1 Introduction

Fractional calculus was born in 1695 as an important branch of mathematics, almost simultaneously with classical calculus. Compared with integer-order derivatives, it has been found that fractional derivatives have the superiority of accuracy and flexibility when used to describe some non-classical 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 integer-order 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 integer-order 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 integer-order calculus is only determined by the local character of the function, while the fractional-order one can gather the global information of the function in the weighted form [13]. In the biological field, Magin argued that the fractional-order 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 fractional-order 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 fractional-order system is still a problem, which has not been researched thoroughly. It is more meaningful to investigate the fractional-order survival red blood cells model instead of the integer-order 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 fractional-order systems, we develop a fractional-order Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) scheme to control the bifurcation of the fractional-order 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 fractional-order systems. Motivated by the above discussions, we investigate the problem of bifurcation and control for the delayed fractional-order survival red blood cells model in the present paper.

2 Model description

In [21], Wazewska-Czyzewska and Lasota proposed the survival red blood cells model:

$$ \frac{dX}{dt} = - aX(t) + be^{ - cX(t - \tau )},\quad a > 0,b > 0,c > 0,\tau > 0,t \ge 0, $$
(2.1)

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 Grünwald-Letnikov definition, the Riemann-Liouville definition and the Caputo definition are usually used to deal with fractional-order systems. Since the Caputo derivative only requires the initial conditions which are based on integer-order derivative and represents well-understood features of physical situation, it is more applicable to real world problems. Hence, the Caputo fractional-order derivative is employed in this paper.

The Caputo fractional-order derivative is defined as follows:

$$ {}_{e}^{C}D_{t}^{\alpha} f(t) = \frac{1}{\Gamma (d - \alpha )} \int_{e}^{t} (t - \tau )^{d - \alpha - 1}f^{(n)} (\tau )\,d\tau, $$
(2.2)

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 fractional-order derivative is represented as

$$ L\bigl\{ {}_{0}^{C}D_{t}^{\alpha} f(t)\bigr\} = s^{\alpha} F(s) - \sum_{k = 0}^{d - 1} s^{\alpha - k - 1}f^{(k)} (0). $$
(2.3)

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 n-dimensional linear fractional-order systems with multiple time delays can be represented in the following form [22]:

$$ \begin{gathered} \frac{d^{\alpha_{1}}x_{1}}{dt^{\alpha_{1}}} = a_{11}x_{1}(t - \tau_{11}) + a_{12}x_{2}(t - \tau_{12}) + \cdots + a_{1n}x_{n}(t - \tau_{1n}), \\ \frac{d^{\alpha_{2}}x_{2}}{dt^{\alpha_{2}}} = a_{21}x_{1}(t - \tau_{21}) + a_{22}x_{2}(t - \tau_{22}) + \cdots + a_{2n}x_{n}(t - \tau_{2n}), \\ \cdots \\ \frac{d^{\alpha_{n}}x_{n}}{dt^{\alpha_{n}}} = a_{n1}x_{1}(t - \tau_{n1}) + a_{n2}x_{2}(t - \tau_{n2}) + \cdots + a_{nn}x_{n}(t - \tau_{nn}), \end{gathered} $$
(2.4)

with the characteristic equation

$$ \det \left ( \begin{matrix} s^{\alpha_{1}} - a_{11}e^{ - s\tau_{11}} & - a_{12}e^{ - s\tau_{12}} & \cdots & - a_{1n}e^{ - s\tau_{1n}} \\ - a_{21}e^{ - s\tau_{21}} & s^{\alpha_{2}} - a_{22}e^{ - s\tau_{22}} & \cdots & - a_{2n}e^{ - s\tau_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ - a_{n1}e^{ - s\tau_{n1}} & - a_{n2}e^{ - s\tau_{n2}} & \ldots & s^{\alpha_{n}} - a_{nn}e^{ - s\tau_{nn}} \end{matrix} \right ) = 0, $$
(2.5)

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 fractional-order 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 fractional-order 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

$$m = \frac{a}{bc},\qquad \upsilon = \tau bc,\qquad u(t) = cX\biggl( \frac{t}{bc}\biggr). $$

Then we get

$$ \frac{du(t)}{dt} = - mu(t) + e^{ - u(t - \upsilon )},\quad t \ge 0,m > 0,\upsilon > 0. $$
(2.6)

In this paper, we focus on the dynamics of the following fractional-order survival red blood cells model with time delays:

$$ \frac{d^{\alpha} u(t)}{d^{\alpha} t} = - mu(t) + e^{ - u(t - \upsilon )},\quad t \ge 0,m > 0,\upsilon > 0. $$
(2.7)

We can easily see that, for model (2.6), there is a unique \(u^{*}\) satisfying the following equation:

$$ mu^{*} = e^{ - u^{*}}. $$
(2.8)

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 fractional-order 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

$$ \frac{d^{\alpha} y}{dt^{\alpha}} = - my(t) - mu^{*}y(t - \upsilon ), $$
(3.1)

with the characteristic equation

$$ s^{q} + m + mu^{*}e^{ - s\upsilon} = 0. $$
(3.2)

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

$$ \omega^{q}\cos \frac{q\pi}{2} + \omega^{q}\sin \frac{q\pi}{2} \cdot i + m + m \cdot u^{*}(\cos \omega \upsilon - i \sin \omega \upsilon ) = 0. $$
(3.3)

Separating the real and imaginary parts gives

$$ \textstyle\begin{cases} a_{2}\cos \omega \upsilon = a_{1} - \omega^{q}\cos \frac{q\pi}{2}, \\ a_{2}\sin \omega \upsilon = \omega^{q}\sin \frac{q\pi}{2}, \end{cases} $$
(3.4)

where \(a_{1} = - m\), \(a_{2} = mu^{*}\). It can be obtained from (3.4) that

$$ \omega^{2q} - 2a_{1}\omega^{q}\cos \frac{q\pi}{2} + a_{1}^{2} - a_{2}^{2} = 0. $$
(3.5)

Denote

$$h_{1}(\omega ) = \omega^{2q} - 2a_{1} \omega^{q}\cos \frac{q\pi}{2} + a_{1}^{2} - a_{2}^{2}. $$

Lemma 3.1

For equation (3.2), we have the following:

  1. (i)

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

  2. (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:

$$(H_{1}) \quad \frac{P_{1}Q_{1} + P_{2}Q_{2}}{Q_{1}^{2} + Q_{2}^{2}} > 0, $$

where

$$\begin{gathered} P_{1} = mu^{*} \omega_{0}\sin \omega_{0}\upsilon_{0}, \\ P_{2} = mu^{*}\omega_{0}\cos \omega_{0}\upsilon_{0}, \\ Q_{1} = q\omega_{0}^{q - 1}\cos \frac{(q - 1)}{2} \pi - mu^{*}\upsilon \cos \omega_{0}\upsilon_{0}, \\ Q_{2} = q\omega_{0}^{q - 1}\sin \frac{(q - 1)}{2} \pi + mu^{*}\upsilon \sin \omega_{0}\upsilon_{0}. \end{gathered} $$

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

$$\operatorname{Re} \biggl[ \frac{ds}{d\tau} \biggr]_{\omega = \omega_{0},\upsilon = \upsilon_{0}} > 0. $$

Proof

Differentiating equation (3.2) implicitly with respect to υ, we obtain

$$ \frac{ds}{d\upsilon} = \frac{smu^{*}e^{ - s\upsilon}}{qs^{q - 1} - \upsilon mu^{*}e^{ - s\upsilon}}. $$
(3.7)

Hence, we deduce that

$$\operatorname{Re} \biggl[ \frac{ds}{d\upsilon} \biggr]_{\omega = \omega_{0},\upsilon = \upsilon_{0}} = \frac{P_{1}Q_{1} + P_{2}Q_{2}}{Q_{1}^{2} + Q_{2}^{2}}. $$

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:

  1. (i)

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

  2. (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 fractional-order Proportional and Derivative (\(\mathit{PD}^{\alpha} \)) control strategy.

For the delayed fractional-order model (2.7), we propose a single input and output \(\mathit{PD}^{\alpha} \) controller as follows:

$$ \rho (t) = k_{p}\bigl(u(t) - u^{*}\bigr) + k_{d}\frac{d^{\alpha}}{dt^{\alpha}} \bigl(u(t) - u^{*} \bigr), $$
(4.1)

where \(k_{p}\) is the proportional control parameter and \(k_{d}\) is the derivative control parameter.

Hence, the controlled fractional-order survival red blood cells model with time delays becomes

$$ \begin{aligned}[b] \frac{d^{\alpha} u}{dt^{\alpha}} &= - mu(t) + e^{ - u(t - \upsilon )} + \rho (t) \\ &= - mu(t) + e^{ - u(t - \upsilon )} + k_{p}\bigl(u(t) - u^{*} \bigr) + k_{d}\frac{d^{\alpha}}{dt^{\alpha}} \bigl(u(t) - u^{*}\bigr). \end{aligned} $$
(4.2)

Let \(u(t) - u^{*} = y(t)\), The controlled model (4.2) becomes

$$ \begin{aligned}[b] \frac{d^{\alpha} y}{dt^{\alpha}} &= \frac{1}{1 - k_{d}}\bigl[ - my(t) - mu^{*}y(t - \upsilon ) + k_{p}y(t)\bigr] \\ &= \frac{1}{1 - k_{d}}\bigl[(k_{p} - m)y(t) - mu^{*}y(t - \upsilon )\bigr], \end{aligned} $$
(4.3)

with the characteristic equation

$$ s^{q} - \frac{1}{1 - k_{d}}\bigl[(k_{p} - m) - mu^{*}e^{ - s\upsilon} \bigr] = 0. $$
(4.4)

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 fractional-order 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

$$\omega^{q}\biggl(\cos \frac{q\pi}{2} + i\sin \frac{q\pi}{2} \biggr) - \frac{k_{p} - m}{1 - k_{d}} + \frac{mu^{*}}{1 - k_{d}}(\cos \omega \upsilon - i\sin \omega \upsilon ) = 0. $$

Separating the real and imaginary parts, we get

$$ \textstyle\begin{cases} a_{4}\cos \omega \upsilon = a_{3} - \omega^{q}\cos \frac{q\pi}{2}, \\ a_{4}\sin \omega \tau = \omega^{q}\sin \frac{q\pi}{2}, \end{cases} $$
(4.5)

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

$$ \omega^{2q} - 2a_{1}\omega^{q}\cos \frac{q\pi}{2} + a_{3}^{2} - a_{4}^{2} = 0. $$
(4.6)

Denote

$$h_{2}(\omega ) = \omega^{2q} - 2a_{1} \omega^{q}\cos \frac{q\pi}{2} + a_{3}^{2} - a_{4}^{2}. $$

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

$$ \upsilon_{j}^{c} = \frac{1}{\omega_{0}^{c}}\arccos \biggl( \frac{a_{3} - \omega_{0}^{c}\cos \frac{q\pi}{2}}{a_{4}} + 2j\pi \biggr),\quad j = 0,1,2,\ldots, $$
(4.7)

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 fractional-order model’s Hopf bifurcations.

We make the following assumption:

$$(H_{2}) \quad \frac{M_{1}N_{1} + M_{2}N_{2}}{N_{1}^{2} + N_{2}^{2}} > 0 , $$

where

$$\begin{gathered} M_{1} = \frac{\omega_{0}^{c}mu^{*}}{1 - k_{d}}\sin \omega_{0}^{c} \upsilon_{0}^{c}, \\ M_{2} = \frac{\omega_{0}^{c}mu^{*}}{1 - k_{d}}\cos \omega_{0}^{c} \upsilon_{0}^{c}, \\ N_{1} = q\bigl(\omega_{0}^{c} \bigr)^{q - 1}\cos \frac{(q - 1)}{2}\pi - \frac{\upsilon mu^{*}}{1 - k_{d}}\cos \omega_{0}^{c}\upsilon_{0}^{c}, \\ N_{2} = q\bigl(\omega_{0}^{c} \bigr)^{q - 1}\sin \frac{(q - 1)}{2}\pi + \frac{\upsilon mu^{*}}{1 - k_{d}}\sin \omega_{0}^{c}\upsilon_{0}^{c}. \end{gathered} $$

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

$$\operatorname{Re} \biggl[ \frac{ds}{d\upsilon} \biggr]_{\omega = \omega_{0}^{c},\upsilon = \upsilon_{0}^{c}} > 0. $$

Proof

Differentiating equation (4.4) relative to υ, we get

$$ \frac{ds}{d\upsilon} = \frac{\frac{smu^{*}}{1 - k_{d}}e^{ - s\upsilon}}{ qs^{q - 1} - \frac{\upsilon mu^{*}}{1 - k_{d}}e^{ - s\upsilon}}. $$
(4.8)

Hence, we obtain

$$\operatorname{Re} \biggl[ \frac{ds}{d\upsilon} \biggr]_{\omega = \omega_{0}^{c},\upsilon = \upsilon_{0}^{c}} = \frac{M_{1}N_{1} + M_{2}N_{2}}{N_{1}^{2} + N_{2}^{2}}. $$

 □

Theorem 4.1

For model (4.2), when \(\vert a_{3} \vert < a_{4}\), we get the following results.

  1. (i)

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

  2. (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 fractional-order 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.

Figure 1
figure 1figure 1

Waveform plot and phase portrait of the fractional-order model ( 2.7 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), and the initial value \(\pmb{u_{0} = 1}\). The equilibrium \(u^{*}\) is asymptotically stable when \(\upsilon = 18 < \upsilon_{0} = 19.6749\).

Full size image
Figure 2
figure 2figure 2

Waveform plot and phase portrait of the fractional-order model ( 2.7 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), and the initial value \(\pmb{u_{0} = 1}\). A Hopf bifurcation occurs and the periodic oscillation bifurcates from \(u^{*}\), where \(\upsilon = 21 > \upsilon_{0} = 19.6749\).

Full size image

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.

Table 1 Bifurcation point \(\pmb{\upsilon_{0} }\) versus fractional order α for model ( 2.7 )
Full size table

In order to make a comparison with the uncontrolled fractional-order 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.

Figure 3
figure 3figure 3

Waveform plot and phase portrait of the controlled fractional-order model ( 4.2 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), \(\pmb{k_{p} = - 0.05}\), \(\pmb{k_{d} = 0.5}\) and the initial value \(\pmb{u_{0} = 2}\). The equilibrium \(u^{*}\) is asymptotically stable when \(\upsilon = 24 < \upsilon_{0}^{c} = 27.7129\).

Full size image
Figure 4
figure 4figure 4

Waveform plot and phase portrait of the controlled fractional-order model ( 4.2 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), \(\pmb{k_{p} = - 0.05}\), \(\pmb{k_{d} = 0.5}\) and the initial value \(\pmb{u_{0} = 2}\). A Hopf bifurcation occurs and the periodic oscillation bifurcates from \(u^{*}\), where \(\upsilon = 30 > \upsilon_{0}^{c} = 27.7129\).

Full size image

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.

Figure 5
figure 5figure 5

Waveform plot and phase portrait of the controlled fractional-order model ( 4.2 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), \(\pmb{k_{p} = 0.05}\), \(\pmb{k_{d} = 0.5}\) and the initial value \(\pmb{u_{0} = 2}\). The equilibrium \(u^{*}\) is asymptotically stable when \(\upsilon = 6 < \upsilon_{0}^{c} = 6.1794\).

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Figure 6
figure 6figure 6

Waveform plot and phase portrait of the controlled fractional-order model ( 4.2 ) with \(\pmb{\alpha = 0.92}\), \(\pmb{m = 0.2}\), \(\pmb{k_{p} = 0.05}\), \(\pmb{k_{d} = 0.5}\) and the initial value \(\pmb{u_{0} = 2}\). A Hopf bifurcation occurs and the periodic oscillation bifurcates from \(u^{*}\), where \(\upsilon = 7 > \upsilon_{0}^{c} = 6.1794\).

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6 Conclusions

In this paper, we have studied the Hopf bifurcation of a fractional-order 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 fractional-order red blood cells model, we have designed a fractional-order 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 fractional-order systems with time delays.