membranes, cells and tissues. However, the qualitative theory of bifurcations in a fractionalorder system is still a problem, which has not been researched thoroughly. It is more meaningful to investigate the fractionalorder survival red blood cells model instead of the integerorder counterpart.
For getting better desirable stability domain, we can add some effective controllers, such as hybrid controller [14, 15], state feedback controller [16, 17] and delayed feedback controller [18, 19], for dynamics control in nonlinear systems. As we know, the PID controller consists of the proportional unit P, the integrating unit I and the differential unit D. We can adjust the stability of the systems by setting the three control parameters \(k_{p}\), \(k_{i}\) and \(k_{d}\) in the PID controller. The PID controller is mainly applicable to systems with essentially linear and dynamic characteristics which are not changeable through time [20]. In consideration of the feature of fractionalorder systems, we develop a fractionalorder Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) scheme to control the bifurcation of the fractionalorder survival red blood cells model in this paper. It is worth mentioning that such a control strategy has not been reported in the control of bifurcation for fractionalorder systems. Motivated by the above discussions, we investigate the problem of bifurcation and control for the delayed fractionalorder survival red blood cells model in the present paper.
2 Model description
In [21], WazewskaCzyzewska and Lasota proposed the survival red blood cells model:
where \(X(t)\) represents the number of red blood cells in time t, a is the death rate of red blood cells, b and c describe the production of red blood cells per unit time and τ is the necessary time to produce a red blood cell.
There are many definitions of fractional derivatives. The GrünwaldLetnikov definition, the RiemannLiouville definition and the Caputo definition are usually used to deal with fractionalorder systems. Since the Caputo derivative only requires the initial conditions which are based on integerorder derivative and represents wellunderstood features of physical situation, it is more applicable to real world problems. Hence, the Caputo fractionalorder derivative is employed in this paper.
The Caputo fractionalorder derivative is defined as follows:
where \(d  1 < \alpha < d\), \(d \in N\), and \(\Gamma ( \cdot )\) is the gamma function. The symbol α denotes the value of the fractional order that is usually chosen in the range \(0 < \alpha \le 1\).
The Laplace transformation of the Caputo fractionalorder derivative is represented as
If \(f^{(k)}(0) = 0\), \(k = 0,1,\ldots,d  1\), then \(L\{ {}_{0}^{C}D_{t}^{\alpha} f(t)\} = s^{\alpha} F(s)\).
A class of ndimensional linear fractionalorder systems with multiple time delays can be represented in the following form [22]:
with the characteristic equation
where \(0 < \alpha_{i} \le 1\) (\(i = 1,2,3,\ldots,n\)), and \(d^{\alpha_{i}} / dt^{{\alpha_{i}}} \) is chosen as the Caputo fractional derivative (2.2).
Theorem 2.1
([22])
Given that all the roots of the characteristic equation (2.5) have negative real parts, the zero solution of system (2.4) is Lyapunov globally asymptotically stable.
Remark 2.1
Theorem 2.1 indicates that the stability boundary for the delayed fractionalorder system (2.4) is the imaginary axis.
Remark 2.2
If \(\tau_{ij} = 0\), \(i,j = 1,\ldots,n\), then Theorem 2.1 converts into Matignon criterion [23]: if all the roots λs of the equation \(\det (\lambda I  A) = 0\) satisfy \(\vert \arg (\lambda ) \vert > \alpha \pi / 2\), then the zero solution of system (2.4) is Lyapunov globally asymptotically stable, where \(A = (a_{ij})_{n \times n}\) is the coefficient matrix and \(\lambda = s^{\alpha}\). It can be seen that the stability boundary is described by \(\vert \arg (\lambda ) \vert = \alpha \pi / 2\) (or \(\vert \arg (s) \vert = \pi / 2\)) for the fractionalorder system (2.4) without delays.
Remark 2.3
If all the eigenvalues λs of A satisfy \(\vert \arg (\lambda ) \vert > \alpha \pi / 2\) and the characteristic equation (2.5) has no purely imaginary roots for any \(\tau_{ij} > 0\), \(i,j = 1,\ldots,n\), then the zero solution of system (2.4) is Lyapunov globally asymptotically stable [22].
For model (2.1), we let
Then we get
In this paper, we focus on the dynamics of the following fractionalorder survival red blood cells model with time delays:
We can easily see that, for model (2.6), there is a unique \(u^{*}\) satisfying the following equation:
It can be seen that \(u^{*} > 1\) if and only if \(0 < m < 1/e\).
3 Bifurcation analysis of the uncontrolled model
In this part, we investigate the stability of the fractionalorder survival red blood cells model (2.7), and some existence conditions of Hopf bifurcations are addressed.
Let \(u(t)  u^{*} = y(t)\), then the linearized model (2.7) is
with the characteristic equation
In the following, we investigate the roots distribution of equation (3.2) by regarding the time delay υ as the bifurcation parameter.
Let \(s = \omega (\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\) (\(\omega > 0\)). Then equation (3.2) becomes
Separating the real and imaginary parts gives
where \(a_{1} =  m\), \(a_{2} = mu^{*}\). It can be obtained from (3.4) that
Denote
Lemma 3.1
For equation (3.2), we have the following:

(i)
If \(m \ge 1 / e\), then all roots of the characteristic equation (3.2) have negative real parts.

(ii)
If \(0 < m < 1 / e\), then equation (3.2) has a pair of purely imaginary roots \(\pm \omega_{0}\) when \(\upsilon = \upsilon_{j}\), \(j = 0,1,\ldots \) , where
$$ \upsilon_{j} = \frac{1}{\omega_{0}}\arccos\biggl(\frac{a_{1}  \omega_{0}\cos \frac{q\pi}{2}}{a_{2}} + 2j\pi \biggr),\quad j = 0,1,2,\ldots, $$(3.6)where \(\omega_{0}\) is the unique positive zero of the function \(h_{1}(\omega )\).
Proof
(i) From \(\vert a_{1} \vert > a_{2}\), then \(h_{1}(0) > 0\), and the symmetry axis is \(a_{1}\cos \frac{q\pi}{2} < 0\). Combining \(q > 0\), we can see that equation (3.5) has no real root, so equation (3.2) has no purely imaginary root. This finishes the proof of (i).
(ii) By means of \(\vert a_{1} \vert < a_{2}\), it is easy to see that \(h_{1}(0) < 0\). Combining \(q > 0\), there exists a unique positive number \(\omega_{0}\) such that \(h_{1}(\omega ) = 0\). Then \(\omega_{0}\) is a root of equation (3.5). Hence, for \(\upsilon_{j}\) as defined in (3.6), \((\omega_{0},\upsilon_{j})\) is a root of equation (3.3). It can be seen that \(\pm \omega_{0}\) is a pair of purely imaginary roots of equation (3.2), while \(\upsilon = \upsilon_{j}\), \(j = 0,1,\ldots \) . This completes the proof of (ii). □
Remark 3.1
The conclusion (ii) of Lemma 3.1 gives the onset of Hopf bifurcation of model (2.7).
Here we make the following assumption:
where
Lemma 3.2
Let \(s(\upsilon ) = \varsigma (\upsilon ) + i\omega (\upsilon )\) be the root of equation (3.2). It is easy to see \(\varsigma (\upsilon_{j}) = 0\), \(\omega (\upsilon_{j}) = \omega_{0}\), when \(\upsilon = \upsilon_{j}\). If (\(H_{1}\)) holds, then we have
Proof
Differentiating equation (3.2) implicitly with respect to υ, we obtain
Hence, we deduce that
Obviously, hypothesis (\(H_{1}\)) means that the transversality condition is satisfied. □
Theorem 3.1
For model (2.7), when \(0 < m < 1 / e\), the following results hold:

(i)
The equilibrium \(u^{*}\) of model (2.7) is locally asymptotically stable for \(\upsilon \in [0,\upsilon_{0})\), and unstable when \(\upsilon > \upsilon_{0}\).

(ii)
Model (2.7) undergoes a Hopf bifurcation at the equilibrium \(u^{*}\) when \(\upsilon = \upsilon_{0}\).
Proof
Note that the eigenvalue \(\lambda =  (m + mu^{*}) < 0\) of the linearized system of (3.1) satisfies the inequality \(\vert \arg (\lambda ) \vert > q\pi / 2\) when \(\upsilon = 0\). Therefore, the condition for the Hopf bifurcation is satisfied.
(i) We can find that when \(\upsilon = 0\) the roots of equation (3.2) have negative real parts. In Lemma 3.1, we can see that all the roots of equation (3.2) have negative real parts for \(\upsilon \in [0,\upsilon_{0})\) by the definition of \(\upsilon_{0}\). From Lemma 3.2, this implies that equation (3.2) has at least a positive root when \(\upsilon > \upsilon_{0}\).
(ii) From the above discussion, it is obvious that the occurrence condition of the Hopf bifurcation is satisfied for (2.6). Therefore, near the equilibrium \(u^{*}\), there occurs a Hopf bifurcation when \(\upsilon = \upsilon_{0}\). □
4 Bifurcation analysis of the controlled model
In this part, by choosing the time delay υ as the bifurcation parameter, we are trying to control the Hopf bifurcation of (2.7) based on the fractionalorder Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) control strategy.
For the delayed fractionalorder model (2.7), we propose a single input and output \(\mathit{PD}^{\alpha} \) controller as follows:
where \(k_{p}\) is the proportional control parameter and \(k_{d}\) is the derivative control parameter.
Hence, the controlled fractionalorder survival red blood cells model with time delays becomes
Let \(u(t)  u^{*} = y(t)\), The controlled model (4.2) becomes
with the characteristic equation
Here, we assume the value range of the control parameters: \(k_{p} < m\), \(k_{d} < 1\).
Remark 4.1
Comparing model (4.3) and model (3.1), it is obvious that the controlled fractionalorder model and the uncontrolled one have the same equilibrium point.
Let \(s = \omega (\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\) (\(\omega > 0\)), then equation (4.4) becomes
Separating the real and imaginary parts, we get
where \(a_{3} = \frac{k_{p}  m}{1  k_{d}}\), \(a_{4} = \frac{mu^{*}}{1  k_{d}}\). It can be obtained from (4.5) that
Denote
Lemma 4.1
If \(\vert a_{3} \vert < a_{4}\), equation (4.4) has a pair of purely imaginary roots \(\pm \omega_{0}^{c}\) when \(\upsilon = \upsilon_{j}^{c}\), \(j = 0,1,\ldots \) , where
where \(\omega_{0}^{c}\) is the unique positive zero of the function \(h_{2}(\omega )\).
Proof
By means of \(\vert a_{3} \vert < a_{4}\), it is easy to see that \(h_{2}(0) < 0\). Combining with \(q > 0\), there exists a unique positive number \(\omega_{0}^{c}\) such that \(h_{2}(\omega ) = 0\). Then \(\omega_{0}^{c}\) is a root of (4.6). Hence, for \(\upsilon_{j}^{c}\) as defined in (4.7), \((\omega_{0}^{c},\upsilon_{j}^{c})\) is a root of equation (4.5). It can be seen that \(\pm \omega_{0}^{c}\) is a pair of purely imaginary roots of equation (4.4), while \(\upsilon = \upsilon_{j}^{c}\), \(j = 0,1,\ldots \) . □
Remark 4.2
Lemma 4.1 obtains the onset of the delayed fractionalorder model’s Hopf bifurcations.
We make the following assumption:
where
Lemma 4.2
Let \(s(\upsilon ) = \varsigma (\upsilon ) + i\omega (\upsilon )\) be the root of equation (4.4). It is easy to see \(\varsigma (\upsilon_{j}^{c}) = 0\), \(\omega (\upsilon_{j}^{c}) = \omega_{0}^{c}\) when \(\upsilon = \upsilon_{j}^{c}\). If (\(H_{2}\)) holds, then we have
Proof
Differentiating equation (4.4) relative to υ, we get
Hence, we obtain
□
Theorem 4.1
For model (4.2), when \(\vert a_{3} \vert < a_{4}\), we get the following results.

(i)
The equilibrium \(u^{*}\) of model (4.2) is locally asymptotically stable for \(\upsilon \in [0,\upsilon_{0}^{c})\), and unstable when \(\upsilon > \upsilon_{0}^{c}\).

(ii)
Model (4.2) undergoes a Hopf bifurcation at the equilibrium \(u^{*}\) when \(\upsilon = \upsilon_{0}^{c}\).
Proof
Note that the eigenvalue \(\lambda = \frac{1}{1  k_{d}}[(k_{p}  m)  mu^{*}] = a_{1}  a_{2} < 0\) of the linearized system of (4.3) satisfies the inequality \(\vert \arg (\lambda ) \vert > q\pi / 2\) when \(\upsilon = 0\). Therefore, the condition for the Hopf bifurcation is satisfied.
(i) It is easy to see that all the roots of equation (4.4) with \(\upsilon = 0\) have negative real parts. The definition of \(\upsilon_{0}^{c}\) implies that all the roots of equation (4.4) have negative real parts for \(\upsilon \in [0,\upsilon_{0}^{c})\). From Lemma 4.2, it indicates that equation (4.4) has at least a root with positive real parts when \(\upsilon > \upsilon_{0}^{c}\).
(ii) From the above discussion, it is clear that the occurrence condition of the Hopf bifurcation is satisfied for model (4.2). Therefore, near the equilibrium \(u^{*}\), there occurs a Hopf bifurcation when \(\upsilon = \upsilon_{0}^{c}\). □
5 Numerical simulations
In this section, we provide numerical simulations to confirm our theoretical analysis and display the Hopf bifurcation phenomenon of the delayed fractionalorder model.
For the uncontrolled model (2.7), we take \(\alpha = 0.92\), \(m = 0.2\) used in [7]. Then (2.7) has a positive equilibrium \(u^{*} = 1.3267\). From (3.6), we can obtain \(\upsilon_{0} = 19.6749\). The positive equilibrium point \(u^{*}\) is asymptotically stable when \(\upsilon = 18 < \upsilon_{0} = 19.6749 \) as illustrated in Figure 1, and when \(\upsilon = 21 > \upsilon_{0} = 19.6749\), there occurs a Hopf bifurcation at the positive equilibrium point \(u^{*}\) as demonstrated in Figure 2.
The effect of the order α from 0.9 to 1 on the values of \(\upsilon_{0} \) for model (2.7) is shown in Table 1.
In order to make a comparison with the uncontrolled fractionalorder model (2.7), we discuss the controlled model (4.2) with the parameters \(\alpha = 0.92\), \(m = 0.2\). When we choose the control parameters \(k_{p} =  0.05\), \(k_{d} = 0.5\), from (4.7) we can obtain \(\upsilon_{0}^{c} = 27.7129\). It can be seen that the stable region has been enlarged and the critical value \(\upsilon_{0}^{c}\) has been increased to a larger value than that of the uncontrolled model. This indicates that the \(\mathit{PD}^{\alpha} \) controller can delay the onset of Hopf bifurcations effectively. The equilibrium \(u^{*}\) is asymptotically stable when \(\upsilon = 24 < \upsilon_{0}^{c} = 27.7129\) as illustrated in Figure 3, and when \(\upsilon = 30 > \upsilon_{0}^{c} = 27.7129\), there occurs a Hopf bifurcation at the positive equilibrium point \(u^{*}\) as demonstrated in Figure 4.
Next, we select other control parameters to validate the effectiveness of our proposed \(\mathit{PD}^{\alpha} \) scheme in the bifurcation control. We also take the same parameters \(\alpha = 0.92\), \(m = 0.2\) for the original model (2.7). When \(k_{p} = 0.05\), \(k_{d} = 0.5\), from (4.7) one can obtain \(\upsilon_{0}^{c} = 6.1794\). The critical value \(\upsilon_{0}^{c}\) is smaller than that of the uncontrolled model (2.7), which shows that the \(\mathit{PD}^{\alpha} \) controller can advance the onset of the Hopf bifurcation effectively. The equilibrium \(u^{*}\) is asymptotically stable when \(\upsilon = 6 < \upsilon_{0}^{c} = 6.1794 \) as illustrated in Figure 5, and when \(\upsilon = 7 > \upsilon_{0}^{c} = 6.1794\), there occurs a Hopf bifurcation at the positive equilibrium point \(u^{*}\) as demonstrated in Figure 6.
6 Conclusions
In this paper, we have studied the Hopf bifurcation of a fractionalorder red blood cells model with time delay and have proposed the configuration for the stable region. In order to control the Hopf bifurcation of the delayed fractionalorder red blood cells model, we have designed a fractionalorder Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) controller, which can successfully delay or advance the onset of Hopf bifurcation. Therefore, we can choose appropriate values of the Proportional and Derivative parameters to change the characteristics of Hopf bifurcation embedding in fractionalorder systems with time delays.
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Acknowledgements
This work is partly supported by the National Natural Science Foundation of China (Nos. 61573194, 61573096), the Six Talent Peaks High Level Project of Jiangsu Province (No. 2014ZNDW004) and the Science Foundation of Nanjing University of Posts and Telecommunications (No. NY213095).
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Sun, Q., Xiao, M., Tao, B. et al. Hopf bifurcation analysis in a fractionalorder survival red blood cells model and \(\mathit{PD}^{\alpha} \) control. Adv Differ Equ 2018, 10 (2018). https://doi.org/10.1186/s136620171464z
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DOI: https://doi.org/10.1186/s136620171464z