Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback

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1 International Journal of Automation and Computing 12(2, April 215, DOI: 1.17/s z Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback Chang-Jin Xu 1 Yu-Sen Wu 2 1 Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 554, China 2 School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 47123, China Abstract: In this paper, the problem of controlling chaos in a Sprott E system with distributed delay feedback is considered. By analyzing the associated characteristic transcendental equation, we focus on the local stability and Hopf bifurcation nature of the Sprott E system with distributed delay feedback. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form theory and center manifold theory. Numerical simulations for justifying the theoretical analysis are provided. Keywords: Sprott E system, chaos control, stability, Hopf bifurcation, distributed delay. 1 Introduction Since the pioneering work of Ott et al. [1], the topics of chaos and chaotic control are growing rapidly in many different fields such as ecological system, chemical systems and biological systems, and so forth [2]. We all know that chaotic systems have very complicated dynamical nature, which plays an important role in many fields such as secure communications, information processing, high-performance circuit design for telecommunications [11]. Chaos, which often causes irregular behaviors in practical system, is usually undesirable. In many cases, we wish to avoid and eliminate such behaviors. Recently, many schemes such as Ott et al. [1], feedback and non-feedback control [127], observerbased control [12], active control [13], adaptive control [18, 19], inverse optimal control [2], evolutionary algorithm [21],etc., have been presented to implement the chaos control. For more related work, one can see [22 26]. In 212, Wang and Chen [27] reported the very surprising finding of the following new 3D autonomous chaotic Sprott E system with one stable node or stable focus When a =, it is the Sprott E system [28]. When a =, the stability of the single equilibrium is fundamentally different [29]. Let yz + a =,x 2 y =, 1 4x =, we can obtain that system (1 has only one stable equilibrium E(x,y,z =( 1, 1, 16a ifa>. Interestingly, 4 16 Wang and Chen [29] found that system (1 can generate chaotic phenomenon which is shown in Fig. 1 (Fig. 1 shows the waveform portraits and the phase portraits of system (1. ẋ = y(tz(t+a ẏ = x 2 (t y(t ż =1 4x(t. (1 Regular paper Manuscript received November 2, 213; accepted March 19, 214. This work is supported by National Natural Science Foundation of China (Nos and , Soft Science and Technology Program of Guizhou Province (No. 211LKC23, Natural Science and Technology Foundation of Guizhou Province (No. J[212]21, Governor Foundation of Guizhou Province (No. [212]53 and Natural Science and Technology Foundation of Guizhou Province (214, and Natural Science Innovation Team Project of Guizhou Province (No. [213]14. Recommended by Associate Editor Mohammed Chadli c Institute of Automation, Chinese Academy of Science and Springer-Verlag Berlin Heidelberg 215

2 C. J. Xu and Y. S. Wu / Chaos Control and Bifurcation Behavior for a Sprott E system with 183 Fig. 1 Chaotic attractor of system (1 with a =.6. The initial value is (1.5,1.5,1.5 In this paper, we study the stability, and the local Hopf bifurcation nature for system (1. To the best of our knowledge, it is the first try to introduce continuous distributed time-delayed feedback force to control the chaos of the Sprott E system. The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the equilibrium

3 184 International Journal of Automation and Computing 12(2, April 215 and the occurrence of local Hopf bifurcations of the Sprott E system with distributed delay feedback. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to show the validity of chaotic control. Some main conclusions are drawn in Section 5. 2 Stability and bifurcation analysis In this section, we shall study the stability of the equilibrium and the existence of local Hopf bifurcations. In order to apply feedback control, we add a continuous distributed time delayed force b [z(t z(t + s]k( sds to the third equation of system (1, then system (1 takes the form ẋ = y(tz(t+a ẏ = x 2 (t y(t ż =1 4x(t+b [z(t z(t + s]k( sds (2 a, b >, k(sds =1, sk(sds <+. Obviously, system (2 has the equilibrium point E(x,y,z = ( 1, 1, 6a Let x(t =x(t x, ȳ(t =y(t y, z(t =z(t z and still denote x(t, ȳ(t and z(t byx(t,y(t andz(t, respectively, then (2 becomes ẋ = z y(t+y z(t+y(tz(t ẏ =2x x(t y(t+x 2 (t ż = 4x(t+bz(t b z(t + sk( sds. (3 The linearization of (3 near E(x,y,z isgivenby ẋ = z y(t+y z(t ẏ =2x x(t y(t ż = 4x(t+bz(t b (4 z(t + sk( sds whose characteristic equation appears as ( λ(λ +1 λ b + b k( se λs ds +4y (λ +1 2x z (λ b + b k( se λs ds =. (5 In this paper, we consider the weak kernel case, i.e., k(s = αe αs, α>. As to the general gamma kernel case, we can make a similar analysis. We give the initial condition of system (4 as x(s y(s z(s = φ 1(s φ 2(s φ 3(s, <s. The characteristic equation (5 with the weak kernel case takes the form λ 4 + θ 1(αλ 3 + θ 2(αλ 2 + θ 3(αλ + θ 4(α = (6 θ 1(α =α b +1 θ 2(α =α b 2x z +4y θ 3(α =4y (1 + α 2x z (α b θ 4(α =4y α. In view of the well known Routh-Hurwitz criterion, we can conclude that all the roots of (6 has negative real parts if the following conditions D 1(α =θ 1(α =α b +1> D 2(α =θ 1(αθ 2(α θ 3(α = (α b +1(α b 2x z +4y [4y (1 + α 2x z (α b] > D 3(α =θ 3(αD 2(α θ1(αθ 2 4(α = [4y (1 + α 2x z (α b] {(α b +1(α b 2x z +4y [4y (1 + α 2x z (α b]} 4y α(α b +1 2 > D 4(α =θ 4(αD 3(α =4y αd 3(α > hold true. Based on the analysis above, we can easily obtain the following Theorem 1. Theorem 1. The equilibrium E(x,y,z ofsystem(2 with the weak kernel is locally asymptotically stable if the following conditions are fulfilled: α b +1> (α b +1(α b 2x z +4y (7 (8 [4y (1 + α 2x z (α b] > [4y (1 + α 2x z (α b] {(α b +1(α b 2x z +4y [4y (1 + α 2x z (α b]} 4y α(α b +1 2 >. Let λ i (i =1, 2, 3, 4 be the roots of (6, then we have λ 1 + λ 2 + λ 3 + λ 4 = θ 1(α λ 1λ 2 + λ 1λ 3 + λ 1λ 4 + λ 2λ 3 + λ 2λ 4 + λ 3λ 4 = θ 2(α λ 1λ 2λ 3 + λ 1λ 2λ 4 + λ 1λ 3λ 4 + λ 2λ 3λ 4 = θ 3(α λ 1λ 2λ 3λ 4 = θ 4(α. (9 If there exists an α R + such that D 3(α = and dd 3 (α dα α=α, then by the Routh-Hurwitz criterion, there exists a pair of purely imaginary roots, say λ 1 = λ 2 =iω (ω =, and the other two roots λ 3,λ 4 satisfy: if λ 3,λ 4 are real, then λ 3 <,λ 4 < ; if λ 3,λ 4 are complex conjugate, then Re{λ 3} =Re{λ 4} = θ 1(α 2. It is easy to calculate that d(re(λ 1 dα θ 1(α = 2[θ1(αθ 3 3(α+(θ 1(αθ 2(α 2θ 3(α 2 ] dd 3(α (1 dα α=α thus the Hopf bifurcation occurs near E(x,y,z whenα passes through α.

4 C. J. Xu and Y. S. Wu / Chaos Control and Bifurcation Behavior for a Sprott E system with Direction and stability of bifurcating periodic solutions In this section, by using techniques from normal form and center manifold theory [3], we shall investigate the direction, stability, and period of these periodic solutions bifurcating from the equilibrium E(x,y,z. Let μ = α α, then system (3 undergoes the Hopf bifurcation at the equilibrium E(x,y,z forμ =,andthen±iω are purely imaginary roots of the characteristic equation at the equilibrium E(x,y,z. System (3 can be written as an functional differential equation (FDE in C = C([, ], R 3 as u(t =A(μu t + R(μu t (11 u(t =[x(t,y(t,z(t] T C and u t(θ =u(t + θ = [x(t + θ,y(t + θ,z(t + θ] T C, anda and R are given by and A(μφ(θ = dφ(θ, θ< dθ Lφ(θ+ (12 F (sφ(sds, θ = { R(μφ(θ = [,, ] T, θ< [f 1,f 2, ] T, θ = (13 respectively, φ(θ =[φ 1(θ,φ 2(θ,φ 3(θ] T C and f 1 = φ 2(φ 3(,f 2 = φ 2 1(. For ψ C ([, + ], (R 3, define A dψ(s, s (, + ] ψ(s = ds L T ψ( + F T (tψ( tds, s =. For φ C([, ], R 3 andψ C([, + ], (R 3, define the bilinear form ψ, φ = ψ(φ( θ ξ= ψ(ξ θdη(θφ(ξdξ η(θ = η(θ,, the A = A( and A are adjoint operators. By the discussions in Section 2, we know that ±iω are eigenvalues of A(, and they are also eigenvalues of A corresponding to iω and iω, respectively. Assume that q(θ =(1,a 1,a 2 T e iωθ is the eigenvector of A( corresponding to iω,thenwehavea(q( = iω q(, i.e., z y 1 Lq(+ F (sq(sds = 2x a b a 2 1 a 1 e iωs ds = bk( s a 2 a 1z + a 2y iω 2x a 1 = iω a 1 4+ba 2 ba 2χ (1 iω a 2 χ (1 = k( se iω s ds = α α +iω. We can obtain a 1 = q(θ =[1,a 1,a 2] T e iω θ 2x iω +1, a2 = 4. b bχ (1 iω Assume that q (s =D[1,a 1,a 2] T e iω s ( s<+ is the eigenvector of A ( corresponding to iω, then we have A (q ( = iω q (, i.e., L T q ( + F (sq ( sds = 2x 4 D z Da 1 + y b Da 2 D Da 1 bk( s Da 2 2x a 1 4a 2 z a 1 y + ba 2 bχ (2 χ (2 = We can obtain If we choose a 1 = = k( se iω s ds = iω D iω a 1D iω a 2D q (s =D[1,a 1,a 2] T e iω s e iω s ds = α α iω. z, a 2 = bχ(2 y. 1 iω b +iω 1 D = 1+a 1ā1 + a 2ā2 + ba 2ā2 θe iω θ k( θdθ then <q (s,q(θ 1and<q (s, q(θ. Next, we use the same notations as those in Hassard [3] and we first compute the coordinates to describe the center manifold C at μ =. Letu t be the solution of (11 when μ =. Define z(t = q,u t W (t, θ =u t(θ 2Re{z(tq(θ} (14 on the center manifold C, and we have W (t, θ =W (z(t, z(t,θ (15 z 2 W (z(t, z(t,θ=w (z, z =W W11z z + z 2 W2 2 + (16 and z and z are local coordinates for center manifold C in the direction of q and q. Noting that W is also real

5 186 International Journal of Automation and Computing 12(2, April 215 if u t is real, we consider only real solutions. For solutions u t C of (11, That is ż(t =iω z + q (θf[,w(z, z, θ] + 2Re{zq(θ} def = iω z + q ([f 1,f 2, ] T. g(z, z =g 2 z 2 ż(t =iω z + g(z, z 2 + g11z z + g2 z 2 Let f =[f 1,f 2] T.Hence,wehave 2 +. g(z, z = q (f (z, z =f(,u t= D (a 1a 2 +ā 1 z 2 + D(2Re{a 1ā 2} +2ā 1z z+ [ D(ā 1ā 2 +ā 1 z 2 + D 1 (3 ā1w 2 2 (+ 1 2 ā 1 Then, we get ā2w (3 2 ( W (1 2 (3 (2 ( + ā1w 11 ( + ā2w 11 (+ ] (1 ( + 2W 11 ( z 2 z + high order term. g 2 =2 D (a 1a 2 +ā 1 g 11 =2 D(Re{a 1ā 2} +ā 1 g 2 =2 D(ā 1ā 2 +ā 1 [ 1 (3 g 21 =2 D ā1w 2 2 ( + 1 (3 ā2w 2 2 ( + ā 1W (3 11 ( ā 1 (2 ( + ā2w 11 (+ ] (1 ( + 2W 11 (. W (1 2 Since there exist unknowns W (1 2 (, W(3 2 (, W (1 11 (,W(2 11 (,W(3 11 ( in g21, we still need to compute them. It follows from (11 and (14 that AW 2Re{ q ([f 1,f 2, ] T q(θ}, θ< W = AW 2Re{ q ([f 1,f 2, ] T q(θ} +[f 1,f 2, ] T, θ = def = AW + H(z, z, θ (17 H(z, z, θ =H 2(θ z2 2 + H11(θz z + H2(θ z2 2 Comparing the coefficients, we obtain +. (18 (AW 2iω W 2 = H 2(θ (19 AW 11(θ = H 11(θ. (2 For θ [,, H(z, z, θ = q (f q(θ q ( f q(θ = g(z, zq(θ ḡ(z, z q(θ. (21 Comparing the coefficients of (21 with (18 gives that H 2(θ = g 2q(θ ḡ 2 q(θ (22 H 11(θ = g 11q(θ ḡ 11 q(θ. (23 From (19, (22 and the definition of A, weget Ẇ 2(θ =2iω W 2(θ+g 2q(θ+ g 2 q(θ. (24 Noting that q(θ =q(e iω θ,wehave W 2(θ = ig2 ω q(e iω θ + iḡ2 3ω q(e iω θ + E 1e 2iω θ (25 E 1 =[E (1 1,E(2 1,E(3 1 ] R3 is a constant vector. Similarly, from (2, (23 and the definition of A, we have Ẇ 11(θ =g 11q(θ+ g 11 q(θ (26 W 11(θ = ig11 ω q(e iω θ + iḡ11 ω q(e iω θ + E 2 (27 E 2 =[E (1 2,E(2 2,E(3 2 ] R3 is a constant vector. In what follows, we shall seek appropriate E 1, E 2 in (25, (27, respectively. It follows from the definition of A and (22, (23 that dη(θw 2(θ =2iω W 2( H 2( (28 and dη(θw 11(θ = H 11( (29 η(θ = η(,θ. From (22 and (23, we have H 2( = g 2q( ḡ 2 q( + 2 a 1a 2 1 (3 Re{a 1ā 2} H 11( = g 11q( ḡ 11( q( (31 Noting that ( iω I e iωθ dη(θ q( = ( iω I e iωθ dη(θ q( = and substituting (25 and (3 into (28, we have ( a 1a 2 2iω I e 2iωθ dη(θ E 1 =2 1.

6 C. J. Xu and Y. S. Wu / Chaos Control and Bifurcation Behavior for a Sprott E system with 187 That is 2iω z y 2x 2iω iω b + bχ (3 χ (3 = It follows that k( se 2iω s ds = E 1 =2 α α +2iω. a 1a 2 1 E (1 1 = Δ11 Δ 1, E (2 1 = Δ12 Δ 1, E (3 1 = Δ13 Δ 1 (32 Δ 2 =det z y 2x 4 b + bχ (1 Re{a 1ā 2} z y Δ 21 =2 det b + bχ (1 Δ 22 =2 det Δ 23 =2 det Re{a 1ā 2} y 2x 4 b + bχ (1 z Re{a 1ā 2} 2x 4. From (25, (27, (32 and (33, we can calculate g 21 derive the following values: and Δ 1 =det 2iω z y 2x 2iω iω b + bχ (3 a 1a 2 z y Δ 11 =2 det 1 2iω +1 2iω b + bχ (3 Δ 12 =2 det Δ 13 =2 det 2iω a 1a 2 y 2x 1 4 2iω b + bχ (3 2iω z a 1a 2 2x 2iω Similarly, substituting (26 and (31 into (29, we have That is ( dη(θ E 2 =2 z y 2x 4 b + bχ (1 It follows that Re{a 1ā 2} 1. E 2 =2[ Re{a 1ā 2},, ] T. E (1 2 = Δ21 Δ 2, E (2 2 = Δ22 Δ 2, E (3 2 = Δ23 Δ 2 (33 c 1( = i (g 2g 11 2 g 11 2 g2 2 + g21 2ω 3 2 Re {c1(} μ 2 = Re{λ (α } β 2 =2Re(c 1( T 2 = Im{c1(} + μ2im{λ (α } ω (34 which determine the quantities of bifurcating periodic solutions on the center manifold C at the critical value α, namely, we have the following result. Theorem 2. The periodic solution is supercritical (subcritical if μ 2 > (μ 2 < ; the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable if β 2 < (β 2 > ; the periods of the bifurcating periodic solutions increase (decrease if T 2 > (T 2 <. 4 Computer simulations In this section, we present some numerical results of system (2 to verify the analytical predictions obtained in the previous section. From Section 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcation periodic solutions. Let us consider the following system: ẋ = y(tz(t+.6 ẏ = x 2 (t y(t ż =1 4x(t+.5 (z(t z(t + sk( sds (35 k(s =αe αs,α >. Let yz +.6 =,x 2 y =, 1 4x =. It is easy to see that system (35 has an equilibrium E( 1, 1, 1, 12 and all the conditions indi-

7 188 International Journal of Automation and Computing 12(2, April 215 cated in Theorem 2 are satisfied. When τ =, the equilibrium E( 1, 1, 1, 12 is asymptotically stable. By means of Matlab 7., we get α Thus the equilibrium E( 1, 1, 1, 12 is stable when α < α which is illustrated by the computer simulations (see Fig. 2. When α passes through the critical value α, the equilibrium E( 1, 1, 1, 12 loses its stability and a Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcations from the equilibrium E( 1, 1, 1, 12. It follows from the formulae (34 presented in Section 3 that μ 2 > andβ 2 <. Then the direction of the Hopf bifurcation is α>α,and these bifurcating periodic solutions from E( 1, 1, 1, 12 around α are stable, which are depicted in Fig. 3.

8 C. J. Xu and Y. S. Wu / Chaos Control and Bifurcation Behavior for a Sprott E system with 189 conditions are fulfilled, then the Sprott E system is asymptotically stable for α<α and when α passes through α,a sequence of Hopf bifurcations occur around the equilibrium E(x,y,z, namely, a family of periodic orbits bifurcate from the the equilibrium E(x,y,z, which implies the chaos of this system can be suppress. Some numerical simulations are included to visualize the theoretical findings. Moreover, the control method used in this paper can be applicable to other chaotic systems. We will carry our some relatedworkinnearfuture. Fig. 2 Behavior and phase portraits of system (35 with α =.25 <α The equilibrium E( 1 4, 1 16, 1 4, isasymptotically stable. The initial value is (1.5, 1.5, Conclusions In this paper, a feedback control method is applied to suppress chaotic behavior of a Sprott E system within the chaotic attractor. By adding a continuous distributed time delayed force to the third equation of the Sprott E system, we have made a detailed discussion on the local stability of the equilibrium E(x,y,z and local Hopf bifurcation of the delayed Sprott E system model. We showed that if some

9 19 International Journal of Automation and Computing 12(2, April 215 Fig. 3 Behavior and phase portraits of system (35 with α =.55 >α Hopf bifurcation occurs from the equilibrium E( 1, 1, 1, 12. The initial value is (1.5,1.5,1.5 References [1] E. Ott, C. Grebogi, J. A. York. Controlling chaos. Physical Review Letters, vol. 64, no. 11, pp , 199. [2] X. F. Liao, G. R. Chen, Hopf bifurcations and chaos analysis of chens system with distributed delays. Chaos, Solitons and Fractals, vol. 25, no. 1, pp , 25. [3] G. Corso, F. B. Rizzato. Hamiltonian bifurcation leading to chaos in a low-energy relativistic wave-particle system. Physica D, vol. 8, no. 3, pp , [4] A. E. Gohary, A. S. Ruzaiza. Chaos and adaptive control in two prey, one predator system with nonlinear feedback. Chaos, Solitons and Fractals, vol. 34, no. 2, pp , 27. [5] Y. A. Kuznetsov, S. Rinaldi. Remarks on food chain dynamics. Mathematical Biosciences, vol. 134, no. 1, pp. 1 33, [6] M. C. Varriale, A. A.Gomes. A study of a three species food chain. Ecological Modelling, vol. 11, no. 2, pp , 1998.

10 C. J. Xu and Y. S. Wu / Chaos Control and Bifurcation Behavior for a Sprott E system with 191 [7] F. Y. Wang, G. P. Pang. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons & Fractals, vol. 36, no. 5, pp , 28. [8] J. G. Du, T. W. Huang, Z. H. Sheng, H. B. Zhang. A new method to control chaos in an economic system. Applied Mathematics and Computation, vol. 217, no. 6, pp , 21. [9] M. Sadeghpour, H. Salarieh, A. Alasty. Controlling chaos in tapping mode atomic force microscopes using improved minimum entropy control. Applied Mathematical Modelling, vol. 37, no. 3, pp , 213. [1] A. S. Hegazi, E. Ahmed, A. E. Matouk. On chaos control and synchronization of the commensurate fractional order Liu system. Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 5, pp , 213. [11] G. R. Chen. Controlling Chaos and Bifurcations in Engineering Systems, Boca Raton, FL: CRC Press, [12] X. S. Yang, G. R. Chen. Some observer-based criteria for discrete-time generalized chaos synchronization. Chaos, Solitons & Fractals, vol. 13, no. 6, pp , 22. [13] G. R. Chen, X. Dong. On feedback control of chaotic continuous-time systems. IEEE Transactions on Circuits and Systems, vol. 4, no. 9, pp , [14] M. T. Yassen. Chaos control of Chen chaotic dynamical system. Chaos, Solitons & Fractals, vol. 15, no. 2, pp , 23. [15] H. N. Agiza. Controlling chaos for the dynamical system of coupled dynamos. Chaos, Solitons & Fractals, vol. 13, no. 2, pp , 22. [16] H. T. Zhao, Y. P. Lin, Y. X. Dai. Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain. Applied Mathematics and Computation, vol. 218, no. 5, pp , 211. [17] Y. L. Song, J. J. Wei. Bifurcation analysis for Chen ssystem with delayed feedback and its application to control of chaos. Chaos, Solitons & Fractals, vol. 22, no. 1, pp , 24. [18] E. W. Bai, K. E. Lonngren. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals, vol. 11, no. 7, pp , 2. [19] M. T. Yassen. Adaptive control and synchronization of a modified Chua s circuit system. Applied Mathematics and Computation, vol. 135, no. 1, pp , 21. [2] T. L. Liao, S. H. Lin. Adaptive control and synchronization of Lorenz system. Journal of the Franklin Institute, vol. 336, no. 6, pp , [21] H. Richter, K. J. Reinschke, H. Richter, K. J. Reinschke. Optimization of local control of chaos by an evolutionary algorithm. Physica D, vol. 144, no. 3 4, pp , 2. [22] R. Senkerik, I. Zelinka, D. Davendra, Z. Oplatkova. Utilization of SOMA and differential evolution for robust stabilization of chaotic Logistic equation. Computers & Mathematics with Applications, vol. 6, no. 4, pp , 21. [23] I. Zelinka, M. Chadli, D. Davendra, R. Senkerik, R. Jasek. An investigation on evolutionary reconstruction on continuous chaotic systems. Mathematical and Computer Modelling, vol. 57, no. 1 2, pp. 2 15, 213. [24] M. Chadli, I. Zelinka, T. Youssef. Unknown inputs observer design for fuzzy systems with application to chaotic system reconstruction. Computers and Mathematics with Applications, vol. 66, no. 2, pp , 213. [25] H. A. Yousef, M. Hamdy. Observer-based adaptive fuzzy control for a class of nonlinear time-delay systems. International Journal of Automation and Computing, vol. 1, no. 4, pp , 213. [26] G. D. Zhao, N. Duan. A continuous state feedback controller design for high-order nonlinear systems with polynomial growth nonlinearities. International Journal of Automation and Computing, vol. 1, no. 4, pp , 213. [27] X. Wang, G. R. Chen. A chaotic system with only one stable equilibrium. Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp , 212. [28] J. C. Sprott. Simplest dissipative chaotic flow. Physics Letters A, vol. 228, no. 3-4, pp , [29] X. Wang, G. R. Chen. Constructing a chaotic system with any number of equilibria. Nonlinear Dynamics, vol. 71, no. 3, pp , 213. [3] B. Hassard, D. Kazarino, Y. Wan. Theory and Applications of Hopf bifurcation, Cambridge, UK: Cambridge University Press, Chang-Jin Xu graduated from Huaihua University, China in He received His M. Sc. degree from Kunming University of Science and Technology, China in 24 and his Ph. D. degree from Central South University, China in 21, respectively. He is currently a professor at the Guizhou Key Laboratory of Economics System Simulation in Guizhou University of Finance and Economics, China. He has published about 1 refereed journal papers. He is a reviewer of Mathematical Reviews, Zentralblatt MATH. His research interests include stability and bifurcation theory of delayed differential equation. xcj43@126.com (Corresponding author ORCID id: Yu-Sen Wu graduated from Liaocheng University, China in 24. He received his M. Sc. degree from Central South University, China in 27, and his Ph. D. degree from Central South University, China in 21, respectively. He is currently an associate professor at School of Mathematics and Statistics of Henan University of Science and Technology. He has published about 3 refereed journal papers. He is a reviewer of Mathematical Reviews, Zentralblatt MATH. His research interests include the qualitative theory of ordinary differential equation and computer symbol calculation. wuyusen82714@yahoo.com.cn