# Integer and Fractional GeneralT-System and Its Application to Control Chaos and Synchronization

Authors: Mihaela Neamţu; Anamaria Liţoiu; Petru C. Strain
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We propose a three-dimensional autonomous nonlinear system,
called the generalT system, which has potential applications in secure communications
and the electronic circuit. For the general T system with delayed feedback, regarding the
delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics.
We determine conditions for the existence of the Hopf bifurcations and analyze their direction
and stability. Also, the fractional order general T-system is proposed and analyzed.
We provide some numerical simulations, where the chaos attractor and the dynamics of
the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic
control and synchronization on schemes for the new fractional order chaotic system are
verified by numerical simulations.


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## Body

## 1. Introduction

Lorenz found the first canonical chaotic attractor [1]. During the time, it has been proved that chaos can occur in simple three-dimensional autonomous systems with one, two, and three nonlinearities. Tigan and Opris [2] proposed and analyzed a new chaotic three-dimensional nonlinear system, called T system, which is similar to the Lorenz system. Because T system allows a larger possibility in choosing the parameters of the system, it can display more complex dynamics [3–10].Recently, based on the study of integer order chaos, the fractional order Lorenz system [11] and the fractional order Liu system [12] were introduced. The system with fractional order still shows the chaotic behavior [13, 14].TheT system is described by [2](1)x˙t=ayt-xt,y˙t=b-axt-axtzt,z˙t=-czt+xtyt,which is chaotic when a=2.1, b=30, and c=0.6 [2].Li et al. [15] have proposed a new Lorenz-like chaotic system derived from (1). The nonlinear differential three-dimensional system is given by(2)x˙t=ayt-xt,y˙t=abxt-axtzt,z˙t=-czt+xtyt,which is chaotic when a=5, b=4, and c=2 [15].Chaotic phenomena in electric circuits have been studied with great interest. The electronic circuit for (2) is designed and a chaotic attractor is implemented and verified [15].Yang [16] proposed another new Lorenz-like system. The nonlinear differential three-dimensional system is(3)x˙t=ayt-xt,y˙t=b1xt-b2xtzt,z˙t=-czt+lxt2+kyt2,which is chaotic when a=10, b1=40, b2=1, c=2.5, l=2, and k=2 [16].Based on (1), (2), and (3), we propose a general T system described by the following differential equations:(4)x˙t=ayt-xt,y˙t=b1xt-b2xtzt,z˙t=-czt+dxtyt+lxt2+kyt2,where a,b2,c,d,h,k are positive real parameters, b1 is a real parameter, and a≠0.An experimental electronic circuit for (4) can be described and implemented in a similar way as in [15, 17].System (4) is chaotic when a=10, b1=40, b2=1, c=2.5, d=3, l=0.9, and k=2 (see Figure 1).Figure 1
Phase portrait of the chaotic attractor for the generalT system (4) with a=10, b1=40, b2=1, c=2.5, d=3, l=0.9, and k=2.The dynamics of Lyapunov exponents of system (4) is displayed in Figure 2.Figure 2
Dynamics of Lyapunov exponents of (4) with a=10, b1=40, b2=1, c=2.5, d=3, l=0.9, and k=2.Forb1=b-a, b2=a, d=1, l=0, and k=0, system (4) is given by (1). For b1=ab, b2=a, d=1, l=0, and k=0, system (4) becomes (2). For d=0, system (4) is (3).Systems (1), (2), (3), and (4) are chaotic and using the method from [15] we can verify that they are not equivalent to the Lorenz, Chen, and Lü systems.Time-delayed feedback is a powerful tool to control unstable periodic orbits or control unstable steady states [18]. Following the idea of Pyragas [19], as in [9, 18], we add a delayed force k1(y(t)-y(t-τ)) to the second equation of (4) and we obtain the delayed feedback control system:(5)x˙t=ayt-xt,y˙t=b1xt-b2xtzt+k1yt-yt-τ,z˙t=-czt+dxtyt+lxt2+kyt2,where τ is the time delay.We assume thatτ>0 and k1∈R, which indicates the strength of the feedback [10].The fractional-order generalT system can be described by(6)Dq1xt=ayt-xt,Dq2yt=b1xt-b2xtzt,Dq3zt=-czt+dxtyt+lxt2+kyt2,where Dtqa is defined by [13](7)Dtqa=dqdtq,Re⁡q>01Re⁡q=0∫atds-qRe⁡q<0and q is the derivative order that can be a complex number with Re⁡(q) the real part of q. The numbers a and t are the limits of the operator. There are many definitions for general fractional derivative. The most frequently used ones are the Grunwald-Letnikov definition, the Riemann-Liouville, and the Caputo definitions.As in [13], in this paper we use the Caputo definition for the fractional derivative.In the present paper, we focus on (5) and (6). The aim is to provide a new investigation of the Hopf bifurcation and chaos control on the general T system given by (5) and an analysis of the fractional general T-system as well.For system (5), we consider τ as the bifurcation parameter. When it passes through some certain critical values, the equilibrium will lose its stability and Hopf bifurcation will occur. We study the direction of the Hopf bifurcation, as well as the stability and period of the bifurcating periodic solutions. Moreover, with different values for k1 and τ, we realize the chaos control.The chaotic dynamics in the generalT-system with fractional derivative is taken into account. Some properties are given. Then, synchronization problem of (6) is provided.The paper is structured as follows. Section2 provides the stability of the steady states when there is no time delay. For system (5), Section 3 analyzes the local stability, the existence of the Hopf bifurcation, and the direction and the stability of the Hopf bifurcation. The analysis of system (6) is presented in Section 4. The chaos control of (6) is given in Section 5 and the synchronization in Section 6. Finally, conclusions are drawn in Section 7.


## 2. Existence of Steady States: Stability Analysis for System (4)

The equilibrium of system (4) can be obtained by solving the following algebraic system:(8)ay-x=0,b1x-b2xz=0,-cz+dxy+lx2+ky2=0.Then, we have the following.Proposition 1.
Consider the following:(i)
Ifb1<0, then system (4) has only one real steady state S0(0,0,0).(ii)
Ifb1>0, then system (4) has three real steady states: S0(0,0,0), S-(-x0,-y0,z0), S+(x0,y0,z0), where(9)x0=y0=b1cb2d+l+k,z0=b1b2.In order to analyze the local stability of the above steady states, the Jacobian matrix of (4) is given by(10)Jx,y,z=-aa0b1-b2z0-b2zdy+2lxdx+2ky-c.If b1<0, the characteristic equation of J(S0) is given by(11)λ+cλ2+aλ-ab1=0.The eigenvalues of (11) are λ1=-c, and λ2,3=(1/2)(-a±a2+4ab1).Forb1>0, both S- and S+ have the same characteristic equation; that is(12)λ3+a+cλ2+ac+b1cd+2kd+l+kλ+2ab1c=0.The eigenvalues of (12), which are dependent on parameters a,b1,c,d,l,k, can be obtained by the Cardano formula. Since a,c,d,l,k are all positive real parameters, one can ensure that (12) has at least one eigenvalue with negative real part as b1>0. The other two eigenvalues could be(a)
two negative real roots;(b)
two positive real roots;(c)
two complex-conjugate roots with negative real part;(d)
two complex-conjugate roots with positive real part.Therefore, analyzing the characteristic equation of (4) and using the Routh-Hurwitz theorem, we obtain the following propositions.Proposition 2.
Consider the following:(i)
Ifb1<0, then the steady state S0(0,0,0) of system (4) is locally asymptotically stable.(ii)
Ifb1>0, then the steady state S0(0,0,0) of system (4) is unstable.(iii)
If0<b1<bc and a(d+2l)-c(d+2k)>0, where(13)bc=aa+cd+l+kad+2l-cd+2k,then the steady states S- and S+ of system (4) are locally asymptotically stable (see Figures 3 and 4).(iv)
Ifb1>bc and a(d+2l)-c(d+2k)>0, then the steady states S- and S+ of system (4) are unstable (see Figures 5 and 6).Figure 3
The steady stateS- of system (4) is locally asymptotically stable when b1<bc.Figure 4
The steady stateS+ of system (4) is locally asymptotically stable when b1<bc.Figure 5
The steady stateS- of system (4) is unstable when b1>bc.Figure 6
The steady stateS+ of system (4) is unstable when b1>bc.Using the Hopf bifurcation theorem [9, 16], we have the following.Proposition 3.
Ifb1>0 and b1=bc, then for the steady state S+(orS-) of system (4), the corresponding characteristic equation has three eigenvalues: one negative and one pair of purely imaginary conjugate roots, satisfying Re⁡(dλ/db1)b1=bc≠0; that is, system (4) undergoes a Hopf bifurcation at the steady state S+(orS-) (see Figures 7 and 8).Figure 7
A Hopf bifurcation occurs at the steady stateS- of system (4) when b1=bc.Figure 8
A Hopf bifurcation occurs at the steady stateS+ of system (4) when b1=bc.


## 3. Local Stability and Hopf Bifurcation for System (5)

### 3.1. Local Stability and the Existence of Hopf Bifurcation for System (5)

Consider system (5). When τ=0, it becomes (4). Since the time delay does not change, the delayed feedback system (5) has the same equilibria as system (4).From Proposition1, under the assumption b1>0, system (5) also has three real steady states: S0(0,0,0), S-(-x0,-y0,z0), and S+(x0,y0,z0).Now, we analyze the effect of the time delay on the stability of these steady states.The linearization of system (5) at S0 is(14)u˙1t=au2t-u1t,u˙2t=b1u1t+k1u2t-u2t-τ,u˙3t=-cu3t.The characteristic equation of (14) is(15)λ+cλ2+a-k1λ-ak1-ab1+λ+ak1e-λτ=0.Equation (15) has a negative root λ=-c for all τ>0; thus, we need to analyse the following equation:(16)λ2+a-k1λ-ak1-ab1+λ+ak1e-λτ=0.If there is no delay, (16) becomes(17)λ2+aλ-ab1=0.Letiω(ω>0) be a root of (16). It follows that(18)ω2+ak1+ab1=ak1cos⁡ωτ+ωk1sin⁡ωτ,a-k1ω=ak1sin⁡ωτ-ωk1cos⁡ωτ,which can be written as(19)ω4+Aω2+B=0,where (20)A=2ab1+a2,B=2a2b12k1+b1.As in [9], we consider the following analysis:(1)
If the conditions(21)H1A>0,B>0hold, then (19) has no positive roots.(2)
If the condition(22)H2B<0holds, then (19) has a unique positive root ω+2. Using (19), we have(23)ω+=-A+A2-4B21/2.(3)
If the conditions(24)H3A<0,B>0,A2-4B>0hold, then (19) has two positive roots ω±2. Thus, we have(25)ω±=-A+A2-4B21/2.A stability switch may occur, through the roots±ω±i, where ω± are given by (25). Therefore, from (18), we have(26)τn±=1ω±arccos⁡k1ω±2+aak1+ab1k1a2+ω±2+2nπω±,where n=1,2,… at which (16) has a pair of purely imaginary roots ±ω±i.Consider(27)λτ=μτ+iωτthe root of (16) so that μ(τn±)=0. Using (16) and considering λ=λ(τ), (dλ(τ)/dτ)-1 is given by(28)dλτdτ-1=2λ+a-k1eλτλk1λ+ak1+k1λk1λ+ak1-τλ.From (18) and (25), we obtain(29)Re⁡dλτdτ-1τ=τn±=1k12ω±2+a2·±A2-4B.Thus, ifA2-4B>0, we have(30)Re⁡dλτdτ-1τ=τn+=1k12ω+2+a2A2-4B>0,Re⁡dλτdτ-1τ=τn-=-1k12ω-2+a2A2-4B<0.From the above findings we have the following.Theorem 4.
Letτn±(n=0,1,2,…) be defined by (26) and τ0=min⁡{τ0+,τ0-}.(i)
If(H1) holds, (16) and (17) have the same number of roots with positive real part for all τ>0.(ii)
If either(H2) or (H3) holds, when τ∈[0,τ0),  (16) and (17) have the same number of roots with positive real part. Moreover, if the transversality conditions (30) hold, then a Hopf bifurcation occurs at the steady state S0 in τ=τn± (see Figure 9).Figure 9
A Hopf bifurcation occurs at the steady stateS0 of system (5), when k1=-20.3, τ1=0.415.From the symmetry ofS+ and S-, it is sufficient to analyze the stability of S+. By the linear transformation(31)x1t=xt-x0,y1t=yt-y0,z1t=zt-z0,system (5) becomes(32)x˙1t=ay1t-x1t,y˙1t=-b2x0z1t-b2x1tz1t+k1y1t-y1t-τ,z˙1t=d+2lx0x1t+d+2ky0y1t-cz1t+dx1ty1t+lx1t2+ky1t2.The characteristic equation of system (32) in (0,0,0) is(33)λ3+a2λ2+a1λ+a0+c2λ2+c1λ+c0e-λτ=0,where(34)a2=a+c-k1,a1=ac-a+ck1+b1cd+2kd+l+k,a0=-ack1+2ab1c,c2=k1,c1=k1a+c,c0=k1ac.If there is no delay, (33) becomes(35)λ3+a2+c2λ2+a1+c1λ+a0+c0=0.Letiω(ω>0) be a root of (33). Then, ω satisfies the equation(36)ω6+a22-c22-2a1ω4+a12-2a0a2-c12+2c0c2ω2+a02-c02=0.Ifz=ω2, then (36) becomes(37)hz≔z3+pz2+qz+r=0,where(38)p=a22-c22-2a1,q=a12-2a0a2-c12+2c0c2,r=a02-c02.Therefore, applying the findings from [20] we obtain the following.Proposition 5.
For the polynomial equation (37), we have the following:(i)
Ifr<0, then (37) has at least one positive root.(ii)
Ifr≥0 and Δ=p2-3q≤0, then (37) has no positive roots.(iii)
Ifr≥0 and Δ>0, then (37) has positive roots if and only if z1∗=1/2(-p+Δ)>0 and h(z1∗)≤0.Suppose that (37) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by z1, z2, and z3. Then, (36) has three positive roots ωh=zh, h=1,2,3.By direct computation, we have the following.Theorem 6.
For the simple pairs of conjugate purely imaginary roots of (36), ±iωh, h=1,2,3, we have(39)τhj=1ωharccos⁡c1ωh2ωh2-a1-a2ωh2-a0c2ωh2-c0c2ωh2-c02+c12ωh2+2jπ.We have h=1,2,3, j=0,1,2,…, with(40)τ0≔τh00=min⁡τ10,τ20,τ30,ω0=ωh0.For (33), using Proposition 5 and [21] to (33), we have the following.Proposition 7 (see [10]).
For (33) we have the following:(i)
Ifr≥0 and Δ=p2-3q≤0, then all roots with positive real part of (33) have the same sum to those of the polynomial equation (35) for all τ≥0.(ii)
If eitherr<0 or r≥0, Δ=p2-3q>0, z1∗=(1/3)(-p+Δ)>0, and h(z1∗)≤0, then all roots with positive real parts of (33) have the same sum to those of the polynomial equation (35) for τ∈[0,τ0).Let us denote the root of (33) by λ(τ)=μ(τ)+iω(τ) with μ(τh(j))=0,ω(τh(j))=ωh.Proposition 8.
Ifzh=ωh2 and h′(zh)≠0, then the transversality condition(41)Re⁡dλτdττ=τhj≠0is satisfied and Re⁡(dλ(τ)/dτ)τ=τh(j) and h′(zh) have the same signs.Applying Propositions7 and 8 to (33), we have the following theorems.Theorem 9.
Letτh(j) and τ0 be defined by (39) and (40). Suppose that condition(42)ad+2l-cd+2k>0holds.(1)
If0<b1<bc, then we have the following:
(i)
Ifk1≤b1 and Δ≤0, then (33) has all the roots with negative real part for all τ≥0 and the steady state S+(orS-) of system (5) is locally asymptotically stable.(ii)
Ifk1>b1 or k1≤b1 and Δ>0, z1∗>0 and h(z1∗)≤0, then (33) has all the roots with negative real part for all τ∈[0,τ0) and the steady state S+(orS-) of system (5) is locally asymptotically stable.(iii)
If the conditions of (ii) hold andh′(zh)≠0, then a Hopf bifurcation occurs at the steady state S+(orS-) for τ=τh(j).(2)
Ifb1>bc, then we have the following:
(i)
Ifk1≤b1 and Δ≤0, then (33) has two roots with positive real part for all τ≥0 and steady state S+(orS-) of system (5) is unstable.(ii)
Ifk1>b1 or k1≤b1 and Δ>0, z1∗>0 and h(z1∗)≤0, then (33) has two roots with positive real part for all τ∈[0,τ0) and the steady state S+(orS-) of system (5) is unstable.(iii)
If the conditions of (ii) hold andh′(zh∗)≠0, then a Hopf bifurcation occurs at the steady state S+(orS-) for τ=τh(j).


### 3.2. Direction and Stability of the Hopf Bifurcation

In the previous section, for system (5), we have obtained conditions for the Hopf bifurcations to occur for a sequence of values of τ. Using the techniques from normal form theory and center manifold theory introduced by [22], we determine the direction, the stability, and the periodicity of the bifurcating solutions. Let (x∗,y∗,z∗) be the steady state of (5), where system (5) undergoes Hopf bifurcations at τ=τh and ±iωh are the corresponding pure imaginary roots of the characteristic equation.For convenience, letx1(t)=x(t)-x∗, x2(t)=y(t)-y∗, and x3(t)=z(t)-z∗, x¯i(t)=xi(tτ), τ=τh+μ and dropping the bars for simplification of notation, system (5) can be written as a FDE (functional differential equation) in C=C([-1,0],R3) as follows:(43)u˙t=Lμut+fμ,ut,where u(t)=(u1t,u2t,u3t)T∈R3 and Lμ:C→R, f:R×C→R are given, restrictively, by(44)Lμϕ=τh+μ·-aa00k1-b2x∗d+2ly∗d+2kx∗-cϕ10ϕ20ϕ30+τh+μ·0000-k100000-b2ϕ10ϕ30dϕ10ϕ20+lϕ102+kϕ202.From the previous section, ifμ=0, system (5) undergoes Hopf bifurcations at (x∗,y∗,z∗). By Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0] so that(45)Lμϕ=∫-10dηθ,0ϕθ,ϕ∈C-1,0.We can choose(46)ηθ,μ=τh+μ-aa00k1-b2x∗d+2ly∗d+2kx∗-cδθ-τh+μ0000-k10000δθ+1,where δ is the Dirac delta function.Forϕ∈C([-1,0],(R3)∗) we define(47)Aμϕ=dϕθdθ,θ∈-1,0∫-10dημ,sϕs,θ=0,Rμϕ=0,θ∈-1,0fμ,ϕθ,θ=0.Then, system (43) is equivalent to the abstract differential equation:(48)u˙t=Aμut+Rμut,where ut(θ)=u(t+θ), for θ∈[-1,0].ForΨ∈C1([0,1],R3), we define(49)A∗Ψs=-dΨsds,s∈0,1∫-10dηTt,θΨ-t,s=0and the bilinear form(50)Ψs,ϕθ=Ψ¯0ϕ0-∫θ=-10∫ɛ=0θΨ¯ɛ-θdηθϕɛdɛ,where η(θ)=η(θ,0).Then,A=A(0) and A∗=A∗(0) are adjoining operators. From Section 2, ±ωhτh are eigenvalues of A; thus, they are also eigenvalues of A∗.By direct computation, we obtain that(51)qθ=q0eiθωhτh,with(52)q0=1,α,βT=1,b2+iωhb2,b2x∗+y∗+iωhx∗b2c+iωhthe eigenvector of A corresponding to iωhτh and(53)q∗s=Dq0∗eisωhτhwith (54)q0∗=1,α∗,β∗T=1,-a-iωhc-iωhad+2ld+2kx∗y∗,a-iωhd+2ly∗the eigenvector of A∗ corresponding to -iωhτh, where(55)D=11+α¯α∗+β¯β∗-k1τhα¯α∗eiωhτh.Using the same notations as in [22], we compute the coordinates to describe the center manifold C0 at μ=0.Letut be the solution of (43) when μ=0 and define(56)zt=q∗,ut,Wt,θ=utθ-2Re⁡ztqθ.On the center manifoldC0 we have (57)Wt,θ=Wzt,z¯θ,θ,where (58)Wz,z¯,θ=W20θz22+W11θzz¯+W02θz¯22+W30θz36+⋯and z and z¯ are local coordinates for center manifold C0 in the direction of q∗ and q¯∗. Note that W is real if ut is real. We consider only real solutions. For the solution ut∈C0 of (43), since μ=0, we have(59)z˙t=q∗,u˙t=q∗,Aμut+Rμut=iτhωhz+q¯∗0f0,Wz,z¯,0+2Re⁡zq0=iτhωhz+q¯∗0f0z,z¯.We rewrite this equation as(60)z˙t=iτhωhzt+gzt,z¯t,where(61)gz,z¯=g20z22+g11zz¯+g01z¯22+g21z2z¯2+⋯.Note that(62)utθ=u1tθ,u2tθ,u3tθ=Wt,θ+zqθ+z¯q¯θand q(θ)=(1-α,β)Teiθωhτh.We have(63)u1t0=z+z¯+W2010z22+W1110zz¯+W0210z¯22+⋯,u2t0=αz+α¯z¯+W2020z22+W1120zz¯+W0220z¯22+⋯,u3t0=βz+β¯z¯+W2030z22+W1130zz¯+W0230z¯22+⋯.Using (61), we obtain(64)g20=2D¯τhdβ¯∗α-b2α¯∗β+β¯∗l+β¯∗kα2,g11=D¯τh-b2α∗β+β¯+β∗dα+α¯+2l+2kαα¯,g02=D¯τhdβ¯∗α¯-b2α¯∗β¯+β¯∗l+β∗kα¯2,g21=-b2D¯τhα¯∗2W2030+W2030+2βW1110+β¯W2010+D¯τhβ¯∗dαW1120+dW2020+2αdW1110+dα¯W2020+l2W1110+W2010+k2αW1120+α¯W0220.Since there areW20(0) and W11(0) in g21, we need to compute them.From (43) and (56), we have(65)W˙=x˙t-z˙q-z¯˙q¯=AW-2Re⁡q¯∗0f0qθ,θ∈-1,0AW-2Re⁡q¯∗0f0q0+f0,θ=0=AW+Hz,z¯,θ,where(66)Hz,z¯,θ=H20θz22+H11θzz¯+H02θz¯22+⋯.Expending the above series and comparing the corresponding coefficients, we obtain(67)A-2iτhωhW20θ=-H20θ,AW11θ=-H11θ.From (65), we know that, for θ∈[-1,0),(68)Hz,z¯,θ=-q¯∗0f0qθ-q∗0f¯0q¯0=-gqθ+q¯qθ.Comparing the coefficients with (66), we obtain(69)H20θ=-g20qθ-g¯02q¯θ,H11θ=-g11qθ-g¯11q¯θ.From (67) and (68) and the definition of A, it follows that(70)W˙20θ=2iτhωhW20θ+g20qθ+g¯02q¯θ.Notice thatq(θ)=(1,α,β)Teiθωhτh and(71)W20θ=ig20ωhτhq0eiθωhτh+ig¯023ωhτhq¯0e¯iθωhτh+E1e2iθωhτh,where E1=(E1(1),E1(2),E1(3))T∈R3 is a constant vector.In a similar way, we can obtain(72)W11θ=-ig11ωhτhq0eiθωhτh+ig¯11ωhτhq¯0e¯iθωhτh+E2,where E2=(E2(1),E2(2),E2(3))T∈R3 is a constant vector.From the definition ofA and (67), we obtain(73)∫-10dηθW20θ=2iωhτhW200-H200,(74)∫-10dηθW11θ=-H110,where η(θ)=η(θ,0).By (65), we have(75)H200=-g20q0-g¯02q¯0+2τh0-βα,(76)H110=-g11q0-g¯11q¯0+2τh0-Re⁡βRe⁡α.Substituting (71) and (75) into (73), we obtain(77)2iωhτhI-∫-10e2iθωhτhdηθE1=2τh0-βαwhich leads to(78)2iωh+a-a002iωh-k1+k1e-2ωhτhb2x∗-d+2ly∗-d+2kx∗2iωh+cE1=20-βα,E11=-2acβ+2iωhβ+b2αx∗Δ1,E12=-β2iωh+c+αb2x∗Δ1,E13=α2iωh+a2iωh-k1+k1e-2ωhτhΔ1-βad+2ly∗-βd+2kx∗2iωh+aΔ1,Δ1=det⁡2iωh+a-a002iωh-k1+k1e-2ωhτhb2x∗-d+2ly∗-d+2kx∗2iωh+c.In a similar way, substituting (72) and (76) into (74), we obtain (79)a-a000b2x∗-d+2ly∗-d+2kx∗cE2=20-Re⁡βRe⁡α,E21=E12=-2ab2x∗Re⁡α-2acRe⁡βΔ2,E23=-2aRe⁡βd+2kx∗+d+2ly∗Δ2,Δ2=ab2x∗d+2kx∗+d+2ly∗.Thus, we can determineW20(0) and W11(0) from (71) and (72). Furthermore, we can determine g21. Therefore, each gij in (64) is determined by the parameters and delay in (43). Thus, we can compute the following values:(80)C10=i2τhωhg11g20-2g112-g0223+g212,μ2=-Re⁡C10Re⁡λ′τh,β2=2Re⁡C10,T2=-Im⁡C10+μ2Im⁡λ′τhτhωh.The above quantities characterize the bifurcating periodic solutions in the center manifold at the critical valueτh [22, 23]:(i)
μ=0 is the Hopf bifurcation of system (43).(ii)
μ2 determines the direction of the Hopf bifurcation: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ>τh(τ<τh).(iii)
β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0).(iv)
T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2>0(T2<0).


## 3.1. Local Stability and the Existence of Hopf Bifurcation for System (5)

Consider system (5). When τ=0, it becomes (4). Since the time delay does not change, the delayed feedback system (5) has the same equilibria as system (4).From Proposition1, under the assumption b1>0, system (5) also has three real steady states: S0(0,0,0), S-(-x0,-y0,z0), and S+(x0,y0,z0).Now, we analyze the effect of the time delay on the stability of these steady states.The linearization of system (5) at S0 is(14)u˙1t=au2t-u1t,u˙2t=b1u1t+k1u2t-u2t-τ,u˙3t=-cu3t.The characteristic equation of (14) is(15)λ+cλ2+a-k1λ-ak1-ab1+λ+ak1e-λτ=0.Equation (15) has a negative root λ=-c for all τ>0; thus, we need to analyse the following equation:(16)λ2+a-k1λ-ak1-ab1+λ+ak1e-λτ=0.If there is no delay, (16) becomes(17)λ2+aλ-ab1=0.Letiω(ω>0) be a root of (16). It follows that(18)ω2+ak1+ab1=ak1cos⁡ωτ+ωk1sin⁡ωτ,a-k1ω=ak1sin⁡ωτ-ωk1cos⁡ωτ,which can be written as(19)ω4+Aω2+B=0,where (20)A=2ab1+a2,B=2a2b12k1+b1.As in [9], we consider the following analysis:(1)
If the conditions(21)H1A>0,B>0hold, then (19) has no positive roots.(2)
If the condition(22)H2B<0holds, then (19) has a unique positive root ω+2. Using (19), we have(23)ω+=-A+A2-4B21/2.(3)
If the conditions(24)H3A<0,B>0,A2-4B>0hold, then (19) has two positive roots ω±2. Thus, we have(25)ω±=-A+A2-4B21/2.A stability switch may occur, through the roots±ω±i, where ω± are given by (25). Therefore, from (18), we have(26)τn±=1ω±arccos⁡k1ω±2+aak1+ab1k1a2+ω±2+2nπω±,where n=1,2,… at which (16) has a pair of purely imaginary roots ±ω±i.Consider(27)λτ=μτ+iωτthe root of (16) so that μ(τn±)=0. Using (16) and considering λ=λ(τ), (dλ(τ)/dτ)-1 is given by(28)dλτdτ-1=2λ+a-k1eλτλk1λ+ak1+k1λk1λ+ak1-τλ.From (18) and (25), we obtain(29)Re⁡dλτdτ-1τ=τn±=1k12ω±2+a2·±A2-4B.Thus, ifA2-4B>0, we have(30)Re⁡dλτdτ-1τ=τn+=1k12ω+2+a2A2-4B>0,Re⁡dλτdτ-1τ=τn-=-1k12ω-2+a2A2-4B<0.From the above findings we have the following.Theorem 4.
Letτn±(n=0,1,2,…) be defined by (26) and τ0=min⁡{τ0+,τ0-}.(i)
If(H1) holds, (16) and (17) have the same number of roots with positive real part for all τ>0.(ii)
If either(H2) or (H3) holds, when τ∈[0,τ0),  (16) and (17) have the same number of roots with positive real part. Moreover, if the transversality conditions (30) hold, then a Hopf bifurcation occurs at the steady state S0 in τ=τn± (see Figure 9).Figure 9
A Hopf bifurcation occurs at the steady stateS0 of system (5), when k1=-20.3, τ1=0.415.From the symmetry ofS+ and S-, it is sufficient to analyze the stability of S+. By the linear transformation(31)x1t=xt-x0,y1t=yt-y0,z1t=zt-z0,system (5) becomes(32)x˙1t=ay1t-x1t,y˙1t=-b2x0z1t-b2x1tz1t+k1y1t-y1t-τ,z˙1t=d+2lx0x1t+d+2ky0y1t-cz1t+dx1ty1t+lx1t2+ky1t2.The characteristic equation of system (32) in (0,0,0) is(33)λ3+a2λ2+a1λ+a0+c2λ2+c1λ+c0e-λτ=0,where(34)a2=a+c-k1,a1=ac-a+ck1+b1cd+2kd+l+k,a0=-ack1+2ab1c,c2=k1,c1=k1a+c,c0=k1ac.If there is no delay, (33) becomes(35)λ3+a2+c2λ2+a1+c1λ+a0+c0=0.Letiω(ω>0) be a root of (33). Then, ω satisfies the equation(36)ω6+a22-c22-2a1ω4+a12-2a0a2-c12+2c0c2ω2+a02-c02=0.Ifz=ω2, then (36) becomes(37)hz≔z3+pz2+qz+r=0,where(38)p=a22-c22-2a1,q=a12-2a0a2-c12+2c0c2,r=a02-c02.Therefore, applying the findings from [20] we obtain the following.Proposition 5.
For the polynomial equation (37), we have the following:(i)
Ifr<0, then (37) has at least one positive root.(ii)
Ifr≥0 and Δ=p2-3q≤0, then (37) has no positive roots.(iii)
Ifr≥0 and Δ>0, then (37) has positive roots if and only if z1∗=1/2(-p+Δ)>0 and h(z1∗)≤0.Suppose that (37) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by z1, z2, and z3. Then, (36) has three positive roots ωh=zh, h=1,2,3.By direct computation, we have the following.Theorem 6.
For the simple pairs of conjugate purely imaginary roots of (36), ±iωh, h=1,2,3, we have(39)τhj=1ωharccos⁡c1ωh2ωh2-a1-a2ωh2-a0c2ωh2-c0c2ωh2-c02+c12ωh2+2jπ.We have h=1,2,3, j=0,1,2,…, with(40)τ0≔τh00=min⁡τ10,τ20,τ30,ω0=ωh0.For (33), using Proposition 5 and [21] to (33), we have the following.Proposition 7 (see [10]).
For (33) we have the following:(i)
Ifr≥0 and Δ=p2-3q≤0, then all roots with positive real part of (33) have the same sum to those of the polynomial equation (35) for all τ≥0.(ii)
If eitherr<0 or r≥0, Δ=p2-3q>0, z1∗=(1/3)(-p+Δ)>0, and h(z1∗)≤0, then all roots with positive real parts of (33) have the same sum to those of the polynomial equation (35) for τ∈[0,τ0).Let us denote the root of (33) by λ(τ)=μ(τ)+iω(τ) with μ(τh(j))=0,ω(τh(j))=ωh.Proposition 8.
Ifzh=ωh2 and h′(zh)≠0, then the transversality condition(41)Re⁡dλτdττ=τhj≠0is satisfied and Re⁡(dλ(τ)/dτ)τ=τh(j) and h′(zh) have the same signs.Applying Propositions7 and 8 to (33), we have the following theorems.Theorem 9.
Letτh(j) and τ0 be defined by (39) and (40). Suppose that condition(42)ad+2l-cd+2k>0holds.(1)
If0<b1<bc, then we have the following:
(i)
Ifk1≤b1 and Δ≤0, then (33) has all the roots with negative real part for all τ≥0 and the steady state S+(orS-) of system (5) is locally asymptotically stable.(ii)
Ifk1>b1 or k1≤b1 and Δ>0, z1∗>0 and h(z1∗)≤0, then (33) has all the roots with negative real part for all τ∈[0,τ0) and the steady state S+(orS-) of system (5) is locally asymptotically stable.(iii)
If the conditions of (ii) hold andh′(zh)≠0, then a Hopf bifurcation occurs at the steady state S+(orS-) for τ=τh(j).(2)
Ifb1>bc, then we have the following:
(i)
Ifk1≤b1 and Δ≤0, then (33) has two roots with positive real part for all τ≥0 and steady state S+(orS-) of system (5) is unstable.(ii)
Ifk1>b1 or k1≤b1 and Δ>0, z1∗>0 and h(z1∗)≤0, then (33) has two roots with positive real part for all τ∈[0,τ0) and the steady state S+(orS-) of system (5) is unstable.(iii)
If the conditions of (ii) hold andh′(zh∗)≠0, then a Hopf bifurcation occurs at the steady state S+(orS-) for τ=τh(j).


## 3.2. Direction and Stability of the Hopf Bifurcation

In the previous section, for system (5), we have obtained conditions for the Hopf bifurcations to occur for a sequence of values of τ. Using the techniques from normal form theory and center manifold theory introduced by [22], we determine the direction, the stability, and the periodicity of the bifurcating solutions. Let (x∗,y∗,z∗) be the steady state of (5), where system (5) undergoes Hopf bifurcations at τ=τh and ±iωh are the corresponding pure imaginary roots of the characteristic equation.For convenience, letx1(t)=x(t)-x∗, x2(t)=y(t)-y∗, and x3(t)=z(t)-z∗, x¯i(t)=xi(tτ), τ=τh+μ and dropping the bars for simplification of notation, system (5) can be written as a FDE (functional differential equation) in C=C([-1,0],R3) as follows:(43)u˙t=Lμut+fμ,ut,where u(t)=(u1t,u2t,u3t)T∈R3 and Lμ:C→R, f:R×C→R are given, restrictively, by(44)Lμϕ=τh+μ·-aa00k1-b2x∗d+2ly∗d+2kx∗-cϕ10ϕ20ϕ30+τh+μ·0000-k100000-b2ϕ10ϕ30dϕ10ϕ20+lϕ102+kϕ202.From the previous section, ifμ=0, system (5) undergoes Hopf bifurcations at (x∗,y∗,z∗). By Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0] so that(45)Lμϕ=∫-10dηθ,0ϕθ,ϕ∈C-1,0.We can choose(46)ηθ,μ=τh+μ-aa00k1-b2x∗d+2ly∗d+2kx∗-cδθ-τh+μ0000-k10000δθ+1,where δ is the Dirac delta function.Forϕ∈C([-1,0],(R3)∗) we define(47)Aμϕ=dϕθdθ,θ∈-1,0∫-10dημ,sϕs,θ=0,Rμϕ=0,θ∈-1,0fμ,ϕθ,θ=0.Then, system (43) is equivalent to the abstract differential equation:(48)u˙t=Aμut+Rμut,where ut(θ)=u(t+θ), for θ∈[-1,0].ForΨ∈C1([0,1],R3), we define(49)A∗Ψs=-dΨsds,s∈0,1∫-10dηTt,θΨ-t,s=0and the bilinear form(50)Ψs,ϕθ=Ψ¯0ϕ0-∫θ=-10∫ɛ=0θΨ¯ɛ-θdηθϕɛdɛ,where η(θ)=η(θ,0).Then,A=A(0) and A∗=A∗(0) are adjoining operators. From Section 2, ±ωhτh are eigenvalues of A; thus, they are also eigenvalues of A∗.By direct computation, we obtain that(51)qθ=q0eiθωhτh,with(52)q0=1,α,βT=1,b2+iωhb2,b2x∗+y∗+iωhx∗b2c+iωhthe eigenvector of A corresponding to iωhτh and(53)q∗s=Dq0∗eisωhτhwith (54)q0∗=1,α∗,β∗T=1,-a-iωhc-iωhad+2ld+2kx∗y∗,a-iωhd+2ly∗the eigenvector of A∗ corresponding to -iωhτh, where(55)D=11+α¯α∗+β¯β∗-k1τhα¯α∗eiωhτh.Using the same notations as in [22], we compute the coordinates to describe the center manifold C0 at μ=0.Letut be the solution of (43) when μ=0 and define(56)zt=q∗,ut,Wt,θ=utθ-2Re⁡ztqθ.On the center manifoldC0 we have (57)Wt,θ=Wzt,z¯θ,θ,where (58)Wz,z¯,θ=W20θz22+W11θzz¯+W02θz¯22+W30θz36+⋯and z and z¯ are local coordinates for center manifold C0 in the direction of q∗ and q¯∗. Note that W is real if ut is real. We consider only real solutions. For the solution ut∈C0 of (43), since μ=0, we have(59)z˙t=q∗,u˙t=q∗,Aμut+Rμut=iτhωhz+q¯∗0f0,Wz,z¯,0+2Re⁡zq0=iτhωhz+q¯∗0f0z,z¯.We rewrite this equation as(60)z˙t=iτhωhzt+gzt,z¯t,where(61)gz,z¯=g20z22+g11zz¯+g01z¯22+g21z2z¯2+⋯.Note that(62)utθ=u1tθ,u2tθ,u3tθ=Wt,θ+zqθ+z¯q¯θand q(θ)=(1-α,β)Teiθωhτh.We have(63)u1t0=z+z¯+W2010z22+W1110zz¯+W0210z¯22+⋯,u2t0=αz+α¯z¯+W2020z22+W1120zz¯+W0220z¯22+⋯,u3t0=βz+β¯z¯+W2030z22+W1130zz¯+W0230z¯22+⋯.Using (61), we obtain(64)g20=2D¯τhdβ¯∗α-b2α¯∗β+β¯∗l+β¯∗kα2,g11=D¯τh-b2α∗β+β¯+β∗dα+α¯+2l+2kαα¯,g02=D¯τhdβ¯∗α¯-b2α¯∗β¯+β¯∗l+β∗kα¯2,g21=-b2D¯τhα¯∗2W2030+W2030+2βW1110+β¯W2010+D¯τhβ¯∗dαW1120+dW2020+2αdW1110+dα¯W2020+l2W1110+W2010+k2αW1120+α¯W0220.Since there areW20(0) and W11(0) in g21, we need to compute them.From (43) and (56), we have(65)W˙=x˙t-z˙q-z¯˙q¯=AW-2Re⁡q¯∗0f0qθ,θ∈-1,0AW-2Re⁡q¯∗0f0q0+f0,θ=0=AW+Hz,z¯,θ,where(66)Hz,z¯,θ=H20θz22+H11θzz¯+H02θz¯22+⋯.Expending the above series and comparing the corresponding coefficients, we obtain(67)A-2iτhωhW20θ=-H20θ,AW11θ=-H11θ.From (65), we know that, for θ∈[-1,0),(68)Hz,z¯,θ=-q¯∗0f0qθ-q∗0f¯0q¯0=-gqθ+q¯qθ.Comparing the coefficients with (66), we obtain(69)H20θ=-g20qθ-g¯02q¯θ,H11θ=-g11qθ-g¯11q¯θ.From (67) and (68) and the definition of A, it follows that(70)W˙20θ=2iτhωhW20θ+g20qθ+g¯02q¯θ.Notice thatq(θ)=(1,α,β)Teiθωhτh and(71)W20θ=ig20ωhτhq0eiθωhτh+ig¯023ωhτhq¯0e¯iθωhτh+E1e2iθωhτh,where E1=(E1(1),E1(2),E1(3))T∈R3 is a constant vector.In a similar way, we can obtain(72)W11θ=-ig11ωhτhq0eiθωhτh+ig¯11ωhτhq¯0e¯iθωhτh+E2,where E2=(E2(1),E2(2),E2(3))T∈R3 is a constant vector.From the definition ofA and (67), we obtain(73)∫-10dηθW20θ=2iωhτhW200-H200,(74)∫-10dηθW11θ=-H110,where η(θ)=η(θ,0).By (65), we have(75)H200=-g20q0-g¯02q¯0+2τh0-βα,(76)H110=-g11q0-g¯11q¯0+2τh0-Re⁡βRe⁡α.Substituting (71) and (75) into (73), we obtain(77)2iωhτhI-∫-10e2iθωhτhdηθE1=2τh0-βαwhich leads to(78)2iωh+a-a002iωh-k1+k1e-2ωhτhb2x∗-d+2ly∗-d+2kx∗2iωh+cE1=20-βα,E11=-2acβ+2iωhβ+b2αx∗Δ1,E12=-β2iωh+c+αb2x∗Δ1,E13=α2iωh+a2iωh-k1+k1e-2ωhτhΔ1-βad+2ly∗-βd+2kx∗2iωh+aΔ1,Δ1=det⁡2iωh+a-a002iωh-k1+k1e-2ωhτhb2x∗-d+2ly∗-d+2kx∗2iωh+c.In a similar way, substituting (72) and (76) into (74), we obtain (79)a-a000b2x∗-d+2ly∗-d+2kx∗cE2=20-Re⁡βRe⁡α,E21=E12=-2ab2x∗Re⁡α-2acRe⁡βΔ2,E23=-2aRe⁡βd+2kx∗+d+2ly∗Δ2,Δ2=ab2x∗d+2kx∗+d+2ly∗.Thus, we can determineW20(0) and W11(0) from (71) and (72). Furthermore, we can determine g21. Therefore, each gij in (64) is determined by the parameters and delay in (43). Thus, we can compute the following values:(80)C10=i2τhωhg11g20-2g112-g0223+g212,μ2=-Re⁡C10Re⁡λ′τh,β2=2Re⁡C10,T2=-Im⁡C10+μ2Im⁡λ′τhτhωh.The above quantities characterize the bifurcating periodic solutions in the center manifold at the critical valueτh [22, 23]:(i)
μ=0 is the Hopf bifurcation of system (43).(ii)
μ2 determines the direction of the Hopf bifurcation: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ>τh(τ<τh).(iii)
β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0).(iv)
T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2>0(T2<0).


## 4. Analysis of System (6)

Letq1=q2=q3=q∈(0,1). In this case, the fractional order system is commesurate-order [24].Proposition 10.
The initial value problem of the commensurate order system (6) can be rewritten as follows:(81)DqXt=AXt+xtBXt+ytCXt,X0=x0,y0,z0T,where 0<t≤T, X(t)=(x(t),y(t),z(t))∈R3,(82)A=-aa0b10000-c,B=00000-b2ld0,C=0000000k0.Then, for constant T=min⁡{T∗,ϵΓ(q+1)/f∞1/q}, T∗>0, fractional order system (6) has a unique solution, where Γ(·) is the Gamma function.Proof.
Consider the functionf(X(t))=AX(t)+x(t)BX(t)+y(t)CX(t), which is continuous and bounded on the interval X∈[0,T∗]×[X0-ϵ,X0+ϵ], for any T∗,ϵ>0. For X(t)=(xt,yt,zt)T and Y(t)=(x1t,y1t,z1t)T, where Y(t)∈[0,T∗]×[X0-ϵ,X0+ϵ], we can obtain(83)fXt-fYt≤LXt-Yt,where L=A+(B+C)(2X0+2ϵ)>0.
Therefore, the fractional-order generalT system meets the Lipschitz condition. Then, according to the existence and uniqueness theorem of the fractional-order system [25], the initial value problem of the commensurate order system (81) has a unique solution in the interval T=min⁡{T∗,ϵΓ(q+1)/f∞1/q}.
System (6) is dissipative for a+c>0, because(84)∇Vx,y,z=-a-c,where V(x,y,z)=(1/2)(x2+y2+z2). Meanwhile, it is convergent in an exponential rate (85)dVXtdt=e-a+c.
That is to say, an initial volumeV0 is the volume element at time t contraction of volume element V0e-(a+c)t. When t→∞ all the trajectory of systems will eventually be restricted in a volume element for zero point sets, and incremental dynamic behavior of it will be fixed in an attractor.In order to determine the stability conditions for the steady statesS- and S+, we first consider the integer-order case. Based on Proposition 2, S- and S+ are locally asymptotically stable if and only if 0<b1<bc, where(86)bc=aa+cd+l+kad+2l-cd+2kand a(d+2l)-c(d+2k)>0. It is known that the fractional-order system is at least as stable as their integer-order counterpart, so we have the following conclusion.Proposition 11.
The steady statesS- and S+ are stable with q∈(0,1) for b1<bc.Whenb1>bc, the steady states S- and S+ become unstable in an integer-order system but may be stable in the fractional-order case.


## 5. Chaos Control of System (6)

In this section, we want to control the chaos for the fractional-order generalT system (6) to the steady state denoted by S via feedback control.Ann-dimensional fractional-order system can be described as [13](87)DqXt=fXt,where X(t)∈Rn, q∈(0,1).The system with controller is given by [13](88)DqXt=fXt-KXt-S,where K=diag⁡(k1,k2,…,kn) is the matrix of positive feedback gains and S is the steady state of system (87).The controlled fractional-order generalT system (6) is(89)Dq1xt=ayt-xt-u1t,Dq2yt=b1xt-b2xtzt-u2t,Dq3zt=-czt+dxtyt+lxt2+kyt2-u3t,where ui(t), i=1,2,3, are the external control inputs. The control law of single state variables feedback has the following form [13]:(90)u1t=k1xt-x01,u2t=k2yt-y02,u3t=k3zt-z03,where K=diag⁡(k1,k2,k3) is the matrix of positive feedback gains and S=(x01,y02,z03)T is the steady state of system (6).The characteristic equation of the controlled system (89) evaluated at the steady state S is(91)λ+a+k1λ+k2λ+c+k3=0.The fractional-order Routh-Hurwitz conditions lead to(92)a+k1>0,k2>0,c+k3>0.


## 6. Synchronization of System (6)

The nonlinear control method is used to design control in order to make the drive system (6) and response system state synchronization [14]. Two identical systems are introduced, one is the drive system and the other system added a nonlinear control to be the response system. Corresponding to (6), the drive system is(93)Dq1x1t=ay1t-x1t,Dq2y1t=b1x1t-b2x1tz1t,Dq3z1t=-cz1t+dx1ty1t+lx1t2+ky1t2,and the response system is(94)Dq1x2t=ay2t-x2t+v1t,Dq2y2t=b1x2t-b2x2tz2t+v2t,Dq3z2t=-cz2t+dx2ty2t+lx2t2+ky2t2+v3t,where V(t)=(v1(t),v2(t),v3(t))T is the nonlinear synchronization controller. The drive system and the response system achieve synchronization under the driver of V(t). From (93) and (94), the error system is obtained:(95)Dq1e1t=ae2t-e1t+v1t,Dq2e2t=b1e1t-b2x2te3-b2x1te1+v2t,Dq3e3t=-ce3t+dx2te2t+dy1te1t+lx1t+x2te1t+ky1t+y2te2t+v3t,where e1(t)=x2(t)-x1(t),e2(t)=y2(t)-y1(t),e3=z2(t)-z1(t). We have to find the proper control function vi(t), i=1,2,3, so that the response system (94) globally synchronizes with drive system (93); that is, limt→+∞⁡e(t)=0, where e(t)=(e1(t),e2(t),e3(t))T. According to [14], we propose the following control law for system (94):(96)v1t=ae1t-k1e1t,v2t=-b1e1t-ae1t+b2z1te1t+b2x2te3t-k2e2t,v3t=ce3t-dy1te1t-le1tx1t+x2t-dx2te2t-ky2t+y1te2t,where k1, k2, and k3 are the control parameters.Proposition 12 (see [14]).
For any initial conditions, ifki>0, i=1,2,3 then the drive system and response system will synchronize.


## 7. Numerical Simulations

Now, we illustrate the findings from the previous sections. We have proved that at some critical values of the delay, a family of periodic solutions bifurcate from the steady states of system (5) and the stability of the steady states may be changed.The numerical simulations indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.We consider the delayed feedback control system (5) in the following particular form:(97)x˙t=10yt-xt,y˙t=40xt-xtzt+k1yt-yt-τ,z˙t=-2.5zt+3xtyt+0.9xt2+2yt2,which has three steady states S0=(0,0,0), S-=(-4.11,-4.11,40), and S+=(4.11,4.11,40). When τ=0 and k1=0, system (97) is chaotic.The characteristic equation of system (97) is given by(98)λ3+12.5-k1λ2+143.644-12.5k1λ+2000-10k1+k1λ2+12.5λ+25e-λτ=0.If there is no delay, (98) has a negative root and a pair of complex roots with positive real part.Using Theorem9, we have the following:(i)
For5.0413<k1<28.403, r=105(40-k1)>0, Δ=1.052·105-3101.694k1<0, (98) has two roots with positive real part for all τ≥0, and the steady state S+ (or S-) of system (97) is unstable.(ii)
Fork1<5.0413 chaos can occur.In what follows, we considerk1=-3. In this case: ω1=11.58361053, ω2=13.32108470,(99)τ1j=0.473158+2jω1,τ2j=0.51716+2jω2,j=0,1,2,….Then,τ10<τ20<τ21<τ11.(i)
Forτ∈[0,τ20)∪(τ21,∞), the steady states S+ and S- of system (97) are unstable.(ii)
Forτ∈[τ20,τ21), the steady states S+ and S- of system (97) are locally asymptotically stable.(iii)
Forτ=τhj, for h=1,2, j=0,1,2,…, system (97) undergos a Hopf bifurcation at the steady states S+ and S-.For values ofτ, which satisfy the above conditions, we obtain the dynamical behaviors in Figures 10, 11, 12, and 13.Figure 10
For system (97), there is chaos when k1=-3, τ=0.01.Figure 11
For system (97), there is chaos when k1=-3, τ=0.051.Figure 12
For system (97), an unstable periodic solution bifurcates from the steady state S+ when k1=-3, τ=1.5.Figure 13
For system (97), an unstable periodic solution bifurcates from the steady state S- when k1=-3, τ=1.5.For the numerical simulation of the fractional differential equations (FDE) (6), with a=10, b2=1, c=2.5, d=1, l=2, k=2, and q1=q2=q3=0.99, we use the method from [26, 27]. We obtain Figures 14, 15, and 16.Figure 14
Asymptotic stable solution for the fractional generalT system (6) when b1=10.Figure 15
Periodic solution for the fractional generalT system (6) when b1=16.6.Figure 16
Chaos occurs for the fractional generalT system (6) when b1=40.For the controlled fractional order generalT-system (89), using K=diag⁡(5,5,1) the matrix of positive feedback gains, we obtain Figure 17.Figure 17
Asymptotic stable solution for the controlled fractional generalT system (89), when b1=10.


## 8. Conclusion

In the present paper we introduce a generalizedT-system where the time delay is present. The linear stability is analyzed by using the Routh Hurwitz criterion. The existence of the Hopf bifurcation is studied. Then, the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Chaotic behavior is also taken into account. The numerical simulations show that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.Furthermore, the fractional-order generalT system has been proposed. The dynamics, chaos control as well as synchronization have been investigated.The present study will be continued for the system which describes the financial risk [28]. Also, as in [29], the fractional-order chaotic complex system will be taken into consideration.



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