1 Introduction

Chaos is one of the most prominent features of complex dynamical systems whose state variables are highly dependent on their initial conditions. This property may lead to the divergence of nonlinear chaotic systems; therefore, the analysis of such phenomena is of great importance from both theoretical and practical points of view [1].

Because of the wide appearance of chaotic systems in different fields of study such as secure communications, physics, biology and economy, a growing interest has been devoted to the control and synchronization of chaotic processes [2]. In [3], a new systematic approach within the use of scalar transmitted signal was designed to synchronize a class of hyperchaotic systems. In [4], a synchronization approach was presented for chaotic/hyperchaotic systems by formulating a linear feedback control problem. In [5], two identical hyperchaotic systems were synchronized by using a nonlinear control technique. In [6], a parameter observer was designed in order to identify unknown parameters in hyperchaotic systems, which was needed for designing a state-feedback controller. The author in [7] controlled and synchronized chaotic systems using state-dependent Riccati equations. In [8], a newly introduced fractional-order hyperchaotic system was controlled and synchronized by choosing an appropriate Lyapunov function and using a state-feedback control computation. In [9], Lorenz and Chen systems were considered in the presence of parameter uncertainty and controlled by using an adaptive dual synchronization controller. In [10], an adaptive terminal sliding mode control was applied to the chaotic Chen and hyperchaotic Lorenz systems. In [11], considering the stability theory of Lyapunov, the authors synchronized three nonidentical systems of different dimensions.

Recently, the investigation of chaos in biology has attracted the attention of many scientists due to the unpredictable behaviour of biological processes. In [12], the authors controlled the chaotic behaviour of tumor cells using a non-feedback loop. In [13], the author designed an optimal drug delivery schedule for the chaotic behaviour of tumor cells by minimizing Hamiltonian function; then he controlled the same system considering uncertainty in the chaotic model. In [14], the author analysed the chaotic behaviour of Lotka–Voltera biological systems. In [15], the authors synchronized two fractional chaotic maps by using a feedback control method. In [16], a fuzzy adaptive controller was designed to achieve a projective synchronization for a class of fractional chaotic systems with input nonlinearities.

Nowadays, a noticeable number of sensational researches have considered fractional-order systems due to their memory-oriented features, a fact which makes such systems more realistic to describe real-world dynamics [17–23]. Thus, it is very practical to investigate and analyse chaos in fractional-order systems in order to simulate the complex behaviour of realistic phenomena. In addition, efficient stabilization and synchronization techniques are needed to be developed in order to overcome the chaotic behaviour of such fractional processes. Thus, the problem of chaos control and synchronization for complex fractional-order systems has been of great importance to a wide audience. More to the point, efficient numerical methods should be extended to deal with controversial problems in fractional calculus like modelling and control [24–26]. Inspired by the aforementioned argument, this paper introduces a new fractional model for a chaotic biological oscillator. We study the stability and equilibrium of the new system and analyse its chaotic behaviour by using phase-portrait responses. Then we propose a state-feedback control to stabilize the fractional model and apply an active control strategy to synchronize two identical chaotic oscillators. The applicability of the proposed controllers is also verified through theoretical analysis as well as some simulation results.

The reminder of this paper is organized as follows. In Sect. 2, some preliminary results are presented. In Sect. 3, the new mathematical fractional model of a chaotic biological oscillator is introduced, and its stability and equilibrium point are investigated. Afterwards, a state-feedback control is designed to stabilize the new model, and an active controller is introduced in order to achieve synchronization. Finally, the paper is closed by some concluding remarks in the last section.

2 Preliminaries

In this section, we recall the basic definition of Caputo fractional derivative and introduce its corresponding integral operator. The Laplace transform and the antiderivative property of this definition are also presented in the sequel.

Definition 2.1

([27])

Let \(q \in (0,1)\) and f be an integrable function. Then the qth-order Caputo fractional derivative of f is defined by

$$ {}^{C} \mathscr{D}^{q}f(t)= \frac{1}{\Gamma (1-q)} \int _{0}^{t} \dot{f}(\tau ) (t-\tau )^{-q} \,d\tau . $$
(1)

The associated qth-order Riemann–Liouville fractional integral of f is also defined as

$$ \mathscr{I}^{q} f(t)= \frac{1}{\Gamma (q)} \int _{0}^{t} f(\tau ) (t-\tau )^{q-1} \,d\tau , $$
(2)

whenever the integral exists.

For the qth-order Caputo derivative when \(q \in (0,1)\), the Laplace transform is introduced by

$$ \mathscr{L}\bigl[{}^{C} \mathscr{D}^{q}f(t) \bigr]=s^{q}F(s)-f(0), $$
(3)

where \(F(s)=\mathscr{L}[f(t)]\). In addition, the fractional derivative (1), together with the integral operator (2), satisfies the antiderivative property such that

$$ \mathscr{I}^{q}\bigl[{}^{C} \mathscr{D}^{q}f(t) \bigr]=f(t)-f(0). $$
(4)

3 The new fractional model

In the recent study [28], the authors employed an autonomous model to describe the interaction between enzyme and substrate mechanism. In [28], the aforesaid model was introduced in the form of the following system of ordinary differential equations:

$$ \textstyle\begin{cases} \dot{f}_{1}=f_{2}, \\ \dot{f}_{2}=\alpha f_{3}, \\ \dot{f}_{3}=f_{4}, \\ \dot{f}_{4}=-f_{1}-\beta f_{3}+\gamma f_{2}-f_{4}+\varepsilon {f_{2}}(1-{f_{1}^{2}}+ \eta {f_{1}^{4}}-\kappa {f_{1}^{6}}), \end{cases} $$
(5)

where the coefficients α, β, γ, ε, η, κ are real constants. As mentioned in [28], the model (5) exhibits chaotic attractors when the parameters \((\alpha ,\beta ,\gamma ,\varepsilon ,\eta ,\kappa )\) take the values \((5,24,-0.05,2.001,2.55,1.7)\), respectively. However, the integer-order model (5) suffers from the lack of memory effect, while hereditary property is the intrinsic feature of many complex biological systems. To overcome this drawback, the concept of fractional calculus is used to include memory in the model formulation. To do so, we extend the model (5) by substituting the integer-order derivatives with fractional Caputo ones defined in (1). Moreover, we take into account the new auxiliary parameter θ in order to keep away from dimensional mismatching [29]. Thus, we formulate the new fractional biological oscillator by

$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{1}}=f_{2}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{2}}=\alpha f_{3}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{3}}=f_{4}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{4}}=-f_{1}-\beta f_{3}+\gamma f_{2}-f_{4}+ \varepsilon {f_{2}}(1-{f_{1}^{2}}+\eta {f_{1}^{4}}-\kappa {f_{1}^{6}}). \end{cases} $$
(6)

Meanwhile, please note that the model (6) is the extension of a generalized Van der Pol oscillator; hence, this model can describe various biological and bio-physical phenomena.

3.1 Equilibrium point and stability

Considering the certain values of parameters \((\alpha ,\beta ,\gamma ,\varepsilon ,\eta ,\kappa )=(5,24,-0.05,2.001,2.55, 1.7)\), we obtain the equilibrium point of the model (6) from

$$ \textstyle\begin{cases} f_{2}=0, \\ \alpha f_{3}=0, \\ f_{4}=0, \\ -f_{1}-\beta f_{3}+\gamma f_{2}-f_{4}+\varepsilon {f_{2}}(1-{f_{1}^{2}}+ \eta {f_{1}^{4}}-\kappa {f_{1}^{6}})=0. \end{cases} $$
(7)

Solving the system of algebraic equations (7), we find that the system (6) has a unique equilibrium point \(f^{*}=(f_{1}^{*},f_{2}^{*},f_{3}^{*},f_{4}^{*})=(0,0,0,0)\). Then we compute the Jacobian matrix at the point \(f^{*}\)

$$ J^{*}= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 1.951 & -24 & -1 \end{bmatrix}, $$
(8)

whose eigenvalues are

$$ \lambda _{1,2}=0.2468 \pm 0.4279i,\qquad \lambda _{3,4}=-0.7468 \pm 4.4649i. $$
(9)

Now, we list two well-known lemmas discussing the stability of fractional-order systems.

Lemma 3.1

([30])

The equilibrium point \(f^{*}\) of the fractional-order system (6) is locally asymptotically stable if the following inequality holds:

$$ \bigl\vert {\arg (\lambda _{i})} \bigr\vert > \frac{\pi }{2}q,\quad i=1,2,3,4, $$
(10)

where q is the fractional order, and \(\lambda _{i}\) is the ith eigenvalue of \(J^{*}\) at the point \(f^{*}\).

Lemma 3.2

([31])

Let the characteristic equation of the model (6) at \(f^{*}\) be in the form

$$ \det \bigl(\lambda {I}-J^{*}\bigr)=\lambda ^{4}+a_{3}\lambda ^{3}+a_{2} \lambda ^{2}+a_{1} \lambda ^{1}+a_{0}=0. $$
(11)

Then the equilibrium \(f^{*}\) is locally asymptotically stable for all \(q\in {(0,1)}\) if the corresponding Routh–Hurwitz table has elements with the same sign in its first column.

3.2 Simulation results

In the following, we simulate the complex behaviour of the new modified model (6) for the fractional orders \(q=0.9,0.92,0.94,0.96,0.98,1\). To this end, we consider the initial conditions as

$$ f(0)=\bigl(f_{1}(0),f_{2}(0),f_{3}(0),f_{4}(0) \bigr)=(0.5,0.5,0.5,0.5), $$
(12)

and select the parameters from

$$ (\alpha ,\beta ,\gamma ,\varepsilon ,\eta ,\kappa ,\theta )=(5,24,-0.05,2.001,2.55,1.7,0.99). $$
(13)

Figures 1–6 depict the relevant phase-portraits and verify the existence of complex chaotic attractors for the considered values of the fractional order q.

Figure 1
figure 1

The phase planes of the fractional chaotic system (6) for \(q=0.9\)

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Figure 2
figure 2

The phase planes of the fractional chaotic system (6) for \(q=0.92\)

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Figure 3
figure 3

The phase planes of the fractional chaotic system (6) for \(q=0.94\)

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Figure 4
figure 4

The phase planes of the fractional chaotic system (6) for \(q=0.96\)

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Figure 5
figure 5

The phase planes of the fractional chaotic system (6) for \(q=0.98\)

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Figure 6
figure 6

The phase planes of the fractional chaotic system (6) for \(q=1\)

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Remark 3.1

Concerning the numerical simulation, here we applied the predictor–corrector method [32] in which an FFT algorithm is employed to evaluate the discrete convolutions [33]. The accuracy and the convergence of this scheme were studied in [34]. Also, contrary to the classical implementation in which the computational cost is proportional to \(N^{2}\), here we have \(N\log (N^{2})\) for the computational cost when the solution is evaluated at N time-points.

4 Chaos control and synchronization

In this section, designing a state-feedback controller, we compensate the undesirable chaotic behaviour of the fractional-order biological system (6). Then we introduce an active control approach to synchronize two identical fractional chaotic oscillators.

4.1 Chaos control

In order to control the chaotic behaviour of the fractional-order model (6), we consider the following modified controlled system:

$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{1}}=f_{2}-u_{1}(f_{1}-f_{1}^{*}), \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{2}}=\alpha f_{3}-u_{2}(f_{2}-f_{2}^{*}), \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{3}}=f_{4}-u_{3}(f_{3}-f_{3}^{*}), \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{4}}=-f_{1}-\beta f_{3}+\gamma f_{2}-f_{4}+ \varepsilon {f_{2}}(1-{f_{1}^{2}}+\eta {f_{1}^{4}}-\kappa {f_{1}^{6}})-u_{4}(f_{4}-f_{4}^{*}), \end{cases} $$
(14)

where \(u_{i}\geq {0}\) is the state-feedback control gain. The objective is to design \(u_{i}\) in such a way that the state variables \(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{4}\) of the controlled system (14) converge to the steady-state \(f^{*}=(0,0,0,0)\) asymptotically. Considering the system parameters \((\alpha ,\beta ,\gamma ,\varepsilon ,\eta ,\kappa ,\theta )=(5,24,-0.05,2.001,2.55,1.7, 0.99)\), we compute the Jacobian matrix \(J^{*}\) at the point \(f^{*}=(f_{1}^{*},f_{2}^{*},f_{3}^{*},f_{4}^{*})=(0,0,0,0)\) as

$$ J^{*}= \begin{bmatrix} -u_{1} & 1 & 0 & 0 \\ 0 & -u_{2} & 5 & 0 \\ 0 & 0 & -u_{3} & 1 \\ -1 & 1.951 & -24 & -u_{4}-1 \end{bmatrix}. $$
(15)

Then we derive

$$ \lambda {I} - J^{*}= \begin{bmatrix} \lambda +u_{1} & -1 & 0 & 0 \\ 0 & \lambda +u_{2} & -5 & 0 \\ 0 & 0 & \lambda +u_{3} & -1 \\ 1 & -1.951 & 24 & \lambda +u_{4}+1 \end{bmatrix}, $$
(16)

whose determinant is obtained in the form \(\det (\lambda {I}-J^{*})=\lambda ^{4}+a_{3}\lambda ^{3}+a_{2}\lambda ^{2}+a_{1} \lambda ^{1}+a_{0}=0\) where

$$ \begin{aligned} &a_{3}=1+u_{2}+u_{3}+u_{4}+u_{1}, \\ &a_{2}=24+u_{2}+u_{3}+u_{1}+u_{1} u_{2}+u_{1} u_{3}+u_{3} u_{4}+u_{2} u_{4}+u_{1} u_{4}+u_{2} u_{3}, \\ &a_{1}=-9.755+24 u_{2}+24 u_{1}+u_{2} u_{3} u_{4}+u_{1} u_{2} u_{3}+u_{1} u_{2} u_{4}+u_{1} u_{3} u_{4} \\ &\hphantom{a_{1}={}}{}+u_{1} u_{2}+u_{1} u_{3}+u_{2} u_{3}, \\ &a_{0}=5 +24 u_{1} u_{2}+u_{1} u_{2} u_{3} u_{4}-9.755 u_{1}+u_{1} u_{2} u_{3}. \end{aligned} $$
(17)

For the state-feedback gains \((u_{1},u_{2},u_{3},u_{4})=(0.25,0.25,0.25,0.25)\), we obtain

$$ \det \bigl(\lambda {I}-J^{*}\bigr) = \lambda ^{4} + 2 \lambda ^{3}+ 25.125 \lambda ^{2} + 2.495 \lambda + 4.08078125 = 0, $$
(18)

and the corresponding Routh–Hurwitz table of stability is given in Table 1. As can be seen, we get the satisfaction of stability condition in Lemma 3.2, so the equilibrium \(f^{*}\) of the controlled system (14) is locally asymptotically stable for all \(q\in {(0,1)}\). We depict the controlled states in Figs. 7–12 for the fractional orders \(q=0.9,0.92,0.94,0.96,0.98,1\) and the initial conditions \(f(0)=(0.5,0.5,0.5,0.5)\). Simulation results confirm the validity of the proposed approach to overcome the undesirable behaviour of chaotic attractors.

Figure 7
figure 7

The time-domain responses of the controlled system (14) for \(q=0.9\)

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Figure 8
figure 8

The time-domain responses of the controlled system (14) for \(q=0.92\)

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Figure 9
figure 9

The time-domain responses of the controlled system (14) for \(q=0.94\)

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Figure 10
figure 10

The time-domain responses of the controlled system (14) for \(q=0.96\)

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Figure 11
figure 11

The time-domain responses of the controlled system (14) for \(q=0.98\)

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Figure 12
figure 12

The time-domain responses of the controlled system (14) for \(q=1\)

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Table 1 Routh–Hurwitz table of stability
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4.2 Synchronization

In this section, two identical fractional-order chaotic systems are synchronized by an active control scheme. To do so, the master and the slave systems are, respectively, considered as

$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{1}}=f_{2}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{2}}=\alpha f_{3}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{3}}=f_{4}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{f_{4}}=-f_{1}-\beta f_{3}+\gamma f_{2}-f_{4}+ \varepsilon {f_{2}}(1-{f_{1}^{2}}+\eta {f_{1}^{4}}-\kappa {f_{1}^{6}}), \end{cases} $$
(19)
$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{g_{1}}=g_{2}+u_{1}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{g_{2}}=\alpha g_{3}+u_{2}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{g_{3}}=g_{4}+u_{3}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{g_{4}}=-g_{1}-\beta g_{3}+\gamma g_{2}-g_{4}+ \varepsilon {g_{2}}(1-{g_{1}^{2}}+\eta {g_{1}^{4}}-\kappa {g_{1}^{6}})+u_{4}, \end{cases} $$
(20)

where \(f_{i}\) and \(g_{i}\) are the states of the master and the slave systems, respectively, and \(u_{i}\) is the control input. The objective is to design \(u_{i}\) in such a way that the synchronization error goes to zero asymptotically. To reach this aim, we define the error of synchronization by \(e_{i}(t)=g_{i}(t)-f_{i}(t)\). Then the error dynamics will be

$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}=e_{2}+u_{1}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}{e_{2}}=\alpha e_{3}+u_{2}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}=e_{4}+u_{3}, \\ {}^{C}_{0} {D}_{t}^{\rho _{4}}{e_{4}}=-e_{1}-\beta e_{3}+(\gamma +\varepsilon ) e_{2}-e_{4}- \varepsilon {g_{2}}{g_{1}^{2}}+\varepsilon {f_{2}}{f_{1}^{2}}+ \varepsilon \eta {g_{2}}{g_{1}^{4}} \\ \hphantom{{}^{C}_{0} {D}_{t}^{\rho _{4}}{e_{4}}={}}{}-\varepsilon \eta {f_{2}}{f_{1}^{4}} - \varepsilon \kappa {g_{2}}{g_{1}^{6}}+\varepsilon \kappa {f_{2}}{f_{1}^{6}}+u_{4}. \end{cases} $$
(21)

Considering the control variables as

$$ \textstyle\begin{cases} u_{1}=v_{1}, \\ u_{2}=v_{2}, \\ u_{3}=v_{3}, \\ u_{4}=\varepsilon {g_{2}}{g_{1}^{2}}-\varepsilon {f_{2}}{f_{1}^{2}}- \varepsilon \eta {g_{2}}{g_{1}^{4}}+\varepsilon \eta {f_{2}}{f_{1}^{4}} + \varepsilon \kappa {g_{2}}{g_{1}^{6}}-\varepsilon \kappa {f_{2}}{f_{1}^{6}}+v_{4}, \end{cases} $$
(22)

one attains

$$ \textstyle\begin{cases} {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}=e_{2}+v_{1}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}{e_{2}}=\alpha e_{3}+v_{2}, \\ {\frac{1}{\theta ^{1-q}}} {}^{C} \mathscr{D}^{q}{e_{1}}=e_{4}+v_{3}, \\ {}^{C}_{0} {D}_{t}^{\rho _{4}}{e_{4}}=-e_{1}-\beta e_{3}+(\gamma +\varepsilon ) e_{2}-e_{4}+v_{4}, \end{cases} $$
(23)

where \(v_{i}\) represents the active control input. Now, we select

$$ \begin{bmatrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end{bmatrix} =\mathbf{A} \begin{bmatrix} e_{1} \\ e_{2} \\ e_{3} \\ e_{4} \end{bmatrix} , $$
(24)

where A is a constant matrix and selected in order to satisfy the stability condition in Lemma 3.1 for the error dynamics (23). Note that different choices exist for A; an appropriate one can be

$$ \mathbf{A}= \begin{bmatrix} -1 & -1 & 0 & 0 \\ 0 & -1 & -\alpha & 0 \\ 0 & 0 & -1 & -1 \\ 1 & -(\gamma +\varepsilon ) & \beta & 0 \end{bmatrix} . $$
(25)

In addition, synchronization speed can be controlled by changing the main diagonal of matrix A.

Remark 4.1

Synchronization control can be regarded as a special tracking problem. When the master system is fixed in advance, the future dynamics of the master system are known, which can be considered as preview information. In this case, the speed of synchronization can be improved by using the preview control technique discussed in the new study [35].

To verify the theoretical synchronization approach mentioned above, we consider the initial states as

$$ \textstyle\begin{cases} f(0)=(0.5,0.5,0.5,0.5), \\ g(0)=(2.5,-2,2.5,5.5), \end{cases} $$
(26)

when the parameters and the fractional orders are \((\alpha ,\beta ,\gamma ,\varepsilon ,\eta ,\kappa , \theta )=(5,24,-0.05,2.001, 2.55,1.7,0.99)\) and \(q=0.9,0.92,0.94,0.96,0.98,1\), respectively. Figures 13–18 display the synchronized state variables verifying the proposed active control scheme.

Figure 13
figure 13

The state trajectories of the synchronized systems (19) and (20) for \(q=0.9\)

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Figure 14
figure 14

The state trajectories of the synchronized systems (19) and (20) for \(q=0.92\)

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Figure 15
figure 15

The state trajectories of the synchronized systems (19) and (20) for \(q=0.94\)

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Figure 16
figure 16

The state trajectories of the synchronized systems (19) and (20) for \(q=0.96\)

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Figure 17
figure 17

The state trajectories of the synchronized systems (19) and (20) for \(q=0.98\)

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Figure 18
figure 18

The state trajectories of the synchronized systems (19) and (20) for \(q=1\)

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5 Conclusions

The objective of this paper was to investigate the chaotic behaviour of a fractional biological system and compensate this undesirable behaviour by an efficient control strategy. In the first step, we discussed and explored the stability and the equilibrium of the considered model. Then we sketched the associated chaotic attractors in Figs. 1–6. Next, we designed a stabilizing state-feedback controller to overcome the undesirable chaotic behaviour. Afterwards, we introduced an active control design to synchronize two identical fractional biological oscillators. The validity of the chaos control and the synchronization approach was verified theoretically, and this analysis was also confirmed by some simulations in Figs. 7–18. Finally, as the future direction of this manuscript, we will consider more practical models, which will also be more complicated from the mathematical point of view.