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Control and synchronize a novel hyperchaotic system
Applied Mathematics and Computation 216 (2010) 276–284
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Control and synchronize a novel hyperchaotic system Congxu Zhu School of Information Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Keywords: Hyperchaos control Hybrid projective synchronization Scalar controller Nonlinear feedback controller Lyapunov’s direct method Routh–Hurwitz criterion
a b s t r a c t The control and hybrid projective synchronization (HPS) strategies for a novel hyperchaotic system are investigated. Firstly, the novel hyperchaotic system is controlled to the unsteady equilibrium point or limit cycle via only one scalar controller which includes two state variables. Secondly, based on Lyapunov’s direct method HPS between two novel hyperchaotic systems is studied. A new nonlinear feedback vector controller is designed to guarantee HPS, which can be simplified ulteriorly into a single scalar controller to achieve complete synchronization between two novel hyperchaotic systems. Finally, numerical simulations are given to verify the effectiveness of these strategies. The proposed methods have certain significances for reducing the cost and complexity for controller implementation. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Chaos control and chaos synchronization play a very important role in the study of chaotic systems and have great significance in the application of chaos. Chaos is of fundamental concern in a wide range of fields, including secure communications, optics, chemical and biological systems, and so forth. The desirability, or otherwise, of chaos depends on the particular application. Sometimes chaos effect is undesirable in practice, and it restricts the operating range of many electronic and mechanic devices. In this case, therefore, it is necessary that the chaotic behavior should be controlled, e.g. by driving the chaotic attractors to a specific region of the system or by eliminating chaos entirely through the application of suitable control laws. While chaos is favorable in many other cases. Therefore, both chaos utilization and elimination are important depending on the specific applications. Chaos control is an effective method for both chaos utilization and elimination. Chaos synchronization means making two systems oscillate in a synchronized manner, and it has become a very important goal and a subject of much on-going research due to its important applications in secret communication. For the reason that chaos is very sensitive to its initial condition, chaos control and chaos synchronization were once believed to be impossible until the 1990s when Ott et al. developed the OGY method [1] to suppress chaos, Pecora and Carroll [2] introduced a method to synchronize two identical chaotic systems with different initial conditions. Since then, chaos control and chaos synchronization have attracted a great deal of attention from various fields during the last two decades. Research efforts have devoted to the chaos control and chaos synchronization problems in many physical systems [3–21]. Many potential applications have come true in secure communication [22–24]. However, most of the previous researches on chaos control and chaos synchronization was mostly focused on classical chaotic systems such as the unified system [5], Lorenz system [16–18], Chen system [16], Lü system [18], Chua circuit [19,20], and Genesio–Tesi system [21] and et al. Besides, most of research efforts mentioned above concentrated only on studying some conventional synchronization, such as complete synchronization, anti-synchronization, and projective synchronization. Furthermore, the controllers in many existing chaos control schemes are usually vectorial and more difficult to put into practice than a scalar one. In practical applications,
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the controllers to realize chaos control or synchronization must be simple, efficient and easy to implement. The less the number of the controllers designed in control process, the better the control method is in realization. Recently, Chen et al. [25] proposed a novel hyperchaotic system that takes the following form:
8 x_ ¼ aðy xÞ þ hyz; > > > < y_ ¼ cx dxz þ y þ w; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð1Þ
where x, y, z and w are state variables. a, b, c, d, g and h are positive constant parameters. System (1) has only one unstable equilibrium O(0, 0, 0, 0) and has bigger positive Lyapunov exponents than those already known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos [25]. When a = 35, b = 4.9, c = 25, d = 5, h = 35 and varying g from 10 to 126, or a = 35, c = 25, d = 5, g = 100, h = 35 and ranging b between 3.8 and 11, system (1) exhibits hyperchaos. The hyperchaotic attractor of system (1) with b = 4.9 and g = 100 is shown in Fig. 1. Due to the fact that this new system has bigger positive Lyapunov exponents and more complex dynamics within wide parameter ranges [25], it may have good application prospects. For example, it may be more useful in some fields such as information encryption and secure communication. So, the control and synchronization problems about the novel hyperchaotic system are worth further study. To our best knowledge, controlling chaos in hyperchaotic systems (1) via only one scalar controller and synchronizing systems (1) in the sense of hybrid projective synchronization (HPS) have not been reported. The purpose of this paper is the development of simple controllers for chaos control and hybrid projective synchronization for the novel hyperchaotic system. Single scalar controllers, which only include two states but makes it easy to suppress the system to its unstable equilibrium or a periodic orbit and quasi-periodic orbits, is firstly proposed. And then, based on Lyapunov’s direct method, a nonlinear feedback vector controller is derived to guarantee HPS, which can degenerate into a single scalar controller in the complete synchronization case. Finally, we present numerical simulation results to illustrate the effectiveness of the proposed control. The approaches in this paper have certain significances for reducing the cost and complexity for controller implementation. 2. Hyperchaos control via a single scalar controller 2.1. Equilibrium point control In this subsection, hyperchaotic system (1) will be suppressed to its unstable equilibrium O(0, 0, 0, 0) via only one scalar controller which include two state variables. To this end, we propose an adaptive feedback control scheme. In general, consider the following controlled hyperchaotic system:
X ¼ AX þ fðXÞ þ u;
ð2Þ T
where A ¼ ðai;j Þnn is the constant parameter matrix, X ¼ ½x1 ; x2 ; . . . ; xn is the state variable vector, f(X) is a nonlinear function vector, X0 ¼ 0 is the unstable equilibrium of the hyperchaotic system. u ¼ ½u1 ; u2 ; . . . ; un T is the vector controller. Suppose that aq;q P 0 for q ¼ q1 ; q2 ; . . . ; qm in matrix A, m is the number of non- negative parameters in matrix A. In our controller design scheme, the single scalar controller can be designed as
ui ¼ di;q ½kxq þ gðxq1 ; xq2 ; . . . ; xqm Þ; with the feedback gain k is adapted according to the following update law:
Fig. 1. The hyperchaotic attractor of system (1) with a = 35, c = 25, d = 5, h = 35, b = 4.9 and g = 100.
ð3Þ
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k_ ¼ r
qm X
x2q ;
kð0Þ ¼ 0;
r > 0;
ð4Þ
q¼q1
where dq;q ¼ 1 and di;q ¼ 0 for i–q; gðxq1 ; xq2 ; . . . ; xqm Þ is a scalar function to be suitably designed such that h P i 2 n d 1 2 1 < 0 for a sufficiently large constant k . The value of q and the form of function g() depends on i¼1 xi þ 2r ðk k Þ dt 2 the particular structure of the specific hyperchaotic system. For the particular structure of the novel hyperchaotic system (1), the controlled system is considered as follows:
8 x_ ¼ aðy xÞ þ hyz; > > > < y_ ¼ cx dxz þ y þ w þ u2 ; > _ ¼ xy bz; z > > : _ ¼ gy; w
ð5Þ
where u2 is a scalar controller to be designed for the purpose of controlling system (5) converge to its unstable equilibrium O(0, 0, 0, 0) with arbitrary initial conditions. Our goal is how to realize the globally asymptotical stabilization of the controlled system (5) and the objective is to design a controller to make the controlled system (5) asymptotically and globally stable at the origin, i.e. limt!1 kXðtÞk ¼ 0, where X(t) = [x, y, z, w]T. We choose the controller as follows:
u2 ¼ ky þ ðg 1Þw;
ð6Þ
where k is the feedback gain which is different from the usual linear feedback method, it is in the sense that k is adapted according to the following update law:
k_ ¼ rðy2 þ w2 Þ;
r > 0:
kð0Þ ¼ 0;
ð7Þ
Since system (1) is chaotic, the variables x, y, z and w must be bounded. Suppose that zm is the upper bound of jzj and consider the system parameters are positive constants, then we have the following theorem. Theorem 1. The controlled hyperchaotic system (5) will asymptotically and globally converge to the unstable equilibrium point O(0, 0, 0, 0) under the controller (6) with the update law (7). Proof Let the systems (5) and (7) be the augment system. Introducing a positive definite Lyapunov function as the following form:
VðtÞ ¼
1 2 1 ðk k Þ2 ; ðx þ y2 þ z2 þ w2 Þ þ 2 2r
ð8Þ
where k is a sufficient large constant to be determined. Differentiating V(t) with respect to time t along the solution of system (5), we can obtain
1 _ _ þ ðk k Þk_ VðtÞ ¼ xx_ þ yy_ þ zy_ þ ww
r
¼ xðay ax þ hyzÞ þ yðcx dxz þ y þ gw kyÞ þ zðxy bzÞ þ wðgyÞ þ ðk k Þðy2 þ w2 Þ 2
¼ ax2 ðk 1Þy2 bz ðk kÞw2 þ ½a þ c þ ðh d þ 1Þzxy ¼ XT PX; where
2
x
3
6y 6 X¼6 4z
7 7 7; 5
w
0
0
0
1
B g k 1 0 B P¼B @ 0 0 b
0 0
C C C; A
a
0
g
0
1 2
g ¼ ½a þ c þ ðh d þ 1Þz:
ð9Þ
ð10Þ
0 k k
To ensure that the origin of controlled system (5) is asymptotically stable, the symmetric matrix P should be positive definite, that is, P must satisfy the following conditions:
8 a > 0; > > > < aðk 1Þ g2 > 0; > abðk 1Þ bg2 > 0; > > : ðk kÞ½abðk 1Þ bg2 > 0;
ð11Þ
where a > 0; b > 0; c > 0; d > 0; h > 0 and h > d. If we denote the upper bound of g by gm , then gm ¼ ½a þ c þ ðh dþ 2 1Þzm =2. It can be see from Eq. (11) that all the conditions would be satisfied as long as we 2have k > gm =a þ 1 and k > k. To satisfy this condition, we can choose the undetermined constant k as k > max gm =a þ 1; k so that V(t) is _ positive definite and VðtÞ is negative semidefinite. It follows that the equilibrium point (x = 0, y = 0, z = 0, w = 0, k k* = 0) of the augment system is uniformly stable, i.e. x; y; z; w 2 L1 and k k 2 L1 .
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Since V is positive definite, and V_ is negative semidefinite, then, we have
Z
t
kmin ðPÞkXk2 dt 6
0
Z
t
XT PXdt ¼
0
Z
t
_ ðVÞdt ¼ Vð0Þ VðtÞ 6 Vð0Þ:
0
where kmin ðPÞ is the smallest eigenvalue of the positive matrix P. Thus, we can easily know that x; y; z; w 2 L2 . Next by Bar_ y; _ z_ ; w _ 2 L1 , which in turn implies x, y, z, w ? 0 as t ! 1. This balat’s lemma, for any initial condition, Eq. (5) implies that x; implies that the controlled hyperchaotic system (5) will asymptotically and globally converge to the unstable equilibrium point O(0, 0, 0, 0) under the controller (6) with the update law (7). This completes the proof. h Remark 1. For a general controlled hyperchaotic system X = AX + f(X) + u, where A ¼ ðai;j Þnn is the constant parameter matrix, X ¼ ½x1 ; x2 ; . . . ; xn T is the state variable vector, If 8j ¼ q1 ; q2 ; . . . ; qm 2Z . 9ajj P 0, then the system will be controlled to the unstable equilibrium X0 ¼ 0 via one single controller kxq with a feedback gain k update law including m state variables. In order to demonstrate and verify the performance of the proposed method, a numerical simulation is presented to illustrate the theoretical analysis. In the following numerical simulation, the system parameters (a, b, c, d, g, h) are chosen to be (35, 4.9, 25, 5, 10, 35) with which the system (1) behaves hyperchaotically. The initial conditions are set to be x(0) = 10, y(0) = 5, z(0) = 5, w(0) = 10. The initial condition of the adaptive feedback gain is set to be k(0) = 0, and the constant coefficient r is set to be 1. Fig. 2 shows the time responses of states x, y, z, w for the controlled system (5). From Fig. 2, we can conclude that the novel hyperchaotic system is suppressed to its unstable equilibrium O(0, 0, 0, 0) under the single scalar controller u = ky + (g 1)w with the feedback gain adaptive update law k_ ¼ y2 þ w2 . 2.2. Suppression of hyperchaos to periodic and quasi-periodic orbits In this subsection, hyperchaotic system (1) will be controlled to a periodic orbit and quasi-periodic orbit via only one scalar controller. For the particular structure of the novel hyperchaotic system (1), consider the following controlled hyperchaotic system:
8 x_ ¼ 35ðy xÞ þ 35yz; > > > < y_ ¼ 25x 5xz þ y þ w kðx þ yÞ; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð12Þ
where k(x + y) is the linear feedback scalar controller, the control gain k will be determined to suppress hyperchaos. Constants b and g are two positive system parameters which will cause system (12) in hyperchaotic states within wide parameter ranges if no control is applied. In Ref. [26], Liu derived an equivalent implicit algorithm criterion of Hopf bifurcation on the basis of the Routh–Hurwitz stability criterion, which is stated in terms of the coefficients of the characteristic equations instead of the traditional Hopf bifurcation criterion which is stated in terms of the properties of eigenvalues. By using the implicit algorithm criterion one can determine the parameter k ¼ k0 in which Hopf bifurcation occurs. The Jacobi matrix of the system (12) is
0
35
35
0
0
1
B 25 k 1 k 0 1 C B C J¼B C @ 0 0 b 0 A 0
g
0
0
Fig. 2. The states of system (2) converge to O(0, 0, 0, 0) when x0 = 10, y0 = 5, z0 = 5, w0 = 10, and k(0) = 0.
ð13Þ
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then the characteristic equation of matrix (13) is
ðk þ bÞðp0 þ p1 k þ p2 k2 þ p3 k3 Þ ¼ 0; where p0 ¼ 35; p1 ¼ 70k þ g 910; p3 ¼ 1. Because k1 ¼ b < 0, according to the simple Hopf bifurcation criterion [26], when the following conditions are satisfied:
p1 > 0;
p2 > 0;
p3 > 0;
p1 p2 p0 ¼ 0;
dðp1 p2 p0 Þ jk¼k0 – 0: dk
Then the Hopf bifurcation occurs at k ¼ k0 . Thus, the critical parameter is k0 ¼ 11:5605 when g ¼ 100. Choosing the control gain k ¼ k0 ¼ 11:5605 and the parameters b = 4.9, g = 100, we can obtain the Lyapunov exponents of system (12) by using the Wolf method [27], the four Lyapunov exponents are 0, 0.6658, 0.6669 and 47.7893. From the Lyapunov exponents above, one can judge that the controller k0 ðx þ yÞ can stabilize the hyperchaotic system (12) to a periodic orbit. The periodic orbit to which the system (12) is stabilized is shown in Fig. 3(a). Fig. 3(b) shows the time series corresponding to Fig. 3(a). As k decreased further, system (12) will work in a quasi-periodic state. Choosing the control gain k = 9.12, we find that the four Lyapunov exponents of the controlled system (12) are 0, 0, 0.0505 and 46.9908. Thus, system (12) can be controlled to a quasi-periodic orbit with the gain k = 9.12 (see Fig. 4). 3. Hybrid projective synchronization (HPS) 3.1. HPS via nonlinear control Consider two hyperchaotic systems given by the following Eqs. (14) and (15), respectively:
x_ ¼ f 1 ðxÞ; y_ ¼ f 2 ðyÞ þ uðx; yÞ; n
n
ð14Þ ð15Þ n
where x, y 2 R , f1, f2: R ? R . Assume that Eq. (14) is the drive system, Eq. (15) is the response system, and u(x, y) is the nonlinear control vector. If 8xðt 0 Þ; yðt 0 Þ 2 Rn ; limt!1 jyi ðtÞ ai xi ðtÞj ¼ 0; i ¼ 1; 2; . . . ; n, then the response and drive systems are said to be in HPS. In particular, the drive–response systems achieve complete synchronization when all values of ai are equal to 1. Further, if all values of ai are equal to 1, then the two systems are said to be in anti-synchronization. Our purpose herein is to achieve HPS of two identical hyperchaotic systems based on Lyapunov’s direct method. For the novel hyperchaotic system (1), the drive and response systems are defined as the following Eqs. (16) and (17), respectively:
8 x_ 1 ¼ aðy1 x1 Þ þ hy1 z1 ; > > > < y_ 1 ¼ cx1 dx1 z1 þ y1 þ w1 ; > z_ 1 ¼ x1 y1 bz1 ; > > : _ 1 ¼ gy1 ; w 8 _ x > 2 ¼ aðy2 x2 Þ þ hy2 z2 þ u1 ; > > < y_ ¼ cx dx z þ y þ w þ u ; 2 2 2 2 2 2 2 > _ 2 ¼ x2 y2 bz2 þ u3 ; z > > : _ 2 ¼ gy2 þ u4 ; w
ð16Þ
ð17Þ
where [u1, u2, u3, u4] is the nonlinear vector controller by using it two hyperchaotic systems can be synchronized in the sense of HPS, i.e.,
Fig. 3. A periodic orbit of the controlled system (9) with the control gain k = 11.5605.
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Fig. 4. A quasi-periodic state of the controlled system (9) with the control gain k = 9.12.
8 lim kx2 a1 x1 k ¼ 0; > > t!1 > > > > < lim ky2 a2 y1 k ¼ 0; t!1
ð18Þ
> lim kz2 a3 z1 k ¼ 0; > > t!1 > > > : lim kw2 a4 w1 k ¼ 0: t!1
Subtracting Eq. (16) from Eq. (17) yields error dynamical system between Eqs. (16) and (17):
8 e_ 1 > > > < e_ 2 > e > _3 > : e_ 4
¼ aðe2 e1 Þ þ aða2 a1 Þy1 þ hðy2 z2 a1 y1 z1 Þ þ u1 ; ¼ ce1 þ e2 þ e4 þ cða1 a2 Þx1 þ ða4 a2 Þw1 dðx2 z2 a2 x1 z1 Þ þ u2 ; ¼ be3 þ ðx2 y2 a3 x1 y1 Þ þ u3 ;
ð19Þ
¼ ge2 gða2 a4 Þy1 þ u4 ;
where e1 ¼ x2 a1 x1 ; e2 ¼ y2 a2 y1 ; e3 ¼ z2 a3 z1 , and e4 ¼ w2 a4 w1 . a, b, c, d, g, and h are positive system constant parameters. Here, our goal is to make synchronization between the two hyperchaotic systems by using a nonlinear vector controller [u1, u2, u3, u4], i.e., limt!1 keðtÞk ¼ 0, where e(t) = [e1, e2, e3, e4]T. For the two hyperchaotic systems without control (ui = 0, i = 1, 2, 3, 4), if the initial condition (x1(0), y1(0), z1(0), w1(0)) – (x2(0), y2(0), z2(0), w2(0)), the trajectories of the two identical systems will quickly separate each other and become irrelevant. However, for the two controlled hyperchaotic systems, the two systems will approach synchronization for any initial condition by appropriate control vector. For this end, we propose the following control law for the system (17):
8 u1 > > > < u2 > u3 > > : u4
¼ aða2 a1 Þy1 hða2 a3 a1 Þy1 z1 ; ¼ cða1 a2 Þx1 ða4 a2 Þw1 þ dða1 a3 a2 Þx1 z1 ; ¼ ða1 a2 a3 Þx1 y1 e1 e2 2a2 y1 e1 ke3 ;
ð20Þ
¼ gða2 a4 Þy1 ða2 a4 Þ2 e4 ;
where k is the feedback gain which is different from the usual linear feedback method, it is in the sense that k is adapted according to the following update law:
k_ ¼ r e22 þ e24 ;
kð0Þ ¼ 0;
r > 0:
ð21Þ
Then, we have the following main result. Theorem 2. For any initial conditions, the two systems (16) and (17) are globally and asymptotically hybrid projective synchronized by nonlinear feedback controller (20) with the update law (21). Proof Applying the controller of Eq. (20) to Eq. (19) yields the resulting error dynamics as follows:
8 e_ 1 > > > < e_ 2 > e_ 3 > > : e_ 4
¼ ae2 ae1 þ hðe2 e3 þ a3 z1 e2 þ a2 y1 e3 Þ; ¼ ce1 þ e2 þ e4 dðe1 e3 þ a1 x1 e3 þ a3 z1 e1 Þ; ¼ be3 þ a1 x1 e2 a2 y1 e1 ke3 ; ¼ ge2 ða2 a4 Þ2 e4 :
ð22Þ
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Consider a Lyapunov function candidate as
V¼
2 1 1 2 1 2 1 1 ðk k : e1 þ e2 þ e23 þ e24 Þ þ 2 h d gd 2r
ð23Þ
It is clear that the Lyapunov function, V, is a positive definite function. Then the time derivative of V along the solution of error dynamical system (22) gives
1 1 1 1 V_ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ ðk k Þk_ h d gd r " # a 1 2 ða2 a4 Þ2 ¼ e21 k k e2 ðk þ bÞe23 k þ k e24 ¼ eT Pe; h d gd
ð24Þ
where
3 e1 6e 7 6 27 e ¼ 6 7; 4 e3 5 2
e4
0
a=h ða=h þ c=dÞ=2 0 B ða=h þ c=dÞ=2 ðk k 1=dÞ 0 B p¼B @ 0 0 kþb 0
0
0
1
0 0 0
k þ ða2 a4 Þ2 =ðgdÞ k
C C C: A
Since a > 0, b > 0, d > 0, h > 0, obviously, when k > k þ hða=h þ c=dÞ2 =ð4aÞ þ 1=d, and k > k ða2 a4 Þ2 =ðgdÞ. Then the symmetric matrix P is positive, and therefore V_ is negative definite. Based on Lyapunov’s direct method, this translates to limt!1 kei ðtÞk ¼ 0; i ¼ 1; 2; 3; 4. Thus the response and drive systems are globally and asymptotically hybrid projective synchronized. This completes the proof. h Remark 2. For the complete synchronization case ðai ¼ 1; i ¼ 1; 2; 3; 4Þ, the controller can be simplified ulteriorly into: u1 = 0, u2 = 0, u3 = e1e2 2y1e1 ke3, u4 = 0. Namely, the complete synchronization for the novel hyperchaotic system can be achieved via a single scalar controller u3 = e1e2 2y1e1 ke3. Remark 3. The convergence rate of error signal e3 can be regulated by adjusting the constant coefficient
Fig. 5. Synchronization errors e1, e2, e3, and e4 vs. time t with a1 ¼ 1; a2 ¼ 1; a3 ¼ 4, and a4 ¼ 0:5.
r.
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Fig. 6. State trajectories of drive and response systems. (a) Signals x1 and x2 with a1 ¼ 1, (b) signals y1 and y2 with a2 ¼ 1, (c) signals z1 and z2 with a3 ¼ 4, and (d) signals w1 and w2 with a4 ¼ 0:5.
3.2. Numerical simulations In this subsection, numerical simulations are done with Matlab to demonstrate the effectiveness of the proposed HPS scheme. In numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. The initial states of the drive and response systems are chosen as follows: x1(0) = 10, y1(0) = 1, z1(0) = 5, w1(0) = 18; x2(0) = 8, y2(0) = 3, z2(0) = 18, w2(0) = 7. The hybrid projective parameters are chosen as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ 4 and a4 ¼ 0:5. Thus the initial states of the error system (16) are e1(0) = 2, e2(0) = 2, e3(0) = 2, and e4(0) = 2. The system parameters are chosen as follows: a = 35, b = 4.9, c = 25, d = 5, h = 35 and g = 15. We choose the constant coefficient r ¼ 10. The hybrid projective synchronization errors of systems (16) and (17) are displayed in Fig. 5. From these figures, One can see that all of the synchronization errors converge to 0. For further observations, the state trajectories of the drive and response systems are depicted in Fig. 6. As can be seen, variables x2 vs. x1 display a synchronization phenomenon, variables y2 vs. y1 show anti-synchronization behavior, variable z2 finally converges to four times the value of z1; and variable w2 converges to half of the value of w1. It is clear that the two hyperchaotic systems of (16) and (17) can achieve HPS. 4. Conclusions In this paper, the control and hybrid projective synchronization problems for a novel hyperchaotic system are investigated. Based on Lyapunov’s direct method and applying Routh–Hurwitz criterion, we investigate the chaos control problems at first. Single scalar controllers including two state variables are designed to control the equilibrium point and bifurcation, and detailed controllability conditions are also presented. Strict theoretical proof is put forward and numerical simulation results indicate that the controllers can effectively guide the system trajectories to equilibrium point and periodic orbits. Furthermore, hybrid projective synchronization between two identical new hyperchaotic systems is studied using nonlinear feedback control. A nonlinear vector controller is designed according to Lyapunov’s direct method to guarantee HPS, which includes complete synchronization, anti-synchronization, and projective synchronization. The nonlinear vector controller can be simplified ulteriorly into a single scalar controller u3 = e1e2 2y1e1 ke3 to achieve complete synchronization between two novel hyperchaotic systems. Numerical simulations verified the effectiveness of these strategies. Therefore, the proposed methods have certain significances for reducing the cost and complexity for controller implementation.
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The control methods or ideas in this paper can be extended to control and synchronize the other chaotic systems. However, for the different structure of different hyperchaotic system, the specific strategies and controller forms are different. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 60902044 and the First Batch Fostering Foundation in Training Graduate Students of Central South University. References [1] E. Ott, G. Grebogi, J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. [2] L.M. Pecora, T.L. Carroll, Synchronization in chaotic system, Phys. Rev. Lett. 64 (1990) 821–824. [3] Z.M. Ge, C.H. Yang, The generalized synchronization of a Quantum-CNN chaotic oscillator with different order systems, Chaos Solitons Fractals 35 (2008) 980–990. [4] Z.M. Ge, G.H. Lin, The complete lag, and anticipated synchronization of a BLDCM chaotic system, Chaos Solitons Fractals 34 (2007) 740–764. [5] Y.-C. Hung, T.-L. Liao, J.-J. Yan, Adaptive variable structure control for chaos suppression of unified chaotic systems, Appl. Math. Comput. 209 (2009) 391–398. [6] Z.-M. Ge, J.-K. Lee, Chaos synchronization and parameter identification for gyroscope system, Appl. Math. Comput. 163 (2005) 667–682. [7] M.T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput. 135 (2003) 113–128. [8] J.H. Park, D.H. Ji, S.C. Won, S.M. Lee, H1 synchronization of time-delayed chaotic systems, Appl. Math. Comput. 204 (2008) 170–177. [9] X. Zhou, Y. Wu, Y. Li, H. Xue, Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters, Appl. Math. Comput. 203 (2008) 80–85. [10] M.T. Yassen, On hyperchaos synchronization of a hyperchaotic Lü system, Nonlinear Anal-Theor 68 (2008) 3592–3600. [11] J.H. Park, Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos Solitons Fract. 26 (2006) 959–964. [12] C.-H. Chen, L.-J. Sheu, H.-K. Chen, et al, A new hyper-chaotic system and its synchronization, Nonlinear Anal.: Real World 10 (2009) 2088–2096. [13] C. Zhu, Z. Chen, Feedback control strategies for the Liu chaotic system, Phys. Lett. A 372 (2008) 4033–4036. [14] C. Zhu, Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system, Nonlinear Anal.-Theor. 71 (2009) 2441–2446. [15] C. Zhu, Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters, Appl. Math. Comput. 215 (2009) 557– 561. [16] H.-T. Yau, J.-J. Yan, Chaos synchronization of different chaotic systems subjected to input nonlinearity, Appl. Math. Comput. 197 (2008) 775–788. [17] F. Chen, L. Chen, W. Zhang, Stabilization of parameters perturbation chaotic system via adaptive backstepping technique, Appl. Math. Comput. 200 (2008) 101–109. [18] X.R.g. Shi, Z.L. Wang, Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters, Appl. Math. Comput. 215 (2009) 1711–1717. [19] D.H. Ji, J.H. Park, S.C. Won, Master slave synchronization of Lur’e systems with sector and slope restricted nonlinearities, Phys. Lett. A 373 (2009) 1044– 1050. [20] J.H. Park, D.H. Ji, S.C. Won, S.M. Lee, S.J. Choi, H1 control of Lur’e sysetms with sector and slope restricted nonlinearities, Phys. Lett. A 373 (2009) 3734– 3740. [21] J.H. Park, O.M. Kwon, S.M. Lee, LMI optimization approach to stabilization of Genesio–Tesi chaotic system via dynamic controller, Appl. Math. Comput. 196 (2008) 200–206. [22] K. Murall, M. Lakshmanan, Secure communication using a compound signal from generalized synchronizable chaotic systems, Phys. Lett. A 241 (1998) 303–310. [23] T. Yang, A survey of chaotic secure communication systems, Int. J. Comput. Cognition 2 (2004) 81–130. [24] X. Wang, Q. Zhao, M. Wang, Generalized synchronization of different dimensional neural networks and its applications in secure communication, Mod. Phys. Lett. B 22 (2008) 2077–2084. [25] Z. Chen, Y. Yang, G. Qi, Z. Yuan, A novel hyperchaos system only with one equilibrium, Phys. Lett. A 360 (2007) 696–701. [26] W.M. Liu, Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl. 182 (1994) 250–256. [27] A. Wolf, J. Swift, H. Swinney, et al, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285–317.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Control and synchronize a novel hyperchaotic system Congxu Zhu School of Information Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Keywords: Hyperchaos control Hybrid projective synchronization Scalar controller Nonlinear feedback controller Lyapunov’s direct method Routh–Hurwitz criterion
a b s t r a c t The control and hybrid projective synchronization (HPS) strategies for a novel hyperchaotic system are investigated. Firstly, the novel hyperchaotic system is controlled to the unsteady equilibrium point or limit cycle via only one scalar controller which includes two state variables. Secondly, based on Lyapunov’s direct method HPS between two novel hyperchaotic systems is studied. A new nonlinear feedback vector controller is designed to guarantee HPS, which can be simplified ulteriorly into a single scalar controller to achieve complete synchronization between two novel hyperchaotic systems. Finally, numerical simulations are given to verify the effectiveness of these strategies. The proposed methods have certain significances for reducing the cost and complexity for controller implementation. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Chaos control and chaos synchronization play a very important role in the study of chaotic systems and have great significance in the application of chaos. Chaos is of fundamental concern in a wide range of fields, including secure communications, optics, chemical and biological systems, and so forth. The desirability, or otherwise, of chaos depends on the particular application. Sometimes chaos effect is undesirable in practice, and it restricts the operating range of many electronic and mechanic devices. In this case, therefore, it is necessary that the chaotic behavior should be controlled, e.g. by driving the chaotic attractors to a specific region of the system or by eliminating chaos entirely through the application of suitable control laws. While chaos is favorable in many other cases. Therefore, both chaos utilization and elimination are important depending on the specific applications. Chaos control is an effective method for both chaos utilization and elimination. Chaos synchronization means making two systems oscillate in a synchronized manner, and it has become a very important goal and a subject of much on-going research due to its important applications in secret communication. For the reason that chaos is very sensitive to its initial condition, chaos control and chaos synchronization were once believed to be impossible until the 1990s when Ott et al. developed the OGY method [1] to suppress chaos, Pecora and Carroll [2] introduced a method to synchronize two identical chaotic systems with different initial conditions. Since then, chaos control and chaos synchronization have attracted a great deal of attention from various fields during the last two decades. Research efforts have devoted to the chaos control and chaos synchronization problems in many physical systems [3–21]. Many potential applications have come true in secure communication [22–24]. However, most of the previous researches on chaos control and chaos synchronization was mostly focused on classical chaotic systems such as the unified system [5], Lorenz system [16–18], Chen system [16], Lü system [18], Chua circuit [19,20], and Genesio–Tesi system [21] and et al. Besides, most of research efforts mentioned above concentrated only on studying some conventional synchronization, such as complete synchronization, anti-synchronization, and projective synchronization. Furthermore, the controllers in many existing chaos control schemes are usually vectorial and more difficult to put into practice than a scalar one. In practical applications,
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.053
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277
the controllers to realize chaos control or synchronization must be simple, efficient and easy to implement. The less the number of the controllers designed in control process, the better the control method is in realization. Recently, Chen et al. [25] proposed a novel hyperchaotic system that takes the following form:
8 x_ ¼ aðy xÞ þ hyz; > > > < y_ ¼ cx dxz þ y þ w; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð1Þ
where x, y, z and w are state variables. a, b, c, d, g and h are positive constant parameters. System (1) has only one unstable equilibrium O(0, 0, 0, 0) and has bigger positive Lyapunov exponents than those already known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos [25]. When a = 35, b = 4.9, c = 25, d = 5, h = 35 and varying g from 10 to 126, or a = 35, c = 25, d = 5, g = 100, h = 35 and ranging b between 3.8 and 11, system (1) exhibits hyperchaos. The hyperchaotic attractor of system (1) with b = 4.9 and g = 100 is shown in Fig. 1. Due to the fact that this new system has bigger positive Lyapunov exponents and more complex dynamics within wide parameter ranges [25], it may have good application prospects. For example, it may be more useful in some fields such as information encryption and secure communication. So, the control and synchronization problems about the novel hyperchaotic system are worth further study. To our best knowledge, controlling chaos in hyperchaotic systems (1) via only one scalar controller and synchronizing systems (1) in the sense of hybrid projective synchronization (HPS) have not been reported. The purpose of this paper is the development of simple controllers for chaos control and hybrid projective synchronization for the novel hyperchaotic system. Single scalar controllers, which only include two states but makes it easy to suppress the system to its unstable equilibrium or a periodic orbit and quasi-periodic orbits, is firstly proposed. And then, based on Lyapunov’s direct method, a nonlinear feedback vector controller is derived to guarantee HPS, which can degenerate into a single scalar controller in the complete synchronization case. Finally, we present numerical simulation results to illustrate the effectiveness of the proposed control. The approaches in this paper have certain significances for reducing the cost and complexity for controller implementation. 2. Hyperchaos control via a single scalar controller 2.1. Equilibrium point control In this subsection, hyperchaotic system (1) will be suppressed to its unstable equilibrium O(0, 0, 0, 0) via only one scalar controller which include two state variables. To this end, we propose an adaptive feedback control scheme. In general, consider the following controlled hyperchaotic system:
X ¼ AX þ fðXÞ þ u;
ð2Þ T
where A ¼ ðai;j Þnn is the constant parameter matrix, X ¼ ½x1 ; x2 ; . . . ; xn is the state variable vector, f(X) is a nonlinear function vector, X0 ¼ 0 is the unstable equilibrium of the hyperchaotic system. u ¼ ½u1 ; u2 ; . . . ; un T is the vector controller. Suppose that aq;q P 0 for q ¼ q1 ; q2 ; . . . ; qm in matrix A, m is the number of non- negative parameters in matrix A. In our controller design scheme, the single scalar controller can be designed as
ui ¼ di;q ½kxq þ gðxq1 ; xq2 ; . . . ; xqm Þ; with the feedback gain k is adapted according to the following update law:
Fig. 1. The hyperchaotic attractor of system (1) with a = 35, c = 25, d = 5, h = 35, b = 4.9 and g = 100.
ð3Þ
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C. Zhu / Applied Mathematics and Computation 216 (2010) 276–284
k_ ¼ r
qm X
x2q ;
kð0Þ ¼ 0;
r > 0;
ð4Þ
q¼q1
where dq;q ¼ 1 and di;q ¼ 0 for i–q; gðxq1 ; xq2 ; . . . ; xqm Þ is a scalar function to be suitably designed such that h P i 2 n d 1 2 1 < 0 for a sufficiently large constant k . The value of q and the form of function g() depends on i¼1 xi þ 2r ðk k Þ dt 2 the particular structure of the specific hyperchaotic system. For the particular structure of the novel hyperchaotic system (1), the controlled system is considered as follows:
8 x_ ¼ aðy xÞ þ hyz; > > > < y_ ¼ cx dxz þ y þ w þ u2 ; > _ ¼ xy bz; z > > : _ ¼ gy; w
ð5Þ
where u2 is a scalar controller to be designed for the purpose of controlling system (5) converge to its unstable equilibrium O(0, 0, 0, 0) with arbitrary initial conditions. Our goal is how to realize the globally asymptotical stabilization of the controlled system (5) and the objective is to design a controller to make the controlled system (5) asymptotically and globally stable at the origin, i.e. limt!1 kXðtÞk ¼ 0, where X(t) = [x, y, z, w]T. We choose the controller as follows:
u2 ¼ ky þ ðg 1Þw;
ð6Þ
where k is the feedback gain which is different from the usual linear feedback method, it is in the sense that k is adapted according to the following update law:
k_ ¼ rðy2 þ w2 Þ;
r > 0:
kð0Þ ¼ 0;
ð7Þ
Since system (1) is chaotic, the variables x, y, z and w must be bounded. Suppose that zm is the upper bound of jzj and consider the system parameters are positive constants, then we have the following theorem. Theorem 1. The controlled hyperchaotic system (5) will asymptotically and globally converge to the unstable equilibrium point O(0, 0, 0, 0) under the controller (6) with the update law (7). Proof Let the systems (5) and (7) be the augment system. Introducing a positive definite Lyapunov function as the following form:
VðtÞ ¼
1 2 1 ðk k Þ2 ; ðx þ y2 þ z2 þ w2 Þ þ 2 2r
ð8Þ
where k is a sufficient large constant to be determined. Differentiating V(t) with respect to time t along the solution of system (5), we can obtain
1 _ _ þ ðk k Þk_ VðtÞ ¼ xx_ þ yy_ þ zy_ þ ww
r
¼ xðay ax þ hyzÞ þ yðcx dxz þ y þ gw kyÞ þ zðxy bzÞ þ wðgyÞ þ ðk k Þðy2 þ w2 Þ 2
¼ ax2 ðk 1Þy2 bz ðk kÞw2 þ ½a þ c þ ðh d þ 1Þzxy ¼ XT PX; where
2
x
3
6y 6 X¼6 4z
7 7 7; 5
w
0
0
0
1
B g k 1 0 B P¼B @ 0 0 b
0 0
C C C; A
a
0
g
0
1 2
g ¼ ½a þ c þ ðh d þ 1Þz:
ð9Þ
ð10Þ
0 k k
To ensure that the origin of controlled system (5) is asymptotically stable, the symmetric matrix P should be positive definite, that is, P must satisfy the following conditions:
8 a > 0; > > > < aðk 1Þ g2 > 0; > abðk 1Þ bg2 > 0; > > : ðk kÞ½abðk 1Þ bg2 > 0;
ð11Þ
where a > 0; b > 0; c > 0; d > 0; h > 0 and h > d. If we denote the upper bound of g by gm , then gm ¼ ½a þ c þ ðh dþ 2 1Þzm =2. It can be see from Eq. (11) that all the conditions would be satisfied as long as we 2have k > gm =a þ 1 and k > k. To satisfy this condition, we can choose the undetermined constant k as k > max gm =a þ 1; k so that V(t) is _ positive definite and VðtÞ is negative semidefinite. It follows that the equilibrium point (x = 0, y = 0, z = 0, w = 0, k k* = 0) of the augment system is uniformly stable, i.e. x; y; z; w 2 L1 and k k 2 L1 .
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Since V is positive definite, and V_ is negative semidefinite, then, we have
Z
t
kmin ðPÞkXk2 dt 6
0
Z
t
XT PXdt ¼
0
Z
t
_ ðVÞdt ¼ Vð0Þ VðtÞ 6 Vð0Þ:
0
where kmin ðPÞ is the smallest eigenvalue of the positive matrix P. Thus, we can easily know that x; y; z; w 2 L2 . Next by Bar_ y; _ z_ ; w _ 2 L1 , which in turn implies x, y, z, w ? 0 as t ! 1. This balat’s lemma, for any initial condition, Eq. (5) implies that x; implies that the controlled hyperchaotic system (5) will asymptotically and globally converge to the unstable equilibrium point O(0, 0, 0, 0) under the controller (6) with the update law (7). This completes the proof. h Remark 1. For a general controlled hyperchaotic system X = AX + f(X) + u, where A ¼ ðai;j Þnn is the constant parameter matrix, X ¼ ½x1 ; x2 ; . . . ; xn T is the state variable vector, If 8j ¼ q1 ; q2 ; . . . ; qm 2Z . 9ajj P 0, then the system will be controlled to the unstable equilibrium X0 ¼ 0 via one single controller kxq with a feedback gain k update law including m state variables. In order to demonstrate and verify the performance of the proposed method, a numerical simulation is presented to illustrate the theoretical analysis. In the following numerical simulation, the system parameters (a, b, c, d, g, h) are chosen to be (35, 4.9, 25, 5, 10, 35) with which the system (1) behaves hyperchaotically. The initial conditions are set to be x(0) = 10, y(0) = 5, z(0) = 5, w(0) = 10. The initial condition of the adaptive feedback gain is set to be k(0) = 0, and the constant coefficient r is set to be 1. Fig. 2 shows the time responses of states x, y, z, w for the controlled system (5). From Fig. 2, we can conclude that the novel hyperchaotic system is suppressed to its unstable equilibrium O(0, 0, 0, 0) under the single scalar controller u = ky + (g 1)w with the feedback gain adaptive update law k_ ¼ y2 þ w2 . 2.2. Suppression of hyperchaos to periodic and quasi-periodic orbits In this subsection, hyperchaotic system (1) will be controlled to a periodic orbit and quasi-periodic orbit via only one scalar controller. For the particular structure of the novel hyperchaotic system (1), consider the following controlled hyperchaotic system:
8 x_ ¼ 35ðy xÞ þ 35yz; > > > < y_ ¼ 25x 5xz þ y þ w kðx þ yÞ; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð12Þ
where k(x + y) is the linear feedback scalar controller, the control gain k will be determined to suppress hyperchaos. Constants b and g are two positive system parameters which will cause system (12) in hyperchaotic states within wide parameter ranges if no control is applied. In Ref. [26], Liu derived an equivalent implicit algorithm criterion of Hopf bifurcation on the basis of the Routh–Hurwitz stability criterion, which is stated in terms of the coefficients of the characteristic equations instead of the traditional Hopf bifurcation criterion which is stated in terms of the properties of eigenvalues. By using the implicit algorithm criterion one can determine the parameter k ¼ k0 in which Hopf bifurcation occurs. The Jacobi matrix of the system (12) is
0
35
35
0
0
1
B 25 k 1 k 0 1 C B C J¼B C @ 0 0 b 0 A 0
g
0
0
Fig. 2. The states of system (2) converge to O(0, 0, 0, 0) when x0 = 10, y0 = 5, z0 = 5, w0 = 10, and k(0) = 0.
ð13Þ
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then the characteristic equation of matrix (13) is
ðk þ bÞðp0 þ p1 k þ p2 k2 þ p3 k3 Þ ¼ 0; where p0 ¼ 35; p1 ¼ 70k þ g 910; p3 ¼ 1. Because k1 ¼ b < 0, according to the simple Hopf bifurcation criterion [26], when the following conditions are satisfied:
p1 > 0;
p2 > 0;
p3 > 0;
p1 p2 p0 ¼ 0;
dðp1 p2 p0 Þ jk¼k0 – 0: dk
Then the Hopf bifurcation occurs at k ¼ k0 . Thus, the critical parameter is k0 ¼ 11:5605 when g ¼ 100. Choosing the control gain k ¼ k0 ¼ 11:5605 and the parameters b = 4.9, g = 100, we can obtain the Lyapunov exponents of system (12) by using the Wolf method [27], the four Lyapunov exponents are 0, 0.6658, 0.6669 and 47.7893. From the Lyapunov exponents above, one can judge that the controller k0 ðx þ yÞ can stabilize the hyperchaotic system (12) to a periodic orbit. The periodic orbit to which the system (12) is stabilized is shown in Fig. 3(a). Fig. 3(b) shows the time series corresponding to Fig. 3(a). As k decreased further, system (12) will work in a quasi-periodic state. Choosing the control gain k = 9.12, we find that the four Lyapunov exponents of the controlled system (12) are 0, 0, 0.0505 and 46.9908. Thus, system (12) can be controlled to a quasi-periodic orbit with the gain k = 9.12 (see Fig. 4). 3. Hybrid projective synchronization (HPS) 3.1. HPS via nonlinear control Consider two hyperchaotic systems given by the following Eqs. (14) and (15), respectively:
x_ ¼ f 1 ðxÞ; y_ ¼ f 2 ðyÞ þ uðx; yÞ; n
n
ð14Þ ð15Þ n
where x, y 2 R , f1, f2: R ? R . Assume that Eq. (14) is the drive system, Eq. (15) is the response system, and u(x, y) is the nonlinear control vector. If 8xðt 0 Þ; yðt 0 Þ 2 Rn ; limt!1 jyi ðtÞ ai xi ðtÞj ¼ 0; i ¼ 1; 2; . . . ; n, then the response and drive systems are said to be in HPS. In particular, the drive–response systems achieve complete synchronization when all values of ai are equal to 1. Further, if all values of ai are equal to 1, then the two systems are said to be in anti-synchronization. Our purpose herein is to achieve HPS of two identical hyperchaotic systems based on Lyapunov’s direct method. For the novel hyperchaotic system (1), the drive and response systems are defined as the following Eqs. (16) and (17), respectively:
8 x_ 1 ¼ aðy1 x1 Þ þ hy1 z1 ; > > > < y_ 1 ¼ cx1 dx1 z1 þ y1 þ w1 ; > z_ 1 ¼ x1 y1 bz1 ; > > : _ 1 ¼ gy1 ; w 8 _ x > 2 ¼ aðy2 x2 Þ þ hy2 z2 þ u1 ; > > < y_ ¼ cx dx z þ y þ w þ u ; 2 2 2 2 2 2 2 > _ 2 ¼ x2 y2 bz2 þ u3 ; z > > : _ 2 ¼ gy2 þ u4 ; w
ð16Þ
ð17Þ
where [u1, u2, u3, u4] is the nonlinear vector controller by using it two hyperchaotic systems can be synchronized in the sense of HPS, i.e.,
Fig. 3. A periodic orbit of the controlled system (9) with the control gain k = 11.5605.
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Fig. 4. A quasi-periodic state of the controlled system (9) with the control gain k = 9.12.
8 lim kx2 a1 x1 k ¼ 0; > > t!1 > > > > < lim ky2 a2 y1 k ¼ 0; t!1
ð18Þ
> lim kz2 a3 z1 k ¼ 0; > > t!1 > > > : lim kw2 a4 w1 k ¼ 0: t!1
Subtracting Eq. (16) from Eq. (17) yields error dynamical system between Eqs. (16) and (17):
8 e_ 1 > > > < e_ 2 > e > _3 > : e_ 4
¼ aðe2 e1 Þ þ aða2 a1 Þy1 þ hðy2 z2 a1 y1 z1 Þ þ u1 ; ¼ ce1 þ e2 þ e4 þ cða1 a2 Þx1 þ ða4 a2 Þw1 dðx2 z2 a2 x1 z1 Þ þ u2 ; ¼ be3 þ ðx2 y2 a3 x1 y1 Þ þ u3 ;
ð19Þ
¼ ge2 gða2 a4 Þy1 þ u4 ;
where e1 ¼ x2 a1 x1 ; e2 ¼ y2 a2 y1 ; e3 ¼ z2 a3 z1 , and e4 ¼ w2 a4 w1 . a, b, c, d, g, and h are positive system constant parameters. Here, our goal is to make synchronization between the two hyperchaotic systems by using a nonlinear vector controller [u1, u2, u3, u4], i.e., limt!1 keðtÞk ¼ 0, where e(t) = [e1, e2, e3, e4]T. For the two hyperchaotic systems without control (ui = 0, i = 1, 2, 3, 4), if the initial condition (x1(0), y1(0), z1(0), w1(0)) – (x2(0), y2(0), z2(0), w2(0)), the trajectories of the two identical systems will quickly separate each other and become irrelevant. However, for the two controlled hyperchaotic systems, the two systems will approach synchronization for any initial condition by appropriate control vector. For this end, we propose the following control law for the system (17):
8 u1 > > > < u2 > u3 > > : u4
¼ aða2 a1 Þy1 hða2 a3 a1 Þy1 z1 ; ¼ cða1 a2 Þx1 ða4 a2 Þw1 þ dða1 a3 a2 Þx1 z1 ; ¼ ða1 a2 a3 Þx1 y1 e1 e2 2a2 y1 e1 ke3 ;
ð20Þ
¼ gða2 a4 Þy1 ða2 a4 Þ2 e4 ;
where k is the feedback gain which is different from the usual linear feedback method, it is in the sense that k is adapted according to the following update law:
k_ ¼ r e22 þ e24 ;
kð0Þ ¼ 0;
r > 0:
ð21Þ
Then, we have the following main result. Theorem 2. For any initial conditions, the two systems (16) and (17) are globally and asymptotically hybrid projective synchronized by nonlinear feedback controller (20) with the update law (21). Proof Applying the controller of Eq. (20) to Eq. (19) yields the resulting error dynamics as follows:
8 e_ 1 > > > < e_ 2 > e_ 3 > > : e_ 4
¼ ae2 ae1 þ hðe2 e3 þ a3 z1 e2 þ a2 y1 e3 Þ; ¼ ce1 þ e2 þ e4 dðe1 e3 þ a1 x1 e3 þ a3 z1 e1 Þ; ¼ be3 þ a1 x1 e2 a2 y1 e1 ke3 ; ¼ ge2 ða2 a4 Þ2 e4 :
ð22Þ
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Consider a Lyapunov function candidate as
V¼
2 1 1 2 1 2 1 1 ðk k : e1 þ e2 þ e23 þ e24 Þ þ 2 h d gd 2r
ð23Þ
It is clear that the Lyapunov function, V, is a positive definite function. Then the time derivative of V along the solution of error dynamical system (22) gives
1 1 1 1 V_ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ ðk k Þk_ h d gd r " # a 1 2 ða2 a4 Þ2 ¼ e21 k k e2 ðk þ bÞe23 k þ k e24 ¼ eT Pe; h d gd
ð24Þ
where
3 e1 6e 7 6 27 e ¼ 6 7; 4 e3 5 2
e4
0
a=h ða=h þ c=dÞ=2 0 B ða=h þ c=dÞ=2 ðk k 1=dÞ 0 B p¼B @ 0 0 kþb 0
0
0
1
0 0 0
k þ ða2 a4 Þ2 =ðgdÞ k
C C C: A
Since a > 0, b > 0, d > 0, h > 0, obviously, when k > k þ hða=h þ c=dÞ2 =ð4aÞ þ 1=d, and k > k ða2 a4 Þ2 =ðgdÞ. Then the symmetric matrix P is positive, and therefore V_ is negative definite. Based on Lyapunov’s direct method, this translates to limt!1 kei ðtÞk ¼ 0; i ¼ 1; 2; 3; 4. Thus the response and drive systems are globally and asymptotically hybrid projective synchronized. This completes the proof. h Remark 2. For the complete synchronization case ðai ¼ 1; i ¼ 1; 2; 3; 4Þ, the controller can be simplified ulteriorly into: u1 = 0, u2 = 0, u3 = e1e2 2y1e1 ke3, u4 = 0. Namely, the complete synchronization for the novel hyperchaotic system can be achieved via a single scalar controller u3 = e1e2 2y1e1 ke3. Remark 3. The convergence rate of error signal e3 can be regulated by adjusting the constant coefficient
Fig. 5. Synchronization errors e1, e2, e3, and e4 vs. time t with a1 ¼ 1; a2 ¼ 1; a3 ¼ 4, and a4 ¼ 0:5.
r.
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Fig. 6. State trajectories of drive and response systems. (a) Signals x1 and x2 with a1 ¼ 1, (b) signals y1 and y2 with a2 ¼ 1, (c) signals z1 and z2 with a3 ¼ 4, and (d) signals w1 and w2 with a4 ¼ 0:5.
3.2. Numerical simulations In this subsection, numerical simulations are done with Matlab to demonstrate the effectiveness of the proposed HPS scheme. In numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. The initial states of the drive and response systems are chosen as follows: x1(0) = 10, y1(0) = 1, z1(0) = 5, w1(0) = 18; x2(0) = 8, y2(0) = 3, z2(0) = 18, w2(0) = 7. The hybrid projective parameters are chosen as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ 4 and a4 ¼ 0:5. Thus the initial states of the error system (16) are e1(0) = 2, e2(0) = 2, e3(0) = 2, and e4(0) = 2. The system parameters are chosen as follows: a = 35, b = 4.9, c = 25, d = 5, h = 35 and g = 15. We choose the constant coefficient r ¼ 10. The hybrid projective synchronization errors of systems (16) and (17) are displayed in Fig. 5. From these figures, One can see that all of the synchronization errors converge to 0. For further observations, the state trajectories of the drive and response systems are depicted in Fig. 6. As can be seen, variables x2 vs. x1 display a synchronization phenomenon, variables y2 vs. y1 show anti-synchronization behavior, variable z2 finally converges to four times the value of z1; and variable w2 converges to half of the value of w1. It is clear that the two hyperchaotic systems of (16) and (17) can achieve HPS. 4. Conclusions In this paper, the control and hybrid projective synchronization problems for a novel hyperchaotic system are investigated. Based on Lyapunov’s direct method and applying Routh–Hurwitz criterion, we investigate the chaos control problems at first. Single scalar controllers including two state variables are designed to control the equilibrium point and bifurcation, and detailed controllability conditions are also presented. Strict theoretical proof is put forward and numerical simulation results indicate that the controllers can effectively guide the system trajectories to equilibrium point and periodic orbits. Furthermore, hybrid projective synchronization between two identical new hyperchaotic systems is studied using nonlinear feedback control. A nonlinear vector controller is designed according to Lyapunov’s direct method to guarantee HPS, which includes complete synchronization, anti-synchronization, and projective synchronization. The nonlinear vector controller can be simplified ulteriorly into a single scalar controller u3 = e1e2 2y1e1 ke3 to achieve complete synchronization between two novel hyperchaotic systems. Numerical simulations verified the effectiveness of these strategies. Therefore, the proposed methods have certain significances for reducing the cost and complexity for controller implementation.
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