# Control Scheme for a Fractional-Order Chaotic Genesio-Tesi Model

Authors: Changjin Xu; Peiluan Li; Maoxin Liao; Zixin Liu; Qimei Xiao; Shuai Yuan
Journal: Complexity
 (2019)

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## Abstract

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems.


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## Body

## 1. Introduction

As is known to us, chaos control issue has been widely studied in the last decades because of its potential practical value in various areas. How to control the chaotic phenomenon to serve human beings has become a hot issue in today’s world. Chaos control has attracted much attention of researchers from various fields. In recent years, many chaos control techniques (for example, delayed feedback approach [1], Ott–Grebogi–York (OGY) technique [2], variable structure control [3], observer-based control [4], backstepping design technique [5], and active control [6]) have been proposed. Many excellent fruits have been reported. For example, Kocamaz et al. [7] investigated the chaos control by applying the sliding mode control technique; Chen [8] proposed an adaptive feedback control technique for chaos and hyperchaos control; Din [9] studied the chaos control of a discrete-time prey-predator system by using three different types of feedback control strategies; Singh and Gakkhar [10] controlled the chaos of a food chain model with a time-delayed feedback controller; Yan et al. [11] discussed the chaos control of continuous unified chaotic systems applying discrete rippling sliding mode control. For more knowledge on chaos control, readers can refer to [12–19, 34–44].In 1992, Genesio and Tesi [20] put up the chaotic Genesio-Tesi system:(1)dw1dt=w2,dw2dt=w3,dw3dt=k1w1+k2w2+k3w3+w12,where wii=1,2,3 denote different voltage of electronic components and k1,k2,andk3 are all negative coefficients. Genesio and Tesi [20] investigated the chaotic behavior of model (1) by applying harmonic balance approaches. In 2012, Guan et al. [21] discussed the chaos control of model (1) by designing distributed delay feedback controller. By analyzing the characteristic equation of the controlled Genesio-Tesi model, the sufficient conditions to ensure the stability of the equilibrium point and the existence of Hopf bifurcation for the controlled Genesio-Tesi model are established. In addition, the stability and the direction of bifurcating periodic solution are determined by the centre manifold theorem and normal form theory. In 2009, Sun [22] proposed a tracking control to realize chaos synchronization for the Genesio-Tesi chaotic system (1) based on the time-domain approach. In 2007, Park [23] designed a novel feedback controller to realize exponential synchronization of the Genesio-Tesi chaotic system (1). In 2009, Park [24] further considered the functional projective synchronization problem for the Genesio-Tesi chaotic system (1) and Zhou and Chen [25] dealt with the Hopf bifurcation and Si’lnikov chaos of Genesio model (1).It is worth mentioning that all the above papers focus only on the integer-order differential models. In recent years, many researchers argue that fractional-order differential equations play a key role in describing the real phenomena and are found to have potential applications in various areas such as physics, economics, physics, heat transfer, and chemical engineering [26–28]. Thus, it is important to deal with the fractional-order differential models. Based on discussion above, we revise model (1) in the following fractional-order form:(2)dσw1dtσ=w2,dσw2dtσ=w3,dσw3dtσ=k1w1+k2w2+k3w3+w12,where wii=1,2,3 denote different voltage of electronic components, k1,k2,andk3 are all negative coefficients, and σ∈0,1 denotes the fractional order. Model (2) is the commensurate fractional-order system. Let σ=0.87 and k1=−6,k2=−2.92,andk3=−1.2, then the chaotic phenomenon will occur in model (2). The fact is shown in the following (see Figures 1 and 2).Figure 1
Dynamics of system (2) when σ=0.87 and k1=−6,k2=−2.92,andk3=−1.2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 2
Maximum Lyapunov exponent of system (2) with respect to the fractional order σ.This paper mainly focuses on two aspects: (a) designing two appropriate controllers to control the chaotic phenomenon of model (2) and (b) revealing the impact of time delay on the stability and bifurcation behavior of the controlled fractional-order Genesio-Tesi chaotic model. The superiority of this paper can be summarized as follows:(a)
A new fractional-order Genesio-Tesi chaotic model is proposed(b)
Two controllers are designed to control the chaotic phenomenon of the fractional-order Genesio-Tesi chaotic model(c)
The advantages and disadvantages of two controllers are comparedThe outline of this paper is organized as follows: In Section2, some elementary knowledge on fractional-order differential is prepared. In Section 3, two different control techniques are designed to control the chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model. In Section 4, two examples are given to illustrate the theoretical analysis. In Section 5, we end this manuscript with conclusion.


## 2. Elementary Knowledge

In this section, we list several basic results about fractional-order differential equations.Definition 1 (see [29]).
The Caputo fractional-order derivative is defined as follows:(3)Dσql=1Γn−σ∫l0lqmsl−sσ−m+1ds,where ql∈l0,∞,R, l≥l0, and m is a positive integer such that σ∈m−1,m. If σ∈0,1, then(4)Dσql=1Γ1−σ∫l0lq′sl−sσds.Definition 2 (see [30]).
The fractional-order system is given as follows:(5)dσwdtσ=gw,w0=w0,where σ∈0,1 and w∈Rn. w∗ is said to the equilibrium point of system (5) if gw∗=0.Lemma 1 (see [31]).
Denoteλii=1,2,…,n the root of the characteristic equation of the autonomous system(6)Dσwt=Aw,w0=w0,where σ∈0,1,w∈Rn,andA∈Rn×n. Then, system (6) is asymptotically stable and also stable ⟺ argλi≥σπ/2i=1,2,…,n, and those critical eigenvalues that satisfy argλi=σπ/2i=1,2,…,n have geometric multiplicity one.Lemma 2 (see [32]).
For the system(7)Dσwt=Bwt+Cwt−ρ,where σ∈0,1,w∈Rn,andB,C∈Rn×n, the characteristic equation takes the following form:(8)detsσI−B−Ce−sρ=0.The zero solution of system (7) is asymptotically stable if all the roots of equation (8) are negative real roots.Remark 1.
In Lemma 1 and Lemma 2, we say that system (6) (or system (7)) is asymptotically stable (or stable), implying that the fixed point E of system (6) (or system (7)) is asymptotically stable (or stable).


## 3. Chaos Control via Time-Delayed Feedback Controllers

It is easy to see that system (2) has two unique equilibrium points:(9)E1=0,0,0,E2=−k1,0,0.A long time ago, there were many publications that handle the chaos control issue of integer-order differential models by applying time-delayed feedback controllers. But the studies on the chaos control of fractional-order chaotic models via time-delayed feedback controllers are very rare. To make up the deficiency, in this paper, we will design two time-delayed feedback controllers to eliminate the chaotic behavior of system (2). In this paper, we are concerned only with the equilibrium point E1=0,0,0. The other equilibrium point E2=−k1,0,0 is easily analyzed in a similar way.


### 3.1. Adding Time-Delayed Feedback Controller to the First Equation of Model (2)

In this section, we add a time-delayed feedback controllerα1w1t−w1t−ϱ to the first equation of model (2), and then system (2) takes the following form:(10)dσw1dtσ=w2+α1w1t−w1t−ϱ,dσw2dtσ=w3,dσw3dtσ=k1w1+k2w2+k3w3+w12,where α1 is the gain coefficient and ϱ is the time delay. The linear system of system (10) near the equilibrium point E1=0,0,0 reads(11)dσw1dtσ=α1w1+w2−α1w1t−ϱ,dσw2dtσ=w3,dσw3dtσ=k1w1+k2w2+k3w3.The corresponding characteristic equation of (11) takes the following form:(12)detsσ−α1+α1e−sϱ−100sσ−1−k1−k2sσ−k3.Hence,(13)A1s+A2se−sϱ=0,where(14)A1s=s3σ−α1+k3s2σ+α1k3−k2sσ+k2α1−k1,A2s=α1s2σ−α1k3sσ−k2α1.Suppose thats=iχ=χcosπ/2+isinπ/2 is the root of (13), then we get(15)A2Rχcosχϱ+A2Iϕsinχϱ=−A1Rχ,A2Iχcosχϱ−A2Rϕsinχϱ=−A1Iχ,where(16)A1Rχ=χ3σcos3σπ2−α1+k3χ2σcosσπ+α1k3−k2χσcosσπ2+k2α1−k1,A1Iχ=χ3σsin3σπ2−α1+k3χ2σsinσπ+α1k3−k2χσsinσπ2,A2Rχ=α1χ2σcosσπ−α1k3χσcosσπ2−k2α1,A2Iχ=α1χ2σsinσπ−α1k3χσsinσπ2.In view of (15), we have(17)cosχσ=−A1RχA2Rχ+A1IχA2IχA2R2χ+A2I2χ,sinχσ=−A1RχA2Iχ−A1IχA2RχA2R2χ+A2I2χ.Let(18)p1=cos3σπ2,p2=−α1+k3cosσπ,p3=α1k3−k2cosσπ2,p4=k2α1−k1,p5=sin3σπ2,p6=−α1+k3sinσπ,p7=α1k3−k2sinσπ2,p8=α1cosσπ,p9=−α1k3cosσπ2,p10=−k2α1,p11=α1sinσπ,p12=−α1k3sinσπ2.Hence,(19)A1Rχ=p1χ3σ+p2χ2σ+p3χσ+p4,A1Iχ=p5χ3σ+p6χ2σ+p7χσ,A2Rχ=p8χ2σ+p9χσ+p10,A2Iχ=p11χ2σ+p12χσ.By (19), we have(20)A1RχA2Rχ+A1IχA2Iχ2+A1RχA2Iχ−A1IχA2Rχ2=A2R2χ+A2I2χ2.In addition,(21)A1RχA2Rχ+A1IχA2Iχ2=q1χ5σ+q2χ4σ+q3χ3σ+q4χ2σ+q5χσ+q62,A1RχA2Iχ−A1IχA2Rχ2=q7χ5σ+b8χ4σ+q9χ3ϑ+q10χ2σ+q11χσ2,A2R2χ+A2I2χ2=q12χ4q+q13χ3ϑ+q14χ2σ+q15χσ2,where(22)q1=p1p8+p5p11,q2=p2p8+p1p9+p6p11+p5p12,q3=p3p8+p2p9+p1p10+p7p11+p6p12,q4=p4p8+p3p9+p2p10+p7p12,q5=p4p9+p3p10,q6=p4p10,q7=p1p11−p5p8,q8=p2p11+p1p12−p5p9−p6p8,q9=p3p11+p2p12−p5p10−p6p9−p7p8,q10=p4p11+p3p12−p6p10−p7p9,q11=p4p12−p7p10,q12=p82+p112,q13=2p8p9+p11p12,q14=2p8p10+p122,q15=2p9p10.By (20), we have(23)ξ1χ10σ+ξ2χ9σ+ξ3χ8σ+ξ4χ7σ+ξ5χ6σ+ξ6χ5σ+ξ7χ4σ+ξ8χ3σ+ξ9χ2σ+ξ10χσ+ξ11=0,where(24)ξ1=q12+q72,ξ2=2q1q2+2q7q8,ξ3=q22+b82−q122+2q1q3+2q7q9,ξ4=2q2q3+q1q2+q7q10+q8q9−q12q13,ξ5=q32+2q1q5+2q2q4+q92+2q7q11+2q8q10−q132−2q12q14,ξ6=2q1q6+2q2q5+2q3q4+2q9q10−2q12q15−2q13q14,ξ7=q42+2q3q5+q102+2q8q11+2q9q11−q142−2q13q15,ξ8=2q3q6+2q4q5+2q10q11−2q14q15,ξ9=q52+2q4q6+q112−q152,ξ10=2q5q6,ξ11=q62.Let(25)Wχ=ξ1χ10σ+ξ2χ9σ+ξ3χ8σ+ξ4χ7σ+ξ5χ6σ+ξ6χ5σ+ξ7χ4σ+ξ8χ3σ+ξ9χ2σ+ξ10χσ+ξ11=0,(26)Vμ=ξ1μ10+ξ2μ9+ξ3μ8+ξ4μ7+ξ5μ6+ξ6μ5+ξ7μ4+ξ8μ3+ξ9μ2+ξ10μ+ξ11.Lemma 3.
(i) Assume thatξj>0j=1,2,…,11, then the root that has zero real parts does not exist in (13).(ii) Assume that ξ11<0 and ∃μ0>0 which satisfies Vμ0<0, then (13) possesses at least two pairs of purely imaginary roots.Proof of Lemma 3(i).
According to (25), we have(27)dWχdχ=10σξ1χ10σ−1+9σξ2χ9σ−1+8σξ3χ8σ−1+7σξ4χ7σ−1+6σξ5χ6σ−1+5σξ6χ5σ−1+4σξ7χ4σ−1+3σξ8χ3σ−1+2σξ9χ2σ−1+σξ10χσ−1.
In view ofξi>0i=1,2,…,10, we have dWχ/dχ>0, ∀ χ>0. In view of W0=ξ11>0, we know that (26) does not possess positive real roots. Furthermore, s=0 is not the root of (13). This proves Lemma 3(i).Proof of Lemma 3(ii).
According toV0=ξ11>0,Vε0<0ε0>0, and limτ⟶+∞Vτ/dτ=+∞, ∃ ε01∈0,ε0 and ε02∈ε0,+∞ such that Vε01=Vε02=0. Hence, (26) possesses at least two positive real roots and (13) possesses at least two pairs of purely imaginary roots. This proves Lemma 3 (ii).Suppose that (23) possesses ten positive real roots ζjj=1,2,…,10. In view of (17), we get(28)ϱmh=1χmarccos−A1RχmA2Rχm+A1IχmA2IχmA2R2χm+A2I2χm+2hπ,where h=0,1,2,…,andm=1,2,…,10. Hence, ±iχm is a pair of purely imaginary roots of (13) when ϱ=ϱmh. Let(29)ϱ0=minm=1,2,…,11ϱm0,χ0=χϱ=ϱ0.Next, the hypothesisℋ1 is given: K1S1+K2S2>0, where(30)K1=3σχ03σ−1cos3σ−1π2+2σα1+k3χ02σ−1cos2σ−1π2+σα1k3−k2χ0σ−1cosσ−1π2+2σα1χ02σ−1cos2σ−1π2−σα1k3χ0σ−1cosσ−1π2cosχ0ϱ0+2σα1χ02σ−1sin2σ−1π2−σα1k3χ0σ−1sinσ−1π2sinχ0ϱ0,K2=3σχ03σ−1sin3σ−1π2+2σα1+k3χ02σ−1sin2σ−1π2+σα1k3−k2χ0σ−1sinσ−1π2+2σα1χ02σ−1cos2σ−1π2−σα1k3χ0σ−1cosσ−1π2sinχ0ϱ0+2σα1χ02σ−1sin2σ−1π2−σα1k3χ0σ−1sinσ−1π2cosχ0ϱ0,S1=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0sinχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0cosχ0ϱ0,S2=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0cosχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0sinχ0ϱ0.Lemma 4.
Suppose thatsϱ=γϱ+iϕϱ is the root of (13) around ϱ=ϱ0 such that γϱ0=0,ϕϱ0=ϕ0, then Reds/dϱϱ=ϱ0,ϕ=ϕ0>0.Proof of Lemma 4.
In view of (13), we get(31)dA1sdϱ+dA2sdϱe−sϱ−e−sϱdsdϱϱ+sA2s=0.
Since(32)dA1sdϱ=3σs3σ−1dsdϱ−2σα1+k3s2σ−1dsdϱ+σα1k3−k2sσ−1dsdϱ=3σs3σ−1−2σα1+k3s2σ−1+σα1k3−k2sσ−1dsdϱ,dA2sdϱ=2α1σs2σ−1−α1k3σsσ−1dsdϱ,we obtain(33)dsdϱ−1=P1ϱP2ϱ−ϱs,where(34)P1s=3σs3σ−1−2σα1+k3s2σ−1+σα1k3−k2sσ−1+2α1σs2σ−1−α1k3σsσ−1e−sϱ,P2s=A2sse−sϱ.
Thus,(35)Redsdϱϱ=ϱ0,ψ=ψ0=ReP1ϱP2ϱϱ=ϱ0,ψ=ψ0=K1S1+K2S2S12+S22.
It follows fromℋ1 that(36)Redsdϱ−1ρ=ρ0,ψ=ψ0>0.
This proves Lemma 4.Next, the hypothesisℋ2 is given:(37)k3<0,k2k3>−2k1,k1<0.Lemma 5.
Ifϱ=0 and ℋ2 holds true, then system (13) is locally asymptotically stable.Proof of Lemma 5.
Under the conditionϱ=0, (13) becomes(38)λ3−k3λ2−k2λ−k1=0.
In view ofℋ2, argλi>σπ/2i=1,2,3, where λi is the root of (38). Thus, Lemma 5 holds true, implying the proof.The following results are established by above discussion.Theorem 1 (under the condition of Lemma 3(ii)).
Suppose thatℋ1 and ℋ2 hold true, then (i) the equilibrium point of system (10) at the origin is locally asymptotically stable if 0≤ϱ<ϱ0 and (ii) a Hopf bifurcation will happen near the equilibrium point E1 if ϱ=ϱ0.


### 3.2. Adding Time-Delayed Feedback Controller to the Second Equation of Model (2)

In this section, we add a time-delayed feedback controllerα2w2t−w2t−ρ to the second equation of model (2), and then system (2) becomes(39)dσw1dtσ=w2,dσw2dtσ=w3+α2w2t−w2t−ρ,dσw3dtσ=k1w1+k2w2+k3w3+w12,where α2 is the gain coefficient and ρ is the time delay. The linear equation of equation (39) around E1=0,0,0 is(40)dσw1dtσ=w2,dσw2dtσ=α2w2+w3−α2w2t−ρ,dσw3dtσ=k1w1+k2w2+k3w3.The characteristic equation of (40) takes the following form:(41)detsσ−100sσ−α2+α2e−sρ−1−k11−k2sρ−k3.Hence,(42)ℬ1s+ℬ2se−sρ=0,where(43)ℬ1s=s3σ−α2+k3+1s2σ+α2k3+k3−k2sσ−k1,ℬ2s=α2s2σ−α2k3sσ.Assume thats=iϑ=ϑcosπ/2+isinπ/2 is the root of (42), then we have(44)ℬ2Rϑcosϑρ+ℬ2Iϑsinζρ=−ℬ1Rϑ,ℬ2Iϑcosϑρ−ℬ2Rϕsinϑρ=−ℬ1Iϑ,where(45)ℬ1Rϑ=ϑ3σcos3σπ2−α2+k3+1ϑ2σcosσπ+α2k3+k3−k2ϑσcosσπ2−k1,ℬ1Iϑ=ϑ3σsin3σπ2−α2+k3+1ϑ2σsinσπ+α2k3+k3−k2ϑσsinσπ2,ℬ2Rϑ=α2ϑ2ϑcosϑπ−α2k3ϑσcosσπ2,ℬ2Iϑ=α2ϑ2ϑsinϑπ−α2k3ϑσsinσπ2.According to (44), we get(46)cosϑρ=−ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑℬ2R2ϑ+ℬ2I2ϑ,sinϑρ=−ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑℬ2R2ϑ+ℬ2I2ϑ.Let(47)s1=cos3σπ2,s2=−α2+k3+1cosσπ,s3=α2k3+k3−k2cosσπ2,s4=−k1,s5=sin3σπ2,s6=−α2+k3+1sinσπ,s7=α2k3+k3−k2sinσπ2,s8=α2cosϑπ,s9=−α2k3cosσπ2,s10=α2sinϑπ,s11=−α2k3sinσπ2.Then,(48)ℬ1Rϑ=s1ϑ3σ+s2ϑ2σ+s3ϑσ+s4,ℬ1Iϑ=s5ϑ3σ+s6ϑ2σ+s7ϑσ,ℬ2Rϑ=s8ϑ2σ+s9ϑσ,ℬ2Iϑ=s10ϑ2σ+s11ϑσ.By (46), we have(49)ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑ2+ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑ2=ℬ2R2ϑ+ℬ2I2ϑ2.Furthermore,(50)ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑ2=t1ϑ5σ+t2ϑ4σ+t3ϑ3σ+t4ϑ2σ+t5ϑσ2,ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑ2=t5ϑ5σ+t6ϑ4σ+t7ϑ3ϑ+t8ϑ2σ+t9ϑσ2,ℬ2R2ϑ+ℬ2I2χ2=t10ϑ4σ+t11ϑ3σ+t12ϑ2σ+t13ϑσ2,where(51)t1=s1s8+s5s10,t2=s2s8+s1s9+s6s10+s5s11,t3=s3s8+s2s9+s7s10+s6s11,t4=s4s8+s3s9+s7s11,t5=s4s9,t6=s1s10−s5s8,t7=s2s10+s1s11−s5s9−s6s8,t8=s3s10+s2s11−s6s9−s7s8,t9=s4s10+s3s11−s7s9,t10=s4s11,t11=s82+s102,t12=2s8s9+s10s11,t13=s112.In view of (49), we have(52)κ1ϑ10σ+κ2ϑ9σ+κ3ϑ8σ+κ4ϑ7σ+κ5ϑ6σ+κ6ϑ5σ+κ7ϑ4σ+κ8ϑ3σ+κ9ϑ2σ=0,where(53)κ1=t12+t62,κ2=2t1t2+2t6t7,κ3=t22+t72−t112+2t1t3+2t6t8,κ4=2t2t3+t1t2+t6t9+t7t8−t11t12,κ5=t32+2t2t4+t82+2t6t10+2t7t9−t122−2t11t13,κ6=2t3t4+2t8t9,κ7=t42+t92+2t7t10+2t8t10−t132,κ8=2t9t10+2t4t5,κ9=t52+t102.By (52), we have(54)κ1ϑ8σ+κ2ϑ7σ+κ3ϑ6σ+κ4ϑ5σ+κ5ϑ4σ+κ6ϑ3σ+κ7ϑ2σ+κ8ϑσ+κ9=0.Let(55)Kϑ=κ1ϑ8σ+κ2ϑ7σ+κ3ϑ6σ+κ4ϑ5σ+κ5ϑ4σ+κ6ϑ3σ+κ7ϑ2σ+κ8ϑσ+κ9,(56)Mζ=κ1ζ8+κ2ζ7+κ3ζ6+κ4ζ5+κ5ζ4+κ6ζ3+κ7ζ2+κ8ζ1+κ9.Lemma 6.
(i) Suppose thatκj>0j=1,2,…,9, then the root that has zero real parts does not exist in (42).(ii) Suppose that κ9<0 and ∃η0>0 which satisfies Mη0<0, then (42) possesses at least two pairs of purely imaginary roots.Proof of Lemma 6(i).
In view of (55), we have(57)dKϑdχ=8κ1σϑ8σ−1+7κ2σϑ7σ−1+6κ3σϑ6σ−1+5κ4σϑ5σ−1+4κ5σϑ4σ−1+3κ6σϑ3σ−1+2κ7σϑ2σ−1+κ8σϑσ−1.
In view ofκi>0i=1,2,…,9, we have dKϑ/dϑ>0, ∀ ϑ>0. In view of K0=κ9>0, we know that (56) does not possess positive real roots. Furthermore, s=0 is not the root of (42). This proves Lemma 6(i).Proof of Lemma 6(ii).
In view ofM0=κ9>0,Mη0<0η0>0, and limε⟶+∞Mε/dε=+∞, ∃ η01∈0,η0 and η02∈η0,+∞ such that Mη01=Mη02=0, Hence, (55) has at least two positive real roots and (42) has at least two pairs of purely imaginary roots. This proves Lemma 6(ii).Assume that (54) has eight positive real roots ϑjj=1,2,…,8. By (46), we have(58)ρkι=1ϑkarccos−ℬ1Rϑmℬ2Rϑk+ℬ1Iϑkℬ2Iϑkℬ2R2ϑk+ℬ2I2ϑk+2ιπ,where ι=0,1,2,…,andk=1,2,…,8. Hence, ±iϑk is a pair of purely imaginary roots of (42) when ρ=ρkι. Let(59)ρ0=mink=1,2,…,8ρk0,ϑ0=ϑρ=ρ0.Next, we give the hypothesisℋ3 as follows:ℱ1G1+ℱ2G2>0, where(60)ℱ1=3σϑ03σ−1cos3σ−1π2−2σα2+k3+1ϑ02σ−1cos2σ−1π2+σα2k3+k3−k2ϑ0σ−1cosσ−1π2+2σα2ϑ02σ−1cos2σ−1π2−α2k3ϑ0σ−1cosσ−1π2cosϑ0ρ0+2σα2ϑ02σ−1sin2σ−1π2−−α2k3ϑ0σ−1sinσ−1π2sinϑ0ρ0,ℱ2=3σϑ03σ−1sin3σ−1π2−2σα2+k3+1ϑ02σ−1sin2σ−1π2+σα2k3+k3−k2ϑ0σ−1sinσ−1π2+2σα2ϑ02σ−1cos2σ−1π2−α2k3ϑ0σ−1cosσ−1π2sinϑ0ρ0+2σα2ϑ02σ−1sin2σ−1π2−−α2k3ϑ0σ−1sinσ−1π2cosϑ0ρ0,G1=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0sinχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0cosχ0ϱ0,G2=α2ϑ02σcosσπ−α2k3χ0σcosσπ2−α2k3χ0cosϑ0ρ0+α2ϑ02σsinσπ−α2k3ϑ0σsinσπ2−α2k3ϑ0sinϑ0ρ0.Lemma 7.
Assume thatsρ=ζρ+iϖρ is the root of (42) around ρ=ρ0 such that ζρ0=0,ϖρ0=ϖ0, then Reds/dρρ=ρ0,ϖ=ϖ0>0.Proof of Lemma 7.
By (42), we have(61)dℬ1sdρ+dℬ2sdρe−sρ−e−sρdsdρρ+sℬ2s=0.
Since(62)dℬ1sdρ=3σs3σ−1dsdϱ−2σα2+k3+1s2σ−1dsdρ+σα2k3+k3−k2sσ−1dsdρ=3σs3σ−1−2σα2+k3+1s2σ−1++σα2k3+k3−k2sσ−1dsdρ,dℬ2sdρ=2α2σs2σ−1−α2k3σsσ−1dsdρ,we have(63)dsdρ−1=Q1ρQ2ρ−ρs,where(64)Q1s=3σs3σ−1−2σα2+k3+1s2σ−1+σα2k3+k3−k2sσ−1+2α2σs2σ−1−α2k3σsσ−1e−sϱ,Q2s=ℬ2sse−sρ.
Thus,(65)Redsdρρ=ρ0,ϖ=ϖ0=ReQ1ρQ2ρρ=ρ0,ϖ=ϖ0=ℱ1G1+ℱ2G2G12+G22.
Byℋ3, we have(66)Redsdρ−1ρ=ρ0,ϖ=ϖ0>0.
This proves Lemma 7.Next, we give the hypothesisℋ4 as follows:(67)k3+1<0,k2+1k3−k2<−2k1,k1<0.Lemma 8.
Ifρ=0 and ℋ4 holds true, then system (42) is locally asymptotically stable.Proof of Lemma 8.
Ifρ=0, then (42) takes the following form:(68)λ3−k3λ2−k2λ−k1=0.
In view ofℋ4, argλi>σπ/2i=1,2,3, where λi is the root of (68). Therefore, Lemma 8 holds true, implying the proof.Theorem 2 (in addition to the condition of Lemma 6(ii)).
Assume thatℋ3 and ℋ4 are fulfilled, then (i) the equilibrium point of model (39) at the origin is locally asymptotically stable if 0≤ρ<ρ0 and (ii) a Hopf bifurcation will happen near the equilibrium point E1 if ρ=ρ0.Remark 2.
Although there are many papers [17, 18, 21, 23] that deal with the chaos control by applying the time-delayed feedback controller, they only focus on the integer-order differential systems. In this paper, we handle the chaos control for the fractional-order differential model. All the derived results and analysis ways for the chaos control of integer-order differential systems cannot be transferred to (2) to control the chaotic phenomenon. From this viewpoint, we can say that the results of this paper are completely innovative and supplement the earlier investigation.Remark 3.
By computer simulations and maximum Lyapunov exponent, we show that model (2) has chaotic behavior. If the time delay is introduced into the controlled chaotic model (10) or (39), then we can control the chaotic phenomenon of the origin system (2) under some suitable conditions. Also, we can examine the chaotic nature of the fractional-order system by maximum Lyapunov exponent and computer simulations.Remark 4.
Adding the time-delayed feedback controller to the first equation and the second equation of model (2) is only a try. Of course, we can add the controller to the third equation. Through the later analysis, we can check whether this control technique is right or not.Remark 5.
Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesi model have the same form, the dynamical behavior and the research method on stability and Hopf bifurcation are very different because of introduction and the variation of fractional order. According to some previous related papers (see [17–19, 21, 38, 39]), we can see that the dynamical behavior and the research method on stability and Hopf bifurcation for fractional-order differential systems are very different from those for integer-order differential systems.


## 3.1. Adding Time-Delayed Feedback Controller to the First Equation of Model (2)

In this section, we add a time-delayed feedback controllerα1w1t−w1t−ϱ to the first equation of model (2), and then system (2) takes the following form:(10)dσw1dtσ=w2+α1w1t−w1t−ϱ,dσw2dtσ=w3,dσw3dtσ=k1w1+k2w2+k3w3+w12,where α1 is the gain coefficient and ϱ is the time delay. The linear system of system (10) near the equilibrium point E1=0,0,0 reads(11)dσw1dtσ=α1w1+w2−α1w1t−ϱ,dσw2dtσ=w3,dσw3dtσ=k1w1+k2w2+k3w3.The corresponding characteristic equation of (11) takes the following form:(12)detsσ−α1+α1e−sϱ−100sσ−1−k1−k2sσ−k3.Hence,(13)A1s+A2se−sϱ=0,where(14)A1s=s3σ−α1+k3s2σ+α1k3−k2sσ+k2α1−k1,A2s=α1s2σ−α1k3sσ−k2α1.Suppose thats=iχ=χcosπ/2+isinπ/2 is the root of (13), then we get(15)A2Rχcosχϱ+A2Iϕsinχϱ=−A1Rχ,A2Iχcosχϱ−A2Rϕsinχϱ=−A1Iχ,where(16)A1Rχ=χ3σcos3σπ2−α1+k3χ2σcosσπ+α1k3−k2χσcosσπ2+k2α1−k1,A1Iχ=χ3σsin3σπ2−α1+k3χ2σsinσπ+α1k3−k2χσsinσπ2,A2Rχ=α1χ2σcosσπ−α1k3χσcosσπ2−k2α1,A2Iχ=α1χ2σsinσπ−α1k3χσsinσπ2.In view of (15), we have(17)cosχσ=−A1RχA2Rχ+A1IχA2IχA2R2χ+A2I2χ,sinχσ=−A1RχA2Iχ−A1IχA2RχA2R2χ+A2I2χ.Let(18)p1=cos3σπ2,p2=−α1+k3cosσπ,p3=α1k3−k2cosσπ2,p4=k2α1−k1,p5=sin3σπ2,p6=−α1+k3sinσπ,p7=α1k3−k2sinσπ2,p8=α1cosσπ,p9=−α1k3cosσπ2,p10=−k2α1,p11=α1sinσπ,p12=−α1k3sinσπ2.Hence,(19)A1Rχ=p1χ3σ+p2χ2σ+p3χσ+p4,A1Iχ=p5χ3σ+p6χ2σ+p7χσ,A2Rχ=p8χ2σ+p9χσ+p10,A2Iχ=p11χ2σ+p12χσ.By (19), we have(20)A1RχA2Rχ+A1IχA2Iχ2+A1RχA2Iχ−A1IχA2Rχ2=A2R2χ+A2I2χ2.In addition,(21)A1RχA2Rχ+A1IχA2Iχ2=q1χ5σ+q2χ4σ+q3χ3σ+q4χ2σ+q5χσ+q62,A1RχA2Iχ−A1IχA2Rχ2=q7χ5σ+b8χ4σ+q9χ3ϑ+q10χ2σ+q11χσ2,A2R2χ+A2I2χ2=q12χ4q+q13χ3ϑ+q14χ2σ+q15χσ2,where(22)q1=p1p8+p5p11,q2=p2p8+p1p9+p6p11+p5p12,q3=p3p8+p2p9+p1p10+p7p11+p6p12,q4=p4p8+p3p9+p2p10+p7p12,q5=p4p9+p3p10,q6=p4p10,q7=p1p11−p5p8,q8=p2p11+p1p12−p5p9−p6p8,q9=p3p11+p2p12−p5p10−p6p9−p7p8,q10=p4p11+p3p12−p6p10−p7p9,q11=p4p12−p7p10,q12=p82+p112,q13=2p8p9+p11p12,q14=2p8p10+p122,q15=2p9p10.By (20), we have(23)ξ1χ10σ+ξ2χ9σ+ξ3χ8σ+ξ4χ7σ+ξ5χ6σ+ξ6χ5σ+ξ7χ4σ+ξ8χ3σ+ξ9χ2σ+ξ10χσ+ξ11=0,where(24)ξ1=q12+q72,ξ2=2q1q2+2q7q8,ξ3=q22+b82−q122+2q1q3+2q7q9,ξ4=2q2q3+q1q2+q7q10+q8q9−q12q13,ξ5=q32+2q1q5+2q2q4+q92+2q7q11+2q8q10−q132−2q12q14,ξ6=2q1q6+2q2q5+2q3q4+2q9q10−2q12q15−2q13q14,ξ7=q42+2q3q5+q102+2q8q11+2q9q11−q142−2q13q15,ξ8=2q3q6+2q4q5+2q10q11−2q14q15,ξ9=q52+2q4q6+q112−q152,ξ10=2q5q6,ξ11=q62.Let(25)Wχ=ξ1χ10σ+ξ2χ9σ+ξ3χ8σ+ξ4χ7σ+ξ5χ6σ+ξ6χ5σ+ξ7χ4σ+ξ8χ3σ+ξ9χ2σ+ξ10χσ+ξ11=0,(26)Vμ=ξ1μ10+ξ2μ9+ξ3μ8+ξ4μ7+ξ5μ6+ξ6μ5+ξ7μ4+ξ8μ3+ξ9μ2+ξ10μ+ξ11.Lemma 3.
(i) Assume thatξj>0j=1,2,…,11, then the root that has zero real parts does not exist in (13).(ii) Assume that ξ11<0 and ∃μ0>0 which satisfies Vμ0<0, then (13) possesses at least two pairs of purely imaginary roots.Proof of Lemma 3(i).
According to (25), we have(27)dWχdχ=10σξ1χ10σ−1+9σξ2χ9σ−1+8σξ3χ8σ−1+7σξ4χ7σ−1+6σξ5χ6σ−1+5σξ6χ5σ−1+4σξ7χ4σ−1+3σξ8χ3σ−1+2σξ9χ2σ−1+σξ10χσ−1.
In view ofξi>0i=1,2,…,10, we have dWχ/dχ>0, ∀ χ>0. In view of W0=ξ11>0, we know that (26) does not possess positive real roots. Furthermore, s=0 is not the root of (13). This proves Lemma 3(i).Proof of Lemma 3(ii).
According toV0=ξ11>0,Vε0<0ε0>0, and limτ⟶+∞Vτ/dτ=+∞, ∃ ε01∈0,ε0 and ε02∈ε0,+∞ such that Vε01=Vε02=0. Hence, (26) possesses at least two positive real roots and (13) possesses at least two pairs of purely imaginary roots. This proves Lemma 3 (ii).Suppose that (23) possesses ten positive real roots ζjj=1,2,…,10. In view of (17), we get(28)ϱmh=1χmarccos−A1RχmA2Rχm+A1IχmA2IχmA2R2χm+A2I2χm+2hπ,where h=0,1,2,…,andm=1,2,…,10. Hence, ±iχm is a pair of purely imaginary roots of (13) when ϱ=ϱmh. Let(29)ϱ0=minm=1,2,…,11ϱm0,χ0=χϱ=ϱ0.Next, the hypothesisℋ1 is given: K1S1+K2S2>0, where(30)K1=3σχ03σ−1cos3σ−1π2+2σα1+k3χ02σ−1cos2σ−1π2+σα1k3−k2χ0σ−1cosσ−1π2+2σα1χ02σ−1cos2σ−1π2−σα1k3χ0σ−1cosσ−1π2cosχ0ϱ0+2σα1χ02σ−1sin2σ−1π2−σα1k3χ0σ−1sinσ−1π2sinχ0ϱ0,K2=3σχ03σ−1sin3σ−1π2+2σα1+k3χ02σ−1sin2σ−1π2+σα1k3−k2χ0σ−1sinσ−1π2+2σα1χ02σ−1cos2σ−1π2−σα1k3χ0σ−1cosσ−1π2sinχ0ϱ0+2σα1χ02σ−1sin2σ−1π2−σα1k3χ0σ−1sinσ−1π2cosχ0ϱ0,S1=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0sinχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0cosχ0ϱ0,S2=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0cosχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0sinχ0ϱ0.Lemma 4.
Suppose thatsϱ=γϱ+iϕϱ is the root of (13) around ϱ=ϱ0 such that γϱ0=0,ϕϱ0=ϕ0, then Reds/dϱϱ=ϱ0,ϕ=ϕ0>0.Proof of Lemma 4.
In view of (13), we get(31)dA1sdϱ+dA2sdϱe−sϱ−e−sϱdsdϱϱ+sA2s=0.
Since(32)dA1sdϱ=3σs3σ−1dsdϱ−2σα1+k3s2σ−1dsdϱ+σα1k3−k2sσ−1dsdϱ=3σs3σ−1−2σα1+k3s2σ−1+σα1k3−k2sσ−1dsdϱ,dA2sdϱ=2α1σs2σ−1−α1k3σsσ−1dsdϱ,we obtain(33)dsdϱ−1=P1ϱP2ϱ−ϱs,where(34)P1s=3σs3σ−1−2σα1+k3s2σ−1+σα1k3−k2sσ−1+2α1σs2σ−1−α1k3σsσ−1e−sϱ,P2s=A2sse−sϱ.
Thus,(35)Redsdϱϱ=ϱ0,ψ=ψ0=ReP1ϱP2ϱϱ=ϱ0,ψ=ψ0=K1S1+K2S2S12+S22.
It follows fromℋ1 that(36)Redsdϱ−1ρ=ρ0,ψ=ψ0>0.
This proves Lemma 4.Next, the hypothesisℋ2 is given:(37)k3<0,k2k3>−2k1,k1<0.Lemma 5.
Ifϱ=0 and ℋ2 holds true, then system (13) is locally asymptotically stable.Proof of Lemma 5.
Under the conditionϱ=0, (13) becomes(38)λ3−k3λ2−k2λ−k1=0.
In view ofℋ2, argλi>σπ/2i=1,2,3, where λi is the root of (38). Thus, Lemma 5 holds true, implying the proof.The following results are established by above discussion.Theorem 1 (under the condition of Lemma 3(ii)).
Suppose thatℋ1 and ℋ2 hold true, then (i) the equilibrium point of system (10) at the origin is locally asymptotically stable if 0≤ϱ<ϱ0 and (ii) a Hopf bifurcation will happen near the equilibrium point E1 if ϱ=ϱ0.


## 3.2. Adding Time-Delayed Feedback Controller to the Second Equation of Model (2)

In this section, we add a time-delayed feedback controllerα2w2t−w2t−ρ to the second equation of model (2), and then system (2) becomes(39)dσw1dtσ=w2,dσw2dtσ=w3+α2w2t−w2t−ρ,dσw3dtσ=k1w1+k2w2+k3w3+w12,where α2 is the gain coefficient and ρ is the time delay. The linear equation of equation (39) around E1=0,0,0 is(40)dσw1dtσ=w2,dσw2dtσ=α2w2+w3−α2w2t−ρ,dσw3dtσ=k1w1+k2w2+k3w3.The characteristic equation of (40) takes the following form:(41)detsσ−100sσ−α2+α2e−sρ−1−k11−k2sρ−k3.Hence,(42)ℬ1s+ℬ2se−sρ=0,where(43)ℬ1s=s3σ−α2+k3+1s2σ+α2k3+k3−k2sσ−k1,ℬ2s=α2s2σ−α2k3sσ.Assume thats=iϑ=ϑcosπ/2+isinπ/2 is the root of (42), then we have(44)ℬ2Rϑcosϑρ+ℬ2Iϑsinζρ=−ℬ1Rϑ,ℬ2Iϑcosϑρ−ℬ2Rϕsinϑρ=−ℬ1Iϑ,where(45)ℬ1Rϑ=ϑ3σcos3σπ2−α2+k3+1ϑ2σcosσπ+α2k3+k3−k2ϑσcosσπ2−k1,ℬ1Iϑ=ϑ3σsin3σπ2−α2+k3+1ϑ2σsinσπ+α2k3+k3−k2ϑσsinσπ2,ℬ2Rϑ=α2ϑ2ϑcosϑπ−α2k3ϑσcosσπ2,ℬ2Iϑ=α2ϑ2ϑsinϑπ−α2k3ϑσsinσπ2.According to (44), we get(46)cosϑρ=−ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑℬ2R2ϑ+ℬ2I2ϑ,sinϑρ=−ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑℬ2R2ϑ+ℬ2I2ϑ.Let(47)s1=cos3σπ2,s2=−α2+k3+1cosσπ,s3=α2k3+k3−k2cosσπ2,s4=−k1,s5=sin3σπ2,s6=−α2+k3+1sinσπ,s7=α2k3+k3−k2sinσπ2,s8=α2cosϑπ,s9=−α2k3cosσπ2,s10=α2sinϑπ,s11=−α2k3sinσπ2.Then,(48)ℬ1Rϑ=s1ϑ3σ+s2ϑ2σ+s3ϑσ+s4,ℬ1Iϑ=s5ϑ3σ+s6ϑ2σ+s7ϑσ,ℬ2Rϑ=s8ϑ2σ+s9ϑσ,ℬ2Iϑ=s10ϑ2σ+s11ϑσ.By (46), we have(49)ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑ2+ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑ2=ℬ2R2ϑ+ℬ2I2ϑ2.Furthermore,(50)ℬ1Rϑℬ2Rϑ+ℬ1Iϑℬ2Iϑ2=t1ϑ5σ+t2ϑ4σ+t3ϑ3σ+t4ϑ2σ+t5ϑσ2,ℬ1Rϑℬ2Iϑ−ℬ1Iϑℬ2Rϑ2=t5ϑ5σ+t6ϑ4σ+t7ϑ3ϑ+t8ϑ2σ+t9ϑσ2,ℬ2R2ϑ+ℬ2I2χ2=t10ϑ4σ+t11ϑ3σ+t12ϑ2σ+t13ϑσ2,where(51)t1=s1s8+s5s10,t2=s2s8+s1s9+s6s10+s5s11,t3=s3s8+s2s9+s7s10+s6s11,t4=s4s8+s3s9+s7s11,t5=s4s9,t6=s1s10−s5s8,t7=s2s10+s1s11−s5s9−s6s8,t8=s3s10+s2s11−s6s9−s7s8,t9=s4s10+s3s11−s7s9,t10=s4s11,t11=s82+s102,t12=2s8s9+s10s11,t13=s112.In view of (49), we have(52)κ1ϑ10σ+κ2ϑ9σ+κ3ϑ8σ+κ4ϑ7σ+κ5ϑ6σ+κ6ϑ5σ+κ7ϑ4σ+κ8ϑ3σ+κ9ϑ2σ=0,where(53)κ1=t12+t62,κ2=2t1t2+2t6t7,κ3=t22+t72−t112+2t1t3+2t6t8,κ4=2t2t3+t1t2+t6t9+t7t8−t11t12,κ5=t32+2t2t4+t82+2t6t10+2t7t9−t122−2t11t13,κ6=2t3t4+2t8t9,κ7=t42+t92+2t7t10+2t8t10−t132,κ8=2t9t10+2t4t5,κ9=t52+t102.By (52), we have(54)κ1ϑ8σ+κ2ϑ7σ+κ3ϑ6σ+κ4ϑ5σ+κ5ϑ4σ+κ6ϑ3σ+κ7ϑ2σ+κ8ϑσ+κ9=0.Let(55)Kϑ=κ1ϑ8σ+κ2ϑ7σ+κ3ϑ6σ+κ4ϑ5σ+κ5ϑ4σ+κ6ϑ3σ+κ7ϑ2σ+κ8ϑσ+κ9,(56)Mζ=κ1ζ8+κ2ζ7+κ3ζ6+κ4ζ5+κ5ζ4+κ6ζ3+κ7ζ2+κ8ζ1+κ9.Lemma 6.
(i) Suppose thatκj>0j=1,2,…,9, then the root that has zero real parts does not exist in (42).(ii) Suppose that κ9<0 and ∃η0>0 which satisfies Mη0<0, then (42) possesses at least two pairs of purely imaginary roots.Proof of Lemma 6(i).
In view of (55), we have(57)dKϑdχ=8κ1σϑ8σ−1+7κ2σϑ7σ−1+6κ3σϑ6σ−1+5κ4σϑ5σ−1+4κ5σϑ4σ−1+3κ6σϑ3σ−1+2κ7σϑ2σ−1+κ8σϑσ−1.
In view ofκi>0i=1,2,…,9, we have dKϑ/dϑ>0, ∀ ϑ>0. In view of K0=κ9>0, we know that (56) does not possess positive real roots. Furthermore, s=0 is not the root of (42). This proves Lemma 6(i).Proof of Lemma 6(ii).
In view ofM0=κ9>0,Mη0<0η0>0, and limε⟶+∞Mε/dε=+∞, ∃ η01∈0,η0 and η02∈η0,+∞ such that Mη01=Mη02=0, Hence, (55) has at least two positive real roots and (42) has at least two pairs of purely imaginary roots. This proves Lemma 6(ii).Assume that (54) has eight positive real roots ϑjj=1,2,…,8. By (46), we have(58)ρkι=1ϑkarccos−ℬ1Rϑmℬ2Rϑk+ℬ1Iϑkℬ2Iϑkℬ2R2ϑk+ℬ2I2ϑk+2ιπ,where ι=0,1,2,…,andk=1,2,…,8. Hence, ±iϑk is a pair of purely imaginary roots of (42) when ρ=ρkι. Let(59)ρ0=mink=1,2,…,8ρk0,ϑ0=ϑρ=ρ0.Next, we give the hypothesisℋ3 as follows:ℱ1G1+ℱ2G2>0, where(60)ℱ1=3σϑ03σ−1cos3σ−1π2−2σα2+k3+1ϑ02σ−1cos2σ−1π2+σα2k3+k3−k2ϑ0σ−1cosσ−1π2+2σα2ϑ02σ−1cos2σ−1π2−α2k3ϑ0σ−1cosσ−1π2cosϑ0ρ0+2σα2ϑ02σ−1sin2σ−1π2−−α2k3ϑ0σ−1sinσ−1π2sinϑ0ρ0,ℱ2=3σϑ03σ−1sin3σ−1π2−2σα2+k3+1ϑ02σ−1sin2σ−1π2+σα2k3+k3−k2ϑ0σ−1sinσ−1π2+2σα2ϑ02σ−1cos2σ−1π2−α2k3ϑ0σ−1cosσ−1π2sinϑ0ρ0+2σα2ϑ02σ−1sin2σ−1π2−−α2k3ϑ0σ−1sinσ−1π2cosϑ0ρ0,G1=α1χ02σcosσπ−α1k3χ0σcosσπ2−α1k2χ0sinχ0ϱ0+α1χ02σsinσπ−α1k3χ0σsinσπ2−α1k2χ0cosχ0ϱ0,G2=α2ϑ02σcosσπ−α2k3χ0σcosσπ2−α2k3χ0cosϑ0ρ0+α2ϑ02σsinσπ−α2k3ϑ0σsinσπ2−α2k3ϑ0sinϑ0ρ0.Lemma 7.
Assume thatsρ=ζρ+iϖρ is the root of (42) around ρ=ρ0 such that ζρ0=0,ϖρ0=ϖ0, then Reds/dρρ=ρ0,ϖ=ϖ0>0.Proof of Lemma 7.
By (42), we have(61)dℬ1sdρ+dℬ2sdρe−sρ−e−sρdsdρρ+sℬ2s=0.
Since(62)dℬ1sdρ=3σs3σ−1dsdϱ−2σα2+k3+1s2σ−1dsdρ+σα2k3+k3−k2sσ−1dsdρ=3σs3σ−1−2σα2+k3+1s2σ−1++σα2k3+k3−k2sσ−1dsdρ,dℬ2sdρ=2α2σs2σ−1−α2k3σsσ−1dsdρ,we have(63)dsdρ−1=Q1ρQ2ρ−ρs,where(64)Q1s=3σs3σ−1−2σα2+k3+1s2σ−1+σα2k3+k3−k2sσ−1+2α2σs2σ−1−α2k3σsσ−1e−sϱ,Q2s=ℬ2sse−sρ.
Thus,(65)Redsdρρ=ρ0,ϖ=ϖ0=ReQ1ρQ2ρρ=ρ0,ϖ=ϖ0=ℱ1G1+ℱ2G2G12+G22.
Byℋ3, we have(66)Redsdρ−1ρ=ρ0,ϖ=ϖ0>0.
This proves Lemma 7.Next, we give the hypothesisℋ4 as follows:(67)k3+1<0,k2+1k3−k2<−2k1,k1<0.Lemma 8.
Ifρ=0 and ℋ4 holds true, then system (42) is locally asymptotically stable.Proof of Lemma 8.
Ifρ=0, then (42) takes the following form:(68)λ3−k3λ2−k2λ−k1=0.
In view ofℋ4, argλi>σπ/2i=1,2,3, where λi is the root of (68). Therefore, Lemma 8 holds true, implying the proof.Theorem 2 (in addition to the condition of Lemma 6(ii)).
Assume thatℋ3 and ℋ4 are fulfilled, then (i) the equilibrium point of model (39) at the origin is locally asymptotically stable if 0≤ρ<ρ0 and (ii) a Hopf bifurcation will happen near the equilibrium point E1 if ρ=ρ0.Remark 2.
Although there are many papers [17, 18, 21, 23] that deal with the chaos control by applying the time-delayed feedback controller, they only focus on the integer-order differential systems. In this paper, we handle the chaos control for the fractional-order differential model. All the derived results and analysis ways for the chaos control of integer-order differential systems cannot be transferred to (2) to control the chaotic phenomenon. From this viewpoint, we can say that the results of this paper are completely innovative and supplement the earlier investigation.Remark 3.
By computer simulations and maximum Lyapunov exponent, we show that model (2) has chaotic behavior. If the time delay is introduced into the controlled chaotic model (10) or (39), then we can control the chaotic phenomenon of the origin system (2) under some suitable conditions. Also, we can examine the chaotic nature of the fractional-order system by maximum Lyapunov exponent and computer simulations.Remark 4.
Adding the time-delayed feedback controller to the first equation and the second equation of model (2) is only a try. Of course, we can add the controller to the third equation. Through the later analysis, we can check whether this control technique is right or not.Remark 5.
Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesi model have the same form, the dynamical behavior and the research method on stability and Hopf bifurcation are very different because of introduction and the variation of fractional order. According to some previous related papers (see [17–19, 21, 38, 39]), we can see that the dynamical behavior and the research method on stability and Hopf bifurcation for fractional-order differential systems are very different from those for integer-order differential systems.


## 4. Two Examples

In this section, we use a new predictor-corrector method (NPCM) [33] to carry out computer simulations.Example 1.
The following model is given:(69)d0.87w1dt0.87=w2+α1w1t−w1t−ϱ,d0.87w2dt0.87=w3,d0.87w3dt0.87=−6w1−2.92w2−1.2w3+w12.
Obviously, model (69) possesses the zero equilibrium point. Let α1=2.2, the critical frequency χ0=1.2355, and the bifurcation point ϱ0=2.08. We can easily check that the hypotheses of Theorem 1 are fulfilled. Figure 3 indicates that when ϱ∈0,2.08, the zero equilibrium point of model (69) is locally asymptotically stable. Figure 4 shows that when ϱ∈2.08,+∞, a Hopf bifurcation happens for model (69). Both cases illustrate that the chaotic behavior can be controlled by applying the designed time-delayed controller 2.2w2t−w2t−ϱ. The bifurcation diagram is presented in Figure 5.Figure 3
Computer simulations of model (69) when ϱ=1.9<ϱ0=2.08 and the initial value is (1, 1, 1).
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(b)
(c)
(d)Figure 4
Computer simulations of model (69) when ϱ=2.3>ϱ0=2.08 and the initial value is (1, 1, 1).
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(b)
(c)
(d)Figure 5
Bifurcation diagram with respect to the time delayϱ for model (69). The horizontal axis is ϱ, and the vertical axis is w1t.Example 2.
The following model is given:(70)dσw1dtσ=w2,dσw2dtσ=w3+α2w2t−w2t−ρ,dσw3dtσ=k1w1+k2w2+k3w3+w12.
It is easy to see that model (70) possesses the zero equilibrium point. Let α2=2.2, the critical frequency ϑ0=2.3511, and the bifurcation point ρ0=0.77. We can easily check that the hypotheses of Theorem 1 are satisfied. Figure 6 indicates that when ρ∈0,0.77, the zero equilibrium point of model (70) is locally asymptotically stable. Figure 7 shows that when ρ∈0.77,+∞, a Hopf bifurcation happens for model (70). Both cases manifest that the chaotic behavior of model (70) can be suppressed by using the time-delayed feedback controller 2.2w2t−w2t−ρ.Figure 6
Computer simulations of model (70) when ρ=0.70<ρ0=0.77 and the initial value is (1, 1, 1).
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(b)
(c)
(d)Figure 7
Computer simulations of model (70) when ρ=0.82>ρ0=0.77 and the initial value is (1, 1, 1).
(a)
(b)
(c)
(d)Remark 6.
According to the computer simulation results of model (69) and model (70), we can see that when the time delay remains in a suitable range, the controller chaotic model is locally asymptotically stable, and when the time delay crosses a certain critical value, then a Hopf bifurcation appears near the equilibrium point of the controlled model. Based on the simulation results above, we can conclude that the time-delayed feedback controller in model (70) is more effective for the chaos control than that in model (69) since the delay in model (70) is less than the delay in model (69).


## 5. Conclusions

In real life, chaos control has become a hot issue. During the past decades, a lot of control techniques are proposed to suppress the chaotic phenomenon. But all the control strategies are only concerned with the integer-order differential chaotic systems. In this paper, we adopt two time-delayed feedback controllers to control the same fractional-order chaotic Genesio-Tesi model. The research results show that both time-delayed feedback controllers can effectively control the chaotic behavior of the fractional-order chaotic Genesio-Tesi model. By analyzing the characteristic equation of the controlled fractional-order chaotic Genesio-Tesi model, we establish two sufficient conditions to ensure the stability and the appearance of Hopf bifurcation of the involved chaotic Genesio-Tesi model. In addition, the control technique can be widely applied in numerous fractional-order chaotic models.



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