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- Open Access
Hopf bifurcation analysis in a fractional-order survival red blood cells model and \(\mathit{PD}^{\alpha} \) control
- Qingshan Sun^{1},
- Min Xiao^{1}Email author,
- Binbin Tao^{1},
- Guoping Jiang^{1},
- Jinde Cao^{2, 3},
- Fuchen Zhang^{4} and
- Chengdai Huang^{5}
https://doi.org/10.1186/s13662-017-1464-z
© The Author(s) 2018
- Received: 19 October 2017
- Accepted: 28 December 2017
- Published: 10 January 2018
Abstract
In this paper, we put forward a fractional-order 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 fractional-order 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 fractional-order survival red blood cells model becomes controllable to achieve desirable behaviors. Finally, numerical examples are presented to demonstrate the theoretical analysis.
Keywords
- Hopf bifurcation
- bifurcation control
- time delays
- \(\mathit{PD}^{\alpha} \) controller
- survival red blood cells model
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
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.
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].
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.
Lemma 3.1
- (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 \(\upsilon = \upsilon_{j}\), \(j = 0,1,\ldots \) , wherewhere \(\omega_{0}\) is the unique positive zero of the function \(h_{1}(\omega )\).$$ \upsilon_{j} = \frac{1}{\omega_{0}}\arccos\biggl(\frac{a_{1} - \omega_{0}\cos \frac{q\pi}{2}}{a_{2}} + 2j\pi \biggr),\quad j = 0,1,2,\ldots, $$(3.6)
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). □
Lemma 3.2
Proof
Theorem 3.1
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.
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.
Lemma 4.1
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.
Lemma 4.2
Proof
Theorem 4.1
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.
Bifurcation point \(\pmb{\upsilon_{0} }\) versus fractional order α for model ( 2.7 )
Fractional order α | Bifurcation point \(\boldsymbol{\upsilon_{0}}\) |
---|---|
0.9 | 21.5435 |
0.92 | 19.6749 |
0.94 | 17.9960 |
0.96 | 16.4929 |
0.98 | 15.1312 |
1.0 | 13.9045 |
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.
Declarations
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. 2014-ZNDW-004) and the Science Foundation of Nanjing University of Posts and Telecommunications (No. NY213095).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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