On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system

In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results.


Introduction
For a long time, researches on fractional-order calculus were mainly concentrated in the field of pure mathematics [1]. The main reason for abandoning fractional-order models in practical application was their computational complexity. With the development of computer technology, the application of fractional-order calculus has attracted the interest of many researchers in different fields, including mathematics, physics, chemistry, engineering, and even financial and social sciences. It has been found in some fields that using fractional-order is more appropriate than employing integer-order to describe the dynamical behavior and characteristics [2][3][4][5][6] of models, such as the classical predator-prey system [7,8], the fractional-order model of the dengue virus infection [9], the simulated tidal wave fractional-order model [10], the fractional-order viscoelastic non-Newtonian fluid model [11], and so on. It is proved that the fractional-order dynamical system can more accurately respond to the time variation of general nature [12][13][14].
In order to describe those phenomena whose evolutions not only depend on the state of current time, but also on the state at a previous time, the time delay needs to be introduced into many differential equations which are defined as delayed differential equation (DDE).
DDEs arise in many fields, for example, metal cutting, epidemiology, neuroscience, population dynamics [15], biological systems [16,17], financial system [18], and traffic models [19]. The time delay is often considered as the parameter of fractional-order systems according to the actual background of the problems [20][21][22]. In 1992, Pyragas originally proposed the delayed feedback controller to study the control issue of nonlinear autonomous differential equations [23]. Now the delayed feedback has become an adjustment mechanism that is widely used in many nonlinear control systems. For example, in order to stabilize the unstable periodic orbit by the difference between the current state and the delay state, a suitable delayed feedback controller can be designed to achieve the desired dynamic behavior [24][25][26].

y(t) = c(z(t)y(t)), z(t) = ay(t)bz(t) + x(t)y(t),
where a, b, c, d are constants. Then Bhalekar [28] further explored the forming mechanism of the BG system. In recent years, some valuable results about the BG system have been obtained. Aqeel and Ahmad [29] studied the Hopf bifurcation and chaos of the integerorder BG system. Deshpande et al. [30] found that the fractional-order BG system allows chaotic solutions and the fractional-order could be regarded as the control parameter of chaos. Shahzad et al. [31] added a single time delay into the third equation of (1.1) and studied a delay Bhalekar-Gejji chaotic system of the form:

y(t) = d(z(t)y(t)), z(t) = ay(t)bz(tτ ) + x(t)y(t).
(1.2) They derived some algebraic sufficient conditions that guarantee the globally and asymptotically stable synchronization and antisynchronization between two identical time delay Bhalekar-Gejji chaotic systems. To the best of our knowledge, there are few literature sources discussing the Hopf bifurcation of the fractional-order BG system with time delay.
This motivates us to investigate the effect of time-delay and fractional-order on the occurrence of the Hopf bifurcation and to introduce an appropriate delayed state-feedback controller to control the Hopf bifurcation.
In this paper, we consider the fractional-order time delay Bhalekar-Gejji system of the form: where q i ∈ (0, 1] (i = 1, 2, 3) and a, b, c, d are parameters; c is generally taken as positive, while d is a negative real number; τ ≥ 0 is the time delay.
The main purpose of this paper is to seek for the conditions of the occurrence of Hopf bifurcation for system (1.3) by using time delay as the bifurcation parameter based on the approach of stability analysis [32]. Especially, we give a semianalytical verification of the reasonability of linearized approximation of system (1.3) from the viewpoint of stability. In addition, we also design a delayed feedback controller to control the emergence of Hopf bifurcation and further study the effect of feedback gain on the bifurcation control of the proposed system. This paper is organized as follows. In Sect. 2, we introduce the relevant preliminary knowledge of fractional calculus and fractional-order dynamical system. In Sect. 3, we analyze system (1.3) to get the conditions of the occurrence of Hopf bifurcation and the value range of delay in which Hopf bifurcation appears. We also verify the reasonability of the linearized approximation by the equivalence of stability of equilibrium points between the original system (1.3) and its linearized system. In Sect. 4, the delayed feedback controller is added to system (1.3) to control the Hopf bifurcation. In Sect. 5, numerical simulations are performed to verify the validity of the theoretical results by choosing appropriate values of the constants a, b, c, d, τ . Finally, necessary conclusions and a discussion are presented in Sect. 6.

Preliminaries
In this section, some preliminary knowledge of fractional calculus and fractional-order dynamical system are introduced. In fact, the concept of fractional derivative has many classical definitions. This paper is based on the most widely used definition of Caputo fractional derivative.
Definition 2 ( [1,33]) The Caputo fractional derivative can be written as where x(t) ∈ C n ([t 0 , ∞), R). In particular, if 0 < α ≤ 1, (2.2) can be written as For brevity, in what follows, we use the notation D α x(t) to denote the Caputo fractionalorder derivative operator C t 0 D α t x(t).
denotes the real part of the complex number and s refers to the eigenvalue of the associated characteristic equation of the linearized system.

Reasonability of linearized approximation and Hopf bifurcation for fractional-order delay BG system
In this paper, we only consider the nonzero real equilibrium points of the fractional-order delay BG system (1.3). Hence, we need to assume that (H 1 ) d(ba) > 0. It is obvious that system (1.3) has two nonzero real equilibrium points: We use the delay τ as a bifurcation parameter to find the conditions on the occurrence of Hopf bifurcation at the equilibria of system (1.3).
For brevity, the nonzero equilibrium point is denoted as (x * , y * , z * ). Using the transformations The linearized system of (3.2) at the origin is

Analysis of reasonability of linearized approximation
Since the stability change of an equilibrium involves the appearance of Hopf bifurcation, we need to verify the reasonability of the above linearized approximation by the equivalence of stability of equilibrium for systems (3.2) and (3.3). Following a similar idea as in [36], we prove the equivalence of stability of equilibrium for systems (3.2) and (3.3) in the sense that where the initial values are taken as , we obtain the error system (3.5) We have two basic assertions.
Thus, by Assertions (a) and (b), we verified the reasonability of the above linearized approximation from the viewpoint of stability of equilibrium.

Hopf bifurcation analysis
The linearized system of (3.2) at the origin can be expressed as It is easy to obtain the associated characteristic equation by using Laplace transform on system (3.11): Equation (3.12) can be equivalently rewritten as Assume that s = iω = ω(cos π 2 + i sin π 2 ) is a root of Eq. (3.13), ω > 0. Substituting s = iω into Eq. (3.13) and separating the real and imaginary parts, then it results in where α i (i = 1, 2, 3, 4) are defined in Appendix A.
Substituting s = iω = ω(cos π 2 + i sin π 2 ) into A(s), B(s), and letting A 1 , A 2 and B 1 , B 2 be the real and imaginary parts of A(s), B(s), respectively, it can be deduced from Eq. (3.17) that Basing on the aforementioned analysis, we get holds if the following assumption is satisfied: Next, to verify Assumption (C1) in Definition 4, we need the following lemma. Basing on Definition 4, we achieve the first primary theorem of this paper.

Delayed feedback control of fraction-order delay BG systems
In this section, a delayed feedback controller k[y(t)y(tτ )] is added to the second equation of uncontrolled system (1.3), and then the delay feedback control system can be acquired as

D q 3 z(t) = ay(t)bz(tτ ) + x(t)y(t).
(4.1) For the sake of revealing the relationship between the controller and Hopf bifurcation, we still use the delay τ as a parameter in Eq. (4.1). Analogous to the previous analysis, by performing transformations u(t) = x(t)x * , v(t) = y(t)y * , w(t) = z(t)z * , with the help of the linearized scheme, the linearization of the controlled system (4.1) has the form: By multiplying e sτ on both sides of Eq. (4.4), it is obvious that  Consistent with the previous section, with the formula Q 2 1 (ω) + Q 2 2 (ω) = 1, we can calculate ω easily. We might as well suppose that ω i (i = 1, 2, . . . , n) are positive solutions, and can get the same four cases of τ (k) i (k = 0, 1, 2, . . . ) as in Sect. 3. Next we define the bifurcation point . . , n, k = 0, 1, 2, . . . , (4.8) where ω i corresponds to min{τ (k) i }. It needs to be noticed that the calculation of τ (k) i and ω i is relying on the feedback grain coefficient k (see Appendix C).
Differentiating both sides of Eq. (4.5) with respect to τ , one obtains where F i (s) are the derivatives of F i (s) (i = 1, 2, 3). Therefore, where C(s) = s F 2 (s)e -sτ + 2F 3 (s)e -2sτ , (4.10) It can be deduced from Eq. (4.9) that where C 1 , C 2 , D 1 , D 2 are the real and imaginary parts of C(s) and D(s), respectively, and the exact expressions are given in Appendix D.
Thus, we obtain the following lemma: holds if the following assumption is satisfied: where C i , D i (i = 1, 2) are defined as in (4.11).
Based on the previous discussion, the following theorem can be concluded.
Remark 2 In this section, we use the same method as in Sect. 3 to discuss the delay feedback control system (4.1). In reality, the Hopf bifurcation points (τ * 0 , ω * 0 ) of system (4.1) can be controlled successfully by changing the feedback gain coefficient k. We will illustrate this fact in the next section by numerical simulations.

Numerical simulations
Adams-Bashforth-Moulton predictor-corrector scheme [39] has been widely used in numerical simulation for fractional-order differential equation. In this section, this method is adopted in two examples to verify the efficiency and feasibility of our theoretical results, in which step-length is taken as h = 0.001.

Example 1
For the convenience of comparison, all the system parameters come from the literature [31]: a = 22, b = 10, c = 10, d = -2.667. Without loss of generality, let q 1 = 0.91, q 2 = 0.98, q 3 = 0.95, then system (1.3) can be changed into One can obtain the critical frequency ω 0 = 9.35083452 and bifurcation point τ 0 = 0.09517856. By Theorem 1, Hopf bifurcation of system (5.1) appears at τ 0 . To better present our results, we give two simulations. One uses τ = 0.0949 < τ 0 = 0.09517856, which is displayed in Figs. 1 and 2. Under this condition, we can see that two nonzero equilibria are asymptotically stable. The other uses τ = 0.0952 > τ 0 = 0.09517856, which is displayed in Figs. 3 and 4. It is apparent that two nonzero equilibria are unstable and Hopf bifurcation occurs. Therefore, Theorem 1 is verified by these simulations.

Example 2
In this example, a linear delayed feedback controller is added to the uncontrolled system (5.1) so as to control the Hopf bifurcation. In order to illustrate the effects of bifurcation control via the proposed controller preferably, three fractional orders and all the system parameters are chosen the same as in Example 1, then the controlled system is shown as For exhibiting the impact of feedback gain coefficient k on the Hopf bifurcation for the controlled system (5.2), we calculate a group of critical frequency ω * 0 and bifurcation points τ * 0 corresponding to ever-increasing k, see Table 1. According to Table 1, system (5.2) is controlled by the delayed feedback controller k[y(t)y(tτ )] effectively. When k increases from negative to positive, τ * 0 decreases gradually. This means that the stable domain wanes and the emergence of Hopf bifurcation is advanced. Moreover, by choosing three different k, Figs. 5-7 illustrate that the effect of Hopf bifurcation control is much better as k decreases.

Conclusions and discussion
In this paper, sufficient conditions on the emergence of Hopf bifurcation have been established for a fractional-order delay Bhalekar-Gejji chaotic system. The delay feedback control issue of Hopf bifurcations for such a system has been investigated by theoretical analysis and numerical simulation.
This paper mainly focused on three aspects: the reasonability of the linearized approximation, delay-induced Hopf bifurcation, and delay feedback control of Hopf bifurcation for a fractional-order delay Bhalekar-Gejji chaotic system. Comparing to the previous The effect of bifurcation control for the controlled system (5.2) becomes better as the feedback gain k decreases The effect of bifurcation control for the controlled system (5.2) becomes better as the feedback gain k decreases similar works, we semianalytically verified the reasonability of the linearized approximation by the equivalence of stable equilibrium for the converted systems (3.2) and (3.3) under an appropriate prior assumption. To some extent, this provides a theoretical support for the definition of Hopf bifurcation for fractional-order delay systems proposed by [35].
We find that the time delay has an important influence on the stability of the fractionalorder delay Bhalekar-Gejji chaotic system. The delay can be used as the bifurcation parameter to derive the asymptotic stability interval of the system and in which conditions the system will exhibit dynamic behavior such as Hopf bifurcation. In addition, for the fractional-order delay Bhalekar-Gejji chaotic system, the feedback gain can control the bifurcation value and expand the stability range of the system.
In the simulations, we can also observe the stability of equilibrium points of the proposed system. In fact, it is difficult to analytically prove stability of equilibrium points of the original system (1.3), which is not only because of the complexity of the system, but also due to the lack of a well-developed stability theory of nonlinear fractional-order delay systems.
In the subsequent research, we would also like to explore the effect of the fractional order and time delay on the occurrence of chaos for the fractional-order delay Bhalekar-Gejji chaotic system.