A Unified Lorenz-Like System and Its Tracking Control

Commun. Theor. Phys. 63 (2015) 317 324 Vol. 63, No. 3, March 1, 2015 A Unified Lorenz-Like System and Its Tracking Control LI Chun-Lai ( ) 1, and ZHAO Yi-Bo ( ) 2,3 1 College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China 2 School of Electronic & Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China 3 Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Nanjing, Nanjing 210044, China (Received October 8, 2014; revised manuscript received December 30, 2014) Abstract This paper introduces the finding of a unified Lorenz-like system. By gradually tuning the only parameter d, the reported system belongs to Lorenz-type system in the sense defined by Clikovský. Meanwhile, this system belongs to Lorenz-type system, Lü-type system, Chen-type system with d less than, equivalent to and greater than 1.5, respectively, according to the classification defined by Yang. However, this system can only generate a succession of Lorenz-like attractors. Some basic dynamical properties of the system are investigated theoretically and numerically. Moreover, the tracking control of the system with exponential convergence rate is studied. Theoretical analysis and computer simulation show that the proposed scheme can allow us to drive the output variable x 1 to arbitrary reference signals exponentially, and the guaranteed exponential convergence rate can be estimated accurately. PACS numbers: 05.45.-a, 05.45.Ac, 05.45.Pq Key words: unified Lorenz-like system, Lorenz-like attractor, classification, tracking control 1 Introduction Since the now-classic Lorenz system was found in 1963, [1] a great deal of interests have been motivated to seek three-dimensional autonomous chaotic systems with quadratic nonlinear terms. [2 15] Among the discovered chaotic systems, it is particularly worth to recall the Chen system, which is closely related but not topologically equivalent to the Lorenz system. [2] It is also well bestowed to the mention on the Lü system, which is regarded as the transition between the Lorenz system and the Chen system. [3] The continuous investigation along the line finally leads to the introducing of a family of generalized Lorenz systems, [4 6] with the following canonical form [ ] 0 0 0 A 0 ẋ = x + x 1 0 0 1 x, (1) 0 λ 3 0 1 0 where x = [x 1, x 2, x 3 ] T, λ 3 < 0, and A is a 2 2 real matrix [ ] a11 a 12 A =, (2) a 21 a 22 with the eigenvalues λ 1 > 0, λ 2 < 0. This algebraic form contains a number of chaotic systems with the same stability of equilibrium points and similar attractors in shape. According to the classification defined by Clikovský, Lorenz system satisfies a 12 a 21 > 0, Chen system satisfies a 12 a 21 < 0, while Lü system satisfies a 12 a 21 = 0. In this sense of classification, Chen system is the dual system to the Lorenz system, and Lü system is considered as the transition of Lorenz system and Chen system. Lately, by developing a unified Lorenz-type system, another classification method was proposed by Yang. [7] This method classifies the canonical system (1) into three types by the sign of a 11 a 22, i.e., Lorenz-type system with a 11 a 22 > 0, Chen-type system with a 11 a 22 < 0, and Yang system is regarded as a transient system if satisfying a 11 a 22 = 0. [7] It should be declared that, similarly to the definition of Clikovský, the transition is called Lütype system provisionally. Based on the Yang s theory, the three types of system share a 11 < 0 in common. And, compared with the Clikovský s and Yang s methods, the classifications of each such system are determined in the light of its algebraic structure by a 12 a 21 or a 11 a 22, which make identical result for the classification of classical Lorenz system and Chen system. [1 2] Spontaneously, it is interesting and significant to raise the following questions: (i) Whether the classifications defined by Clikovský and Yang are incompatible for some generalized Lorenz systems? For instance, can we construct a chaotic system belonging to Lorenz-type system in the sense of Clikovský, but simultaneously belonging to Chen-type or Lü-type system in the sense of Yang? (ii) If there does exist such a unified system, are the signs of a 11 a 22 essential to the system dynamics? That is to Supported by the Research Foundation of Education Bureau of Hunan Province of China under Grant No. 13C372, Jiangsu Provincial Natural Science Foundation of China under Grant No. 14KJB120007 and the Outstanding Doctoral Dissertation Project of Special Funds under Grant No. 27122 Corresponding author, E-mail: strive123123@163.com c 2015 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

318 Communications in Theoretical Physics say, can this system generate Lorenz-like attractors, while belongs to Lorenz-type system, Chen-type system or the transient system? These questions will be addressed affirmatively in the present paper. On the other hand, research on chaos has emerged to concentrate on more challenging problem of tracking control due to its potential applications in many disciplines such as information science, secure communication, chemical reactor, and so on. Tracking control is a more generalized form of chaos control and synchronization, which can be explained that, for chaotic system the output ultimately follows the given reference signal by designing appropriate control method Especially, it is equated with chaos control when the reference signal is periodic or fixed value, while it evolves into chaos synchronization when the reference signal is produced by chaotic (hyperchaotic) system. There are many works on tracking control.[16 18] However, the existed control designs always hold multiple control functions. And these designs are restricted to local results, i.e., the stability is asymptotic or uniform, which is a weaker property compared with exponential stability. Therefore, it would be significant to study the tracking control with exponential stabilization by designing a single controller. Motivated by the above discussion, a unified Lorenzlike system is reported in this paper. By continuously changing the only parameter d, the proposed system always belongs to the family of Lorenz-type systems defined by C likovsky. However, according to the classifi- Vol. 63 cation defined by Yang, this system respectively belongs to Lorenz-type system with d less than 1.5, Lu -type system (transient system) with d equivalent to 1.5 and Chentype system with d greater than 1.5. Nevertheless, the reported system always generates a variety of cascading Lorenz-like attractors. Then, based on the exponentially stable theory,[19] a proper control scheme with a single control input is proposed to realize tracking control for the reported chaotic system, which can effectively implement the tracking of the output variable x1 to arbitrary reference signals. And to further illustrate the feasibility of proposed method, three numerical examples are introduced from different angles: chaotic control, complete synchronization, and generalized synchronization (or combination synchronization). 2 The Proposed Lorenz-Like System 2.1 System Description The presented system is given by the following threedimensional autonomous ODEs: x 1 = ax1 + x2 x 2 = (c + 10d)x1 (3 2d)x2 x1 x3, x 3 = x3 + x1 x2, (3) where a, c are the positive parameters, d R is the only regulable parameter. In this paper, we set the parameters as a = 3, c = 50. Fig. 1 (Color online) Phase portraits of Lorenz-type system with d = 1.1 on (a) x1 -x2 plane; (b) x2 -x3 plane; (c) x1 -x3 plane and (d) power spectral density.

No. 3 Communications in Theoretical Physics 319 The Lyapunov exponents of system (3) are calculated as 0.646 23 > 0, 0.0, 7.570 364 1 < 0 when d = 1.1. The corresponding Kaplan Yorke dimension of the system is D KY = 2 + (0.646 23 + 0.0)/7.570 364 1 = 2.085 36. Therefore, the Kaplan Yorke dimension is fractional and system (3) is indeed chaotic. The corresponding chaotic phase diagrams and power spectral density are depicted in Fig. 1. It appears from Fig. 1 that the reported system displays complicated dynamical behaviors. It is easy to see the natural symmetry of system (3) under the coordinate transformation (x 1, x 2, x 3 ) ( x 1, x 2, x 3 ), which reveals that the system has rotation symmetry around the x 3 -axis. 2.2 Parameter Region of Chaotic Attractor For system (3), it is noticed that V = ẋ 1 x 1 + ẋ 2 x 2 + ẋ 3 x 3 = a 4 + 2d = 7 + 2d. (4) So, when d < 7/2 = 3.5, system (3) is dissipative, and shrinks to a subset of measure zero volume with an exponential rate 7 + 2d. On the other hand, we construct the following Lyapunov function which yields V (x 1, x 2, x 3 ) = (x 2 1 + x 2 2 + x 2 3)/2, (5) V (x 1, x 2, x 3 ) = x 1 ẋ 1 + x 2 ẋ 2 + x 3 ẋ 3 = ax 2 1 + (c + 1 + 10d)x 1 x 2 (3 2d)x 2 2 x 2 3 ( ax1 = [3 2d (c + 1 + 10d) ) 2 2 x 2 a ] x 2 2 x 2 3. (6) (c + 1 + 10d)2 4a This means that system (3) is globally uniformly asymptotically stable about the zero equilibrium point when 3 2d (c + 1 + 10d) 2 /4a > 0, i.e. 6.4843 < d < 3.9557. Accordingly, system (3) is not chaotic if 6.4843 < d < 3.9557. As discussed above, the permissive parameter region for generating chaotic attractor in system (3) is d ( 3.9557, 3.5) (, 6.4843). Therefore, in the ensuing section, only the parameter area d (1, 2.2) is considered for the analysis of bifurcation. 2.3 Bifurcation Analysis by Varying Parameter d Figures 2(a) and 2(b) show the bifurcation diagram of the peak of state x 3 and Lyapunov exponent spectrums with parameter area d (1, 2.2), demonstrating a perioddoubling route to chaos. The detailed dynamical routes are summarized as below (i) When 1 < d < 1.61, L 1 > 0, L 2 = 0, L 3 < 0, system (3) is chaotic. But there are some periodic windows in the chaotic band. (ii) When 1.61 < d < 1.665, L 1 = 0, L 2 < 0, L 3 < 0, there is a visible period-doubling bifurcation window. (iii) When 1.665 < d < 1.76, L 1 > 0, L 2 = 0, L 3 < 0, system (3) is chaotic. There are, however, some narrow periodic windows in the chaotic band. (iv) When 1.76 < d < 2.2, L 1 = 0, L 2 < 0, L 3 < 0, there exists a wide period-doubling bifurcation window. Fig. 2 (Color online) (a) Bifurcation diagram; and (b) Lyapunov exponent spectrum of system (3) versus parameter d. 2.4 Topological Equivalence The proposed system (3) can be described by ẋ = f(x), (7) and the Lorenz (Chen, or Lü) system is denoted as ẏ = g(y). (8) If systems (7) and (8) are said to be diffeomorphic (topological equivalent), there would exist a diffeomor-

320 Communications in Theoretical Physics Vol. 63 phism x = T (y), such that g(y) = J 1 (y)f(t (y)), (9) where J(y) = dt (y)/dy is the Jacobian matrix of T at the point y. [20 21] Let y 0 and x 0 = T (y 0 ) be the equilibrium points of systems (7) and (8), A(y 0 ) and B(x 0 ) respectively denote the corresponding Jacobian matrices. If systems (7) and (8) are topological equivalent, then we will have A(y 0 ) = J 1 (y)b(y 0 )J(y). Therefore, their characteristic polynomials and eigenvalues should coincide with each other. Based on the concept and techniques of the equilibrium and resultant eigenvalue, it is easy to actually verify that the system (7) is not smoothly equivalent to system (8). Therefore, systems (7) and (8) are not topological equivalent. 3 Equilibrium and Attracting Basin 3.1 Equilibrium and Stability Applying the equilibrium condition to system (3), it is determined that three equilibrium points exist, as below E 0 (0, 0, 0) E ± (± D/a, ± ad, D), where D = c + 10d 3a + 2ad = 41 + 16d. By linearizing system (3) with respect to the equilibrium E 0, we obtain the Jacobian matrix 3 1 0 J E0 = 50 + 10d 2d 3 0, (10) 0 0 1 and the corresponding characteristic equation f(λ) E0 =λ 3 (2d 7)λ 2 (18d + 35)λ 16d 41 = 0, (11) which leads to the eigenvalues of J E0. λ 1 = 1 < 0, λ 2 = d 3 + d 2 + 10d + 50 > 0, λ 3 = d 3 d 2 + 10d + 50 < 0. Obviously, the equilibrium E 0 is a saddle-node with two-dimensional stable manifold and one-dimensional unstable manifold. Similarly, linearizing the system at the equilibrium points E ±, it yields the following characteristic equations f(λ) E± = λ 3 + (7 2d)λ 2 + (59/3 + 10d/3)λ + 73 + 38d = 0. (12) Let the three roots of Eq. (12) denote as λ 1, λ 2, λ 3. From Eq. (12), we have following relationships λ 1 + λ 2 + λ 3 = 2d 7 = Γ 1, λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 = 59/3 + 10d/3 = Γ 2, λ 1 λ 2 λ 3 = 73 38d = Γ 3. (13) The corresponding Hurwitz matrix of Eq. (12) is obtained as Γ 1 Γ 3 0 M = 1 Γ 2 0. (14) 0 Γ 1 Γ 3 Then the principal minors of M can be described as M 1 = Γ 1, M 2 = Γ 1 Γ 2 + Γ 3, M 3 = Γ 2 Γ 3. According to the Routh Hurwitz criterion, the sufficient and necessary condition for Re (λ 1 ), Re (λ 2 ), Re (λ 3 ) < 0 is that all the principal minors of Mare positive. [10] Consequently, we can obtain that the local stable condition of the equilibrium points E ± is 9.1591 < d < 1.0591. Theorem 1 When parameter d varies and passes through the critical value d = d 0 = 1.0591, system (3) undergoes a Hopf bifurcation at the equilibrium points E ±. Proof Suppose that the characteristic equation (12) has a pair of purely conjugate imaginary roots depicted as λ 1,2 = ±iω, and one negative real root λ 3. Since 3 i=1 λ i = 2d 7, we have λ 3 = 2d 7. Substituting λ 1,2 = ±iω into Eq. (12) yields f(λ) E± = (7 2d)ω 2 + 73 + 38d + iω(59/3 + 10d/3 ω 2 ) = 0. (15) Then, one has d = 1.0591, ω = 8.3421. So the first condition for Hopf bifurcation is satisfied. From Eq. (12), we obtain λ d = 2λ 2 + 10λ/3 + 38 3λ 2 + 2(7 2d)λ + (59/3 + 10d/3), (16) which leads to Re ( λ d λ=λ1, d=d 0 ) 0. Thus, the second condition for Hopf bifurcation to exist is also met. Consequently, a Hopf bifurcation can appear at the points E ±. 3.2 Attracting Basin of Equilibrium Attracting basin of equilibrium point is just the set of initial guesses that leads to the corresponding equilibrium, which represents a mathematically-involved subtle issue. The attracting basins for system (3) with three crosssections are shown in Fig. 3. On these sections, the attracting basins of equilibrium points E 0, E +, E for Lütype system (3) with d = 1.5 are indicated by red, blue and green, respectively. It turns out that all the attracting basins of system (3) have special properties. On one hand, they are localized in such a way that each attracting basin is located partly in its region and partly in the region of the other attracting basins. As a result, if the phase point starts from the part of the attracting basin situated in the region of the other one, it will escape to the other basin. However, if it starts from the part of the attracting basin situated in its own region, it will do not change the basin. On the other hand, the boundaries of the attracting basin of system (3) do have a fractal structure, which makes the estimate of the basin boundaries more discommodious.

Communications in Theoretical Physics No. 3 321 Fig. 3 (Color online) Attracting basins of equilibrium E0, E+, E for Lu -type system (3) with cross-sections of (a) x2 = x1 ; (b) x3 = 6; (c) x3 = 40. 4 Classification First of all, for analyzing conveniently, we set parameter area of d as (1,2.2). And we can write the constructed system (3) into the canonical form (1) with matrix a 1 A=. (17) c + 10d (3 2d) On the one hand, we note that a12 a21 = c + 10d = 50 + 10d > 0 holds in system (3), disregarding the value of parameter d. Therefore, the reported system always belongs to the Lorenz-type systems in the sense of C likovsky. On the other hand, the sign of a11 a22 = a(3 2d) = 3(3 2d) is determined by the value of parameter d. Therefore, provided by the method of Yang s classification, this system respectively belongs to Lorenz-type system with d less than 1.5, Lu -type system (transient system) with d equivalent to 1.5 and Chen-type system with d greater than 1.5. Different cases with three typical values of d are summarized in Table 1, which displays the incompatibility of the classifications defined by C likovsky and Yang for the reported Lorenz-like system. The corresponding phase portraits with d = 1.1 are depicted in Figs. 1(a), 1(b), 1(c). The attractors from Lorenz-type to Chentype through Lu -type system defined by Yang s classification are shown in Fig. 4. From Figs. 1 and 4, we know that the presented system generates a variety of cascading Lorenz-like attractors, while belonging to Lorenz-type system, Chen-type system and Lu -type system, respectively. This further demonstrates that the reported system possesses complex dynamics in the similar way of generalized Lorenz algebraic form. Fig. 4 (Color online) Phase portrait of (a) Chen-type system with d = 1.72; (b) Lu -type system with d = 1.5. Table 1 Classification for system (3) with a = 3, c = 50. Parameter d a12 a21 Classification by C elikovsky a11 a22 Classification by Yang Lyapunov Exponents d = 1.1 = 61 > 0 Lorenz-type = 2.4 > 0 Lorenz-type 0.646 23, 0, 7.570 364 d = 1.5 = 65 > 0 Lorenz-type =0 Lu -type 0.524 314, 0, 6.294 264 d = 1.72 = 67.2 > 0 Lorenz-type = 1.32 < 0 Chen-type 0.658 676, 0, 5.794 097

322 Communications in Theoretical Physics Vol. 63 5 Tracking Control of the Proposed Lorenz- Like System 5.1 Control Scheme The controlled chaotic system is given as follows: ẋ 1 = ax 1 + x 2, ẋ 2 = (c + 10d)x 1 (3 2d)x 2 x 1 x 3 + u, ẋ 3 = x 3 + x 1 x 2. (18) Let x r be the arbitrary given reference signal with second derivative. The synchronization error between system (18) and reference signal x r is defined as e = x 1 x r. Our aim is that, by designing suitable control scheme u, the output variable x 1 of system (18) follows the reference signal x r exponentially. That is x 1 x r α e ηt, t 0, where denotes a 2-norm in R 3, α, η are positive numbers. [19] Theorem 2 For the output variable x 1 of controlled system (18) and reference signal x r, if the controller u is designed as u = µ(e + ė) (a 2 + c + 10d)x 1 + (3 + a 2d)x 2 + x 1 x 3 + ẍ r, (19) with µ > 1, then output x 1 will approach the reference signal x r exponentially, and the guaranteed exponential convergence rate can be given by: η = min(µ 1, 1). Proof Choose the candidate Lyapunov function as V (e, ė) = 1 2 e2 + 1 2 (e + ė)2. (20) Taking the time derivative of V (e, ė) gives V (e, ė) = eė + (e + ė)(ė + ë) = (e + ė)(e + ė + ë) e 2 = (e + ė)[e + ė a( ax 1 + x 2 ) + (c + 10d)x 1 (3 2d)x 2 x 1 x 3 + u ẍ r ] e 2 = [(e + ė)(e + ė) µ(e + ė)(e + ė)] e 2 = (µ 1)(e + ė) 2 e 2 2 min(µ 1, 1)[0.5(e + ė) 2 + 0.5e 2 ]. Let η = min(µ 1, 1), it is obtained V (e, ė) 2ηV. Thus, we will deduce e 2ηt V (e, ė) + e 2ηt 2ηV (e, e) = d dt ( e2ηt V (e, ė)) 0. (21) This implies that t 0 d dτ ( e2ητ V (e(τ), ė(τ)))dτ = e 2ηt V (e, ė) V (e(0), ė(0)) 0. (22) From expresses (20) and (22), we have 0.5e 2 V (e, ė) e 2ηt V (e(0), ė(0)). (23) Consequently, it can be concluded that e(t) 2V (e(0), ė(0)) e ηt. (24) Therefore, the output variable x 1 of system (18) will exponentially follow the reference signal x r 5.2 Numerical Simulations In this section, three numerical examples are introduced to demonstrate the effectiveness of the proposed control scheme For comparing conveniently, in all the process of simulation, the ODE45 method is adopted to solving the differential equations, the parameters are set as a = 3, c = 50, d = 1.1, the initial states of the controlled system (18) are taken as x(0) = (0.1, 0.1, 1), and let µ = 3. (i) Control to Periodic Orbit and Fixed Value First, we take the sinusoidal periodic signal and fixed value as the reference signals for realizing chaotic control, respectively. The simulation results for x r = 5 sin t + 5 sin 2t are shown in Figs. 5(a) and 5(b). Figure 5(a) depicts the time evolution of variable x 1 (red line) and reference signal x r (blue line). Figure 5(b) displays the time evolution of error. It is known that the output x 1 is controlled to the periodic signal x r = 5 sin t + 5 sin 2t exponentially. Fig. 5 (Color online) Control to periodic orbit: (a) time evolution of x 1 and x r; (b) error. The simulation results for x r = 10 are shown in Figs. 6(a) and 6(b). Similarly, Fig. 6(a) depicts the time evolution of variable x 1 (red line) and reference signal x r (blue line). Figure 6(b) displays the time evolution of error. It is known

No. 3 Communications in Theoretical Physics 323 from Fig. 6 that the output x 1 converges the fixed value x r = 10 exponentially. Fig. 6 (Color online) Control to fixed value: (a) time evolution of x 1 and x r; (b) error. (ii) Complete Synchronization In this section, the proposed Lorenz-like system, which is described by (25), is chosen as the response system for achieving complete synchronization (self-synchronization). ẏ 1 = ay 1 + y 2, ẏ 2 = (c + 10d)y 1 (3 2d)y 2 y 1 y 3, ẏ 3 = y 3 + y 1 y 2. (25) The synchronization error between systems (18) and (25) is defined as e = x 1 y 1. Figure 7 shows the corresponding simulation results. In Fig. 7(a), the red line denotes the time evolution the output variable x 1 and the blue line denotes the time evolution the corresponding reference signal x r. Figure 7(b) displays the time evolution of synchronization error. As we can see that the output x 1 and signal y 1 have achieved complete synchronization exponentially. Fig. 7 (Color online) Complete synchronization: (a) time evolution of x 1 and x r; (b) synchronization error. (iii) Generalized Synchronization Now, two different systems, four-wing hyperchaotic system and Van der Pol-Duffing chaotic oscillator will be selected as the response systems to illustrate the validity of the proposed scheme The four-wing hyperchaotic system is given as [22] ẏ 1 = ay 1 + fy 2 + ey 2 y 3, ẏ 2 = dy 2 + cy 4 y 1 y 3, ẏ 3 = by 3 + y 1 y 2, ẏ 4 = ky 2. (26) When a = 10, b = 50, c = 3, d = 10, e = 24, f = 2, and k = 10, system (26) displays four-wing hyperchaotic behaviour. The Van der Pol-Duffing oscillator can be given as [23] ż 1 = z 2, ż 2 = µ(1 z 2 1)z 2 +αz 1 βz 3 1 +f cos(ωt). (27) When the parameters are set equal to µ = 0.1, α = β = 1, ω = 1, f = 3, oscillator (27) will behave chaotically. We take the reference signal as x r = y 3 + 2y 4 + 0.5z 1, and the corresponding synchronization error can be defined as e = x 1 y 3 2y 4 0.5z 1. It should be stressed that this kind of generalized synchronization can be seen as modified combination synchronization with one drive system and two response systems. [24 26] The numerical results for generalized synchronization are depicted in Fig. 8. Figure 8(a) represents the time evolution of variable x 1 (red line) and reference signal x r (blue line). Figure 8(b) depicts the time evolution of generalized synchronization error As we can see that the output x 1 follows the reference signal y 3 + 2y 4 + 0.5z 1 exponentially.

324 Communications in Theoretical Physics Vol. 63 Fig. 8 (Color online) Generalized synchronization: (a) Time evolution of x 1 and x r; (b) Synchronization error. 6 Conclusion This paper has reported the new finding of a unified Lorenz-like system. By tuning the only parameter d continuously, the system belongs to Lorenz-type systems defined by Clikovský. However, this system belongs to Lorenz-type system, Lü-type system, and Chen-type system with the selecting of parameter d, according to the classification defined by Yang. This further demonstrates that the reported system possesses complex dynamics in the similar way of generalized Lorenz algebraic form, and some sealed yet to be investigated algebraic characteristics may be existed for reclassifying three-dimensional quadratic autonomous chaotic systems. We hope that our work can constitute a stimulus for the further research for such direction. Moreover, by constructing a new and special Lyapunov function, a feasible control scheme is proposed to acquire tracking control for the reported chaotic system, the presented numerical examples further illustrate that the introduced method is effective and can allow us to drive the output x 1 of the unified Lorenz-like system to arbitrary reference signal exponentially References [1] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 131. [2] G.R. Chen and T. Ueta, Int. J. Bifurcat Chaos 9 (1999) 1465. [3] J.H. Lü and G.R. Chen, Int. J. Bifurcat Chaos 12 (2002) 659. [4] S. Clikovský and A. Vane c ĕk, Kybernetika 30 (1994) 403. [5] S. Clikovský and G.R. Chen, Int. J. Bifurcat Chaos 12 (2002) 1789. [6] S. Clikovský and G.R. Chen, Chaos, Solitons & Fractals 26 (2005) 1271. [7] Q.G. Yang and G.R. Chen, Int. J. Bifurcat Chaos 18 (2008) 1393. [8] C.C. Sun, E.L. Zhao, and Q.C. Xu, Chin. Phys. B 23 (2014) 0505051. [9] Z.Q. Zhang, H.Y. Shao, Z. Wang, and H. Shen, Appl. Math. Comput. 218 (2012) 7614. [10] D. Kim, P.H. Chang, and S.H. Kim, Nonlinear Dyn. 73 (2013) 1883. [11] D. Kim and P.H. Chang, Results Phys. 3 (2013) 14. [12] D. Cafagna and G. Grassi, Commun. Nonlinear Sci. Numer. Simulat 19 (2014) 2919. [13] K.B. Deng, J. Li, and S.M. Yu, Optik 125 (2014) 3071. [14] F.S. Dias, L.F. Mello, and J.G. Zhang, Nonlinear Anal. RWA. 11 (2010) 3491. [15] B. Munmuangsaen and B. Srisuchinwong, Phys. Lett. A 373 (2009) 4038. [16] I.B. Schwartz and I. Triandaf, Phys. Rev. A 46 (1992) 7439. [17] C.L. Li, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 405. [18] I. Ursu, A. Toader, A. Halanay, and S. Balea, Eur. J. Control. 19 (2013) 65. [19] C.C. Yang, Appl. Math. Comput. 217 (2011) 6490. [20] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York (1998). [21] C.L. Li, L. Wu, H.M. Li, and Y.N. Tong, Nonlinear Anal. Model. Control. 18 (2013) 66. [22] C.L. Li, K.L. Su, and D.Q. Wei, Optik 124 (2013) 5807. [23] Y. Susuki, Y. Yokoi, and T. Hikihara, Chaos 17 (2007) 0231081. [24] R.Z. Luo, Y.L. Wang, and S.C. Deng, Chaos 21 (2011) 0431141. [25] B. Zhang and F.Q. Deng, Nonlinear Dyn. 77 (2014) 1519. [26] J.W. Sun, Y. Shen, Q. Yin, and C.J. Xu, Chaos 23 (2013) 0131401.