A Unified Lorenz-Like System and Its Tracking Control
|
|
- Carol McDonald
- 5 years ago
- Views:
Transcription
1 Commun. Theor. Phys. 63 (2015) 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 , China 2 School of Electronic & Information Engineering, Nanjing University of Information Science and Technology, Nanjing , China 3 Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Nanjing, Nanjing , 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: a, Ac, 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 [ ] A 0 ẋ = x + x x, (1) 0 λ 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. 14KJB and the Outstanding Doctoral Dissertation Project of Special Funds under Grant No Corresponding author, strive123123@163.com c 2015 Chinese Physical Society and IOP Publishing Ltd
2 318 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.
3 No. 3 Communications in Theoretical Physics 319 The Lyapunov exponents of system (3) are calculated as > 0, 0.0, < 0 when d = 1.1. The corresponding Kaplan Yorke dimension of the system is D KY = 2 + ( )/ = 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 x 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 (c d)x 1 x 2 (3 2d)x 2 2 x 2 3 ( ax1 = [3 2d (c d) ) 2 2 x 2 a ] x 2 2 x 2 3. (6) (c d)2 4a This means that system (3) is globally uniformly asymptotically stable about the zero equilibrium point when 3 2d (c d) 2 /4a > 0, i.e < d < Accordingly, system (3) is not chaotic if < d < As discussed above, the permissive parameter region for generating chaotic attractor in system (3) is d ( , 3.5) (, ). 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 < 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-
4 320 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 = d. By linearizing system (3) with respect to the equilibrium E 0, we obtain the Jacobian matrix J E0 = d 2d 3 0, (10) 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 d + 50 > 0, λ 3 = d 3 d d + 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 d = 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 < d < Theorem 1 When parameter d varies and passes through the critical value d = d 0 = , 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 d + iω(59/3 + 10d/3 ω 2 ) = 0. (15) Then, one has d = , ω = So the first condition for Hopf bifurcation is satisfied. From Eq. (12), we obtain λ d = 2λ λ/ λ 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.
5 Communications in Theoretical Physics No 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 = 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 = d > 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, d = 1.5 = 65 > 0 Lorenz-type =0 Lu -type , 0, d = 1.72 = 67.2 > 0 Lorenz-type = 1.32 < 0 Chen-type , 0,
6 322 Communications in Theoretical Physics Vol 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 e (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) e 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
7 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 z 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 z 1 exponentially.
8 324 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) [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) [6] S. Clikovský and G.R. Chen, Chaos, Solitons & Fractals 26 (2005) [7] Q.G. Yang and G.R. Chen, Int. J. Bifurcat Chaos 18 (2008) [8] C.C. Sun, E.L. Zhao, and Q.C. Xu, Chin. Phys. B 23 (2014) [9] Z.Q. Zhang, H.Y. Shao, Z. Wang, and H. Shen, Appl. Math. Comput. 218 (2012) [10] D. Kim, P.H. Chang, and S.H. Kim, Nonlinear Dyn. 73 (2013) [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) [13] K.B. Deng, J. Li, and S.M. Yu, Optik 125 (2014) [14] F.S. Dias, L.F. Mello, and J.G. Zhang, Nonlinear Anal. RWA. 11 (2010) [15] B. Munmuangsaen and B. Srisuchinwong, Phys. Lett. A 373 (2009) [16] I.B. Schwartz and I. Triandaf, Phys. Rev. A 46 (1992) [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) [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) [23] Y. Susuki, Y. Yokoi, and T. Hikihara, Chaos 17 (2007) [24] R.Z. Luo, Y.L. Wang, and S.C. Deng, Chaos 21 (2011) [25] B. Zhang and F.Q. Deng, Nonlinear Dyn. 77 (2014) [26] J.W. Sun, Y. Shen, Q. Yin, and C.J. Xu, Chaos 23 (2013)
A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term
ETASR - Engineering, Technology & Applied Science Research Vol., o.,, 9-5 9 A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term Fei Yu College of Information Science
More informationGeneralized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters
Vol 16 No 5, May 2007 c 2007 Chin. Phys. Soc. 1009-1963/2007/16(05)/1246-06 Chinese Physics and IOP Publishing Ltd Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with
More informationDynamical analysis and circuit simulation of a new three-dimensional chaotic system
Dynamical analysis and circuit simulation of a new three-dimensional chaotic system Wang Ai-Yuan( 王爱元 ) a)b) and Ling Zhi-Hao( 凌志浩 ) a) a) Department of Automation, East China University of Science and
More informationA Novel Hyperchaotic System and Its Control
1371371371371378 Journal of Uncertain Systems Vol.3, No., pp.137-144, 009 Online at: www.jus.org.uk A Novel Hyperchaotic System and Its Control Jiang Xu, Gouliang Cai, Song Zheng School of Mathematics
More informationCONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS
International Journal of Bifurcation and Chaos, Vol. 12, No. 6 (22) 1417 1422 c World Scientific Publishing Company CONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS JINHU LÜ Institute of Systems
More informationAnti-synchronization of a new hyperchaotic system via small-gain theorem
Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised
More informationSIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
International Journal of Bifurcation and Chaos, Vol. 23, No. 11 (2013) 1350188 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413501885 SIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
More informationFunction Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping
Commun. Theor. Phys. 55 (2011) 617 621 Vol. 55, No. 4, April 15, 2011 Function Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping WANG Xing-Yuan ( ), LIU
More informationMULTISTABILITY IN A BUTTERFLY FLOW
International Journal of Bifurcation and Chaos, Vol. 23, No. 12 (2013) 1350199 (10 pages) c World Scientific Publishing Company DOI: 10.1142/S021812741350199X MULTISTABILITY IN A BUTTERFLY FLOW CHUNBIAO
More informationRecent new examples of hidden attractors
Eur. Phys. J. Special Topics 224, 1469 1476 (2015) EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02472-1 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Recent new examples of hidden
More informationConstructing a chaotic system with any number of equilibria
Nonlinear Dyn (2013) 71:429 436 DOI 10.1007/s11071-012-0669-7 ORIGINAL PAPER Constructing a chaotic system with any number of equilibria Xiong Wang Guanrong Chen Received: 9 June 2012 / Accepted: 29 October
More informationControlling a Novel Chaotic Attractor using Linear Feedback
ISSN 746-7659, England, UK Journal of Information and Computing Science Vol 5, No,, pp 7-4 Controlling a Novel Chaotic Attractor using Linear Feedback Lin Pan,, Daoyun Xu 3, and Wuneng Zhou College of
More informationComplete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different 4D Nonlinear Dynamical Systems
Mathematics Letters 2016; 2(5): 36-41 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20160205.12 Complete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different
More informationConstruction of a New Fractional Chaotic System and Generalized Synchronization
Commun. Theor. Phys. (Beijing, China) 5 (2010) pp. 1105 1110 c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 6, June 15, 2010 Construction of a New Fractional Chaotic System and Generalized
More informationBackstepping synchronization of uncertain chaotic systems by a single driving variable
Vol 17 No 2, February 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(02)/0498-05 Chinese Physics B and IOP Publishing Ltd Backstepping synchronization of uncertain chaotic systems by a single driving variable
More informationA new four-dimensional chaotic system
Chin. Phys. B Vol. 19 No. 12 2010) 120510 A new four-imensional chaotic system Chen Yong ) a)b) an Yang Yun-Qing ) a) a) Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai
More informationHYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationResearch Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System
Abstract and Applied Analysis Volume, Article ID 3487, 6 pages doi:.55//3487 Research Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System Ranchao Wu and Xiang Li
More informationGenerating a Complex Form of Chaotic Pan System and its Behavior
Appl. Math. Inf. Sci. 9, No. 5, 2553-2557 (2015) 2553 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090540 Generating a Complex Form of Chaotic Pan
More informationTime-delay feedback control in a delayed dynamical chaos system and its applications
Time-delay feedback control in a delayed dynamical chaos system and its applications Ye Zhi-Yong( ), Yang Guang( ), and Deng Cun-Bing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationAdaptive feedback synchronization of a unified chaotic system
Physics Letters A 39 (4) 37 333 www.elsevier.com/locate/pla Adaptive feedback synchronization of a unified chaotic system Junan Lu a, Xiaoqun Wu a, Xiuping Han a, Jinhu Lü b, a School of Mathematics and
More informationMultistability in the Lorenz System: A Broken Butterfly
International Journal of Bifurcation and Chaos, Vol. 24, No. 10 (2014) 1450131 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414501314 Multistability in the Lorenz System: A Broken
More informationGeneralized projective synchronization between two chaotic gyros with nonlinear damping
Generalized projective synchronization between two chaotic gyros with nonlinear damping Min Fu-Hong( ) Department of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China
More informationBifurcation control and chaos in a linear impulsive system
Vol 8 No 2, December 2009 c 2009 Chin. Phys. Soc. 674-056/2009/82)/5235-07 Chinese Physics B and IOP Publishing Ltd Bifurcation control and chaos in a linear impulsive system Jiang Gui-Rong 蒋贵荣 ) a)b),
More informationProjective synchronization of a complex network with different fractional order chaos nodes
Projective synchronization of a complex network with different fractional order chaos nodes Wang Ming-Jun( ) a)b), Wang Xing-Yuan( ) a), and Niu Yu-Jun( ) a) a) School of Electronic and Information Engineering,
More informationConstruction of four dimensional chaotic finance model and its applications
Volume 8 No. 8, 7-87 ISSN: 34-3395 (on-line version) url: http://acadpubl.eu/hub ijpam.eu Construction of four dimensional chaotic finance model and its applications Dharmendra Kumar and Sachin Kumar Department
More informationHopf Bifurcation and Limit Cycle Analysis of the Rikitake System
ISSN 749-3889 (print), 749-3897 (online) International Journal of Nonlinear Science Vol.4(0) No.,pp.-5 Hopf Bifurcation and Limit Cycle Analysis of the Rikitake System Xuedi Wang, Tianyu Yang, Wei Xu Nonlinear
More informationSimplest Chaotic Flows with Involutional Symmetries
International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1450009 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414500096 Simplest Chaotic Flows with Involutional Symmetries
More informationCrisis in Amplitude Control Hides in Multistability
International Journal of Bifurcation and Chaos, Vol. 26, No. 14 (2016) 1650233 (11 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127416502333 Crisis in Amplitude Control Hides in Multistability
More informationANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM. China
Mathematical and Computational Applications, Vol. 9, No., pp. 84-9, 4 ANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM Ping Cai,, Jia-Shi Tang, Zhen-Bo Li College of
More informationHopf Bifurcation of a Nonlinear System Derived from Lorenz System Using Centre Manifold Approach ABSTRACT. 1. Introduction
Malaysian Journal of Mathematical Sciences 10(S) March : 1-13 (2016) Special Issue: The 10th IMT-GT International Conference on Mathematics, Statistics and its Applications 2014 (ICMSA 2014) MALAYSIAN
More informationDynamics at infinity and a Hopf bifurcation arising in a quadratic system with coexisting attractors
Pramana J. Phys. 8) 9: https://doi.org/.7/s43-7-55-x Indian Academy of Sciences Dynamics at infinity and a Hopf bifurcation arising in a quadratic system with coexisting attractors ZHEN WANG,,,IRENEMOROZ
More informationSYNCHRONIZATION CRITERION OF CHAOTIC PERMANENT MAGNET SYNCHRONOUS MOTOR VIA OUTPUT FEEDBACK AND ITS SIMULATION
SYNCHRONIZAION CRIERION OF CHAOIC PERMANEN MAGNE SYNCHRONOUS MOOR VIA OUPU FEEDBACK AND IS SIMULAION KALIN SU *, CHUNLAI LI College of Physics and Electronics, Hunan Institute of Science and echnology,
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
More informationComputers and Mathematics with Applications. Adaptive anti-synchronization of chaotic systems with fully unknown parameters
Computers and Mathematics with Applications 59 (21) 3234 3244 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Adaptive
More informationA New Fractional-Order Chaotic System and Its Synchronization with Circuit Simulation
Circuits Syst Signal Process (2012) 31:1599 1613 DOI 10.1007/s00034-012-9408-z A New Fractional-Order Chaotic System and Its Synchronization with Circuit Simulation Diyi Chen Chengfu Liu Cong Wu Yongjian
More informationA Highly Chaotic Attractor for a Dual-Channel Single-Attractor, Private Communication System
A Highly Chaotic Attractor for a Dual-Channel Single-Attractor, Private Communication System Banlue Srisuchinwong and Buncha Munmuangsaen Sirindhorn International Institute of Technology, Thammasat University
More informationConstructing Chaotic Systems with Total Amplitude Control
International Journal of Bifurcation and Chaos, Vol. 25, No. 10 (2015) 1530025 (14 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127415300256 Constructing Chaotic Systems with Total Amplitude
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationGenerating hyperchaotic Lu attractor via state feedback control
Physica A 364 (06) 3 1 www.elsevier.com/locate/physa Generating hyperchaotic Lu attractor via state feedback control Aimin Chen a, Junan Lu a, Jinhu Lu b,, Simin Yu c a College of Mathematics and Statistics,
More informationHX-TYPE CHAOTIC (HYPERCHAOTIC) SYSTEM BASED ON FUZZY INFERENCE MODELING
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 28 (73 88) 73 HX-TYPE CHAOTIC (HYPERCHAOTIC) SYSTEM BASED ON FUZZY INFERENCE MODELING Baojie Zhang Institute of Applied Mathematics Qujing Normal University
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
More informationA GALLERY OF LORENZ-LIKE AND CHEN-LIKE ATTRACTORS
International Journal of Bifurcation and Chaos, Vol. 23, No. 4 (2013) 1330011 (20 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413300115 A GALLERY OF LORENZ-LIKE AND CHEN-LIKE ATTRACTORS
More informationHyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system
Nonlinear Dyn (2012) 69:1383 1391 DOI 10.1007/s11071-012-0354-x ORIGINAL PAPER Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system Keihui Sun Xuan Liu Congxu Zhu J.C.
More informationStudy on Proportional Synchronization of Hyperchaotic Circuit System
Commun. Theor. Phys. (Beijing, China) 43 (25) pp. 671 676 c International Academic Publishers Vol. 43, No. 4, April 15, 25 Study on Proportional Synchronization of Hyperchaotic Circuit System JIANG De-Ping,
More informationHopf Bifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators
The Open Acoustics Journal 8 9-3 9 Open Access Hopf ifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators Jianping Cai *a and Jianhe Shen b a Department of
More informationHopf bifurcations analysis of a three-dimensional nonlinear system
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 358), 28, Pages 57 66 ISSN 124 7696 Hopf bifurcations analysis of a three-dimensional nonlinear system Mircea Craioveanu, Gheorghe
More informationFinite-time hybrid synchronization of time-delay hyperchaotic Lorenz system
ISSN 1746-7659 England UK Journal of Information and Computing Science Vol. 10 No. 4 2015 pp. 265-270 Finite-time hybrid synchronization of time-delay hyperchaotic Lorenz system Haijuan Chen 1 * Rui Chen
More informationMultistability in symmetric chaotic systems
Eur. Phys. J. Special Topics 224, 1493 1506 (2015) EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02475-x THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Multistability in symmetric
More informationChaos synchronization of complex Rössler system
Appl. Math. Inf. Sci. 7, No. 4, 1415-1420 (2013) 1415 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/070420 Chaos synchronization of complex Rössler
More informationDynamical behaviour of a controlled vibro-impact system
Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and
More informationCoexisting Hidden Attractors in a 4-D Simplified Lorenz System
International Journal of Bifurcation and Chaos, Vol. 24, No. 3 (2014) 1450034 (12 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414500345 Coexisting Hidden Attractors in a 4-D Simplified
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM
ADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM WITH UNKNOWN PARAMETERS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationSynchronizing Chaotic Systems Based on Tridiagonal Structure
Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 008 Synchronizing Chaotic Systems Based on Tridiagonal Structure Bin Liu, Min Jiang Zengke
More informationA New Hyperchaotic Attractor with Complex Patterns
A New Hyperchaotic Attractor with Complex Patterns Safieddine Bouali University of Tunis, Management Institute, Department of Quantitative Methods & Economics, 41, rue de la Liberté, 2000, Le Bardo, Tunisia
More informationBidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme
Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 1049 1056 c International Academic Publishers Vol. 45, No. 6, June 15, 2006 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic
More informationSynchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems
Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 871 876 c International Academic Publishers Vol. 48, No. 5, November 15, 2007 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More informationEffects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate Cascade Two-Photon Lasers
Commun. Theor. Phys. Beijing China) 48 2007) pp. 288 294 c International Academic Publishers Vol. 48 No. 2 August 15 2007 Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of
More informationGLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationGLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi,
More informationStability and Projective Synchronization in Multiple Delay Rössler System
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(29) No.2,pp.27-214 Stability and Projective Synchronization in Multiple Delay Rössler System Dibakar Ghosh Department
More informationSynchronization between different motifs
Synchronization between different motifs Li Ying( ) a) and Liu Zeng-Rong( ) b) a) College of Information Technology, Shanghai Ocean University, Shanghai 201306, China b) Institute of Systems Biology, Shanghai
More informationBIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ HYPERCHAOTIC SYSTEM
Journal of Applied Analysis and Computation Volume 5, Number 2, May 215, 21 219 Website:http://jaac-online.com/ doi:1.11948/21519 BIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ
More informationControlling the Period-Doubling Bifurcation of Logistic Model
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.20(2015) No.3,pp.174-178 Controlling the Period-Doubling Bifurcation of Logistic Model Zhiqian Wang 1, Jiashi Tang
More informationKingSaudBinAbdulazizUniversityforHealthScience,Riyadh11481,SaudiArabia. Correspondence should be addressed to Raghib Abu-Saris;
Chaos Volume 26, Article ID 49252, 7 pages http://dx.doi.org/.55/26/49252 Research Article On Matrix Projective Synchronization and Inverse Matrix Projective Synchronization for Different and Identical
More informationCOMPLEX DYNAMICS AND CHAOS CONTROL IN DUFFING-VAN DER POL EQUATION WITH TWO EXTERNAL PERIODIC FORCING TERMS
International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 9 No. III (September, 2015), pp. 197-210 COMPLEX DYNAMICS AND CHAOS CONTROL IN DUFFING-VAN DER POL EQUATION WITH TWO EXTERNAL
More informationEffect of various periodic forces on Duffing oscillator
PRAMANA c Indian Academy of Sciences Vol. 67, No. 2 journal of August 2006 physics pp. 351 356 Effect of various periodic forces on Duffing oscillator V RAVICHANDRAN 1, V CHINNATHAMBI 1, and S RAJASEKAR
More informationControl and synchronization of Julia sets of the complex dissipative standard system
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 4, 465 476 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.4.3 Control and synchronization of Julia sets of the complex dissipative standard system
More informationADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS
ADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM
International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT), Vol.1, No., June 011 ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM Sundarapandian Vaidyanathan
More informationLotka Volterra Predator-Prey Model with a Predating Scavenger
Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics
More informationChaos suppression of uncertain gyros in a given finite time
Chin. Phys. B Vol. 1, No. 11 1 1155 Chaos suppression of uncertain gyros in a given finite time Mohammad Pourmahmood Aghababa a and Hasan Pourmahmood Aghababa bc a Electrical Engineering Department, Urmia
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationInverse optimal control of hyperchaotic finance system
ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 10 (2014) No. 2, pp. 83-91 Inverse optimal control of hyperchaotic finance system Changzhong Chen 1,3, Tao Fan 1,3, Bangrong
More informationChaos Control of the Chaotic Symmetric Gyroscope System
48 Chaos Control of the Chaotic Symmetric Gyroscope System * Barış CEVHER, Yılmaz UYAROĞLU and 3 Selçuk EMIROĞLU,,3 Faculty of Engineering, Department of Electrical and Electronics Engineering Sakarya
More informationA New Finance Chaotic Attractor
ISSN 1749-3889(print),1749-3897(online) International Journal of Nonlinear Science Vol. 3 (2007) No. 3, pp. 213-220 A New Finance Chaotic Attractor Guoliang Cai +1,Juanjuan Huang 1,2 1 Nonlinear Scientific
More informationStability and hybrid synchronization of a time-delay financial hyperchaotic system
ISSN 76-7659 England UK Journal of Information and Computing Science Vol. No. 5 pp. 89-98 Stability and hybrid synchronization of a time-delay financial hyperchaotic system Lingling Zhang Guoliang Cai
More informationChapter 3. Gumowski-Mira Map. 3.1 Introduction
Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here
More informationBifurcation and chaos in simple jerk dynamical systems
PRAMANA c Indian Academy of Sciences Vol. 64, No. 1 journal of January 2005 physics pp. 75 93 Bifurcation and chaos in simple jerk dynamical systems VINOD PATIDAR and K K SUD Department of Physics, College
More informationChaos, Solitons and Fractals
Chaos, Solitons and Fractals 41 (2009) 962 969 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos A fractional-order hyperchaotic system
More informationA FEASIBLE MEMRISTIVE CHUA S CIRCUIT VIA BRIDGING A GENERALIZED MEMRISTOR
Journal of Applied Analysis and Computation Volume 6, Number 4, November 2016, 1152 1163 Website:http://jaac-online.com/ DOI:10.11948/2016076 A FEASIBLE MEMRISTIVE CHUA S CIRCUIT VIA BRIDGING A GENERALIZED
More informationGeneralized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems
Generalized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems Yancheng Ma Guoan Wu and Lan Jiang denotes fractional order of drive system Abstract In this paper a new synchronization
More informationA MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS
International Journal of Bifurcation and Chaos, Vol. 18, No. 5 (2008) 1567 1577 c World Scientific Publishing Company A MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS ZERAOULIA ELHADJ Department
More informationNew communication schemes based on adaptive synchronization
CHAOS 17, 0114 2007 New communication schemes based on adaptive synchronization Wenwu Yu a Department of Mathematics, Southeast University, Nanjing 210096, China, Department of Electrical Engineering,
More informationAnti-synchronization Between Coupled Networks with Two Active Forms
Commun. Theor. Phys. 55 (211) 835 84 Vol. 55, No. 5, May 15, 211 Anti-synchronization Between Coupled Networks with Two Active Forms WU Yong-Qing ( ï), 1 SUN Wei-Gang (êå ), 2, and LI Shan-Shan (Ó ) 3
More informationADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR Dr. SR Technical University Avadi, Chennai-600 062,
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department
More informationResearch Article Adaptive Control of Chaos in Chua s Circuit
Mathematical Problems in Engineering Volume 2011, Article ID 620946, 14 pages doi:10.1155/2011/620946 Research Article Adaptive Control of Chaos in Chua s Circuit Weiping Guo and Diantong Liu Institute
More informationDynamics of the Fractional-order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation
IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1 Dynamics of the Fractional-order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation Shaobo He, Kehui Sun, and Huihai Wang Abstract Dynamics
More informationDynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd
More informationMulti-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function
electronics Article Multi-Scroll Chaotic Attractors in SC-CNN via Hyperbolic Tangent Function Enis Günay, * and Kenan Altun ID Department of Electrical and Electronics Engineering, Erciyes University,
More information3. Controlling the time delay hyper chaotic Lorenz system via back stepping control
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol 10, No 2, 2015, pp 148-153 Chaos control of hyper chaotic delay Lorenz system via back stepping method Hanping Chen 1 Xuerong
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationLyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops
Chin. Phys. B Vol. 20 No. 4 (2011) 040505 Lyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops Li Qun-Hong( ) and Tan Jie-Yan( ) College of Mathematics
More informationExperimental and numerical realization of higher order autonomous Van der Pol-Duffing oscillator
Indian Journal of Pure & Applied Physics Vol. 47, November 2009, pp. 823-827 Experimental and numerical realization of higher order autonomous Van der Pol-Duffing oscillator V Balachandran, * & G Kandiban
More informationSynchronization of a General Delayed Complex Dynamical Network via Adaptive Feedback
Synchronization of a General Delayed Complex Dynamical Network via Adaptive Feedback Qunjiao Zhang and Junan Lu College of Mathematics and Statistics State Key Laboratory of Software Engineering Wuhan
More information