This paper is devoted to investigate the tracking control and generalized
synchronization of the hyperchaotic Lorenz-Stenflo system using the tracking model
and the feedback control scheme. We suppress the chaos to unstable equilibrium via
three feedback methods, and we achieve three globally generalized synchronization
controls. Novel tracking controllers with corresponding parameter update laws are
designed such that the Lorenz-Stenflo systems can be synchronized asymptotically.
Moreover, numerical simulations are presented to demonstrate the effectiveness,
through the contrast between the orbits before being stabilized and the ones after
being stabilized.
1. Introduction
Study of chaotic control and generalized synchronization has received great attention in the past several decades [1–10]; many hyperchaotic systems have been proposed and studied in the last decade, for example, a new hyperchaotic Rössler system [4], the hyperchaotic L system [5], Chua’s circuit [6], the hyperchaotic Chen system [7, 8], and so forth. Hyperchaotic system has been proposed for secure communication and the presence of more than one positive Lyapunov exponent clearly improves the security of the communication scheme [9–12]. Therefore, hyperchaotic system generates more complex dynamics than the low-dimensional chaotic system, which has much wider application than the low-dimensional chaotic system.
Until now, a variety of approaches have been proposed for the synchronization of low-dimensional chaotic systems, including Q-S method [13, 14], active control [15, 16], adaptive control [17–24], and time-delay feedback control [25]. Recently, Stenflo [26] presented a new hyperchaotic Lorenz-Stenflo (LS) system:
()
In (1), x, y, z, and w are the state variables of the system and a, b, c, and d are real constant parameters. System (1) is generated from the originally three-dimensional Lorenz chaotic system by introducing a new control parameter b and a state variable w.
In this paper, we will consider chaos control and generalized synchronization related to hyperchaotic Lorenz-Stenflo system. we found that the feedback control achieved in the low-dimensional system like many other studies of dynamics in low-dimensional systems. We suppress the hyperchaotic Lorenz-Stenflo system to unstabilize equilibrium via three control methods: linear feedback control, speed feedback control, and doubly-periodic function feedback control. By designing a nonlinear controller, we achieve the generalized synchronization of two Lorenz-Stenflo systems up to a scaling factor. Moreover, numerical simulations are applied to verify the effectiveness of the obtained controllers.
2. The Hyperchaotic Lorenz-Stenflo System
In the following we would like to consider the hyperchaotic cases of system (1). When a = 1.0, b = 0.7, c = 26, and d = 1.5, system (1) exhibits hyperchaotic behavior. Simulated results are depicted in Figures 1 and 2. Figures 1(a)–1(d) depict the projection of the chaotic attractor in different spaces; Figures 2(a)–2(d) depict the states of system (1) before being stabilized, respectively. The volume of the elements in the phase space δX(t) = δxδyδzδw and the divergence of flow (1) are defined by
The states of system (1) before being stabilized: (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
System (1) is dissipative when 2a + b + 1 > 0. Moreover, an exponential contraction rate is given by
()
It is clear that X(t) = X0e−(2a+b+1)t, which implies that the solutions of system (1) are bounded as t − >+∞. It is easy to find the three equilibria E1(0,0, 0,0),
()
To determine the stability of the equilibria point E1(0,0, 0,0), evaluating the Jacobian matrix of system (1) at E1 yields
()
If a = 1.0, b = 0.7, c = 26, and d = 1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (5) are
()
Thus, the equilibria E1 is a saddle point of the hyperchaotic system (1). The Jacobian matrix of system (1) at E2 yields
()
If a = 1.0, b = 0.7, c = 26, and d = 1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (7) are
()
Thus, the equilibria E2 are unstable, and E3 is similar.
3. The Hyperchaotic Control for Lorenz-Stenflo System
In this section, we control the hyperchaotic system (1) such that all trajections converge to the equilibrium point (0,0, 0,0). The controlled hyperchaotic Lorenz-Stenflo system is given by
()
where u1, u2, u3, and u4 are external control inputs which will be suitably derived from the trajectory of the chaotic system (1), specified by (x, y, z, w) to the equilibrium (0,0, 0) of uncontrolled system ui = 0(i = 1,2, 3,4).
3.1. Linear Function Feedback Control
For the modified hyperchaotic Lorenz-Stenflo system (9), if one of the following feedback controllers ui(i = 1,2, 3,4) is chosen for the system (9), then u1 = −k1x1, u2 = −k2y, u3 = −k3z, u4 = −k4w, and where ki’s are feedback coefficients. Therefore controlled system (9) is rewritten as
()
whose Jacobian matrix is
()
The characteristic equation of J is
()
where A = −a − k1, B = −1 − k2, C = −b − k3, D = −a − k4, and E = ABCD − caCD + dBC.
The abbreviated characteristic equation is
()
where R1 = −B − A − D − C, R2 = CD + AB + BD + BC − ca + AD + AC + d, and R3 = −ABD − ABC − BCD + caD + caC − ACD − dC − dB, R4 = E.
According to the Routh-Hurwitz criterion, constraints are imposed as follows:
()
This characteristic polynomial has four roots, all with negative real roots, under the condition of H1 > 0, H2 > 0, H3 > 0, and H4 > 0. Therefore, the equilibria (0,0, 0,0) are the stable manifold Ws and the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable. The concrete dynamics of (9) can be demonstrated by Proposition 1, while the Maple program is demonstrated in Appendix A.
Proposition 1. If one chooses the control coefficients k1 = 8, k2 = 4, k3 = 3, and k4 = 2 and the parameters a = 1.0, b = 0.7, c = 26, and d = 1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0, 0,0).
Proof. When the parameters were selected by the above value, we obtain the Jacobian matrix
which yields the eigenvalues via the compute simulation
()
Thus the zero solution of system (9) is exponentially stable, Proposition 1 is proved.
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (1) using the fourth-order Runge-Kutta scheme with time step 0.01. The parameters and the corresponding feedback coefficients are given by the above value. The initial values are taken as [x(0) = 1, y(0) = 0.7, z(0) = 20, w(0) = 0.1]. The behaviors of the states (x, y, z, w) of the controlled chaotic Lorenz-Stenflo system (1) with time are displayed in Figures 3(a)–3(d).
The states of system (9): (a) x-t, (b) y-t, (c) z-t, (d) w-t.
3.2. Speed Function Feedback Control
Suppose that u1 = u3 = 0; u2 and u4 are of the speed forms u2 = k2(−bz + xy) and u4 = −k4[a(y − x) + dw], where k2 and k4 are speed feedback coefficients. Therefore the controlled chaotic system (9) is rewritten as
()
The Jacobian matrix is
()
The characteristic equation of J is
()
Proposition 2. If one chooses the control coefficients: k2 = 9, k4 = −1 and the parameters a = 1.0, b = 0.7, c = 26, and d = 1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0, 0,0).
Similar to Proposition 1, the proof of Proposition 2 is straightforward and thus is omitted.
In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix
()
The characteristic equation of J changes the following:
()
The eigenvalues of the above equation are
()
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (20) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0) = 1, y(0) = 0.7, z(0) = 20, w(0) = 0.1]. The behaviors of the states (x, y, z, w) of the controlled chaotic Lorenz-Stenflo system (20) with time are displayed in Figures 4(a)–4(d).
The states of system (19): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
3.3. The Doubly Periodic Function Feedback Control
Suppose that u2 = 0, u3 = 0, and u4 = 0; u1 is of the doubly periodic function u1 = k1cn(x, m), where k1 is speed feedback coefficients and 0 < m < 1 is the modulus of Jacobi elliptic function. Therefore the controlled chaotic system (9) is rewritten as
()
The Jacobian matrix is
()
The characteristic equation of J is
()
Proposition 3. If one chooses the control coefficients k1 = −30, m = 0.3 and the parameters a = 1.0, b = 0.7, c = 26, and d = 1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0, 0,0).
The proof of Proposition 3 is the same as Proposition 1, which is straightforward and thus is omitted.
In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix
()
The characteristic equation of J changes the following:
()
The eigenvalues of the above equation are
()
Thus the zero solution of system (9) is exponentially stable; Proposition 3 is proved.
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (25) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0) = 1, y(0) = 0.7, z(0) = 20, w(0) = 0.1]. The behaviors of the states (x, y, z, w) of the controlled chaotic Lorenz-Stenflo system (25) with time are displayed in Figures 5(a)–5(d).
The states of system (25): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
4. Globally Exponential Hyperchaotic Projective Synchronization Control
Consider two chaotic systems given by
()
()
where xm = (x1m, x2m, …xnm) T, ys = (y1s, y2s, …yns) T, f, g ∈ Cr[R+ × Rn, Rn], u ∈ Cr[R+ × Rn × Rn, Rn], and r ≥ 1. R+ comprises the set of non negative real numbers. Assume that (31) is the master system, (32) is the slave system, and u(xm, ys, t) is the control vector. Let the error state be
()
Then the error dynamics of e(t) are defined by
()
The slave and master systems are said to be exponential, hyperchaotic, projective and synchronized if, for allxm(t0), ys(t0) ∈ Rn, and i ∈ Rn, ∥xm − (ai1ys + ai2)ys∥ → 0 as t → ∞.
Lemma 4 (see [27], [28].)The zero solution of the error dynamical system (34) is globally and exponentially stable; the master-slave systems (31) and (32) are globally and exponentially projective, synchronized, if there exists a positive definite quadratic polynomial V = (e1, e2,…en)P(e1, e2,…en) T such that dV/dt = (e1, e2,…en)Q(e1, e2,…en) T. Moreover, the following negative Lyapunov exponent estimation for the error dynamical system (34) holds:
()
where P = PT ∈ Rn×n and Q = QT ∈ Rn×n are both positive definite matrices, λmax (P) and λmin (P) stand for the minimum and maximum eigenvalues of the matrix P, respectively, and λmin (Q) denotes the minimum eigenvalue of the matrix Q.
In the following, we consider the hyperchaotic system (1) as a master system:
as a slave system, where the subscripts “m” and “s” stand for the master system and slave system, respectively. Let the error state be
()
Then the derivative of e(t) along the trajectories of (36) and (37), we obtain the error system:
()
To demonstrate the synchronization control between systems (36) and (37), we have the following cases based on [29–44].
Case 1 (modified projective synchronization control).
Theorem I. When [a11, a12, a21, a22, a31, a32, a41, a42] = [0, m, 0, m, 0, m, 0, m], a > 0 and b > 0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i = 1,2, 3,4) is given for the slave system (36):
()
there exist many possible choices for v1, v2, v3, and v4. Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential modified projective synchronization can be achieved.
The concrete proof of Theorem I can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (42) with Runge-Kutta integration method. The initial values of the drive system and response system are x1(0) = 0.1, y1(0) = 0.1, z1(0) = 20, and w1(0) = 0.1 and x2(0) = 0.2, y2(0) = 0.2, z2(0) = 21, and w2(0) = 0.2 respectively. The parameters are chosen to be m = −2, a = 1.0, b = 0.7, d = 1.5, and c = 26 so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The numerical simulation of the master and the slave systems without active synchronization control law is shown in Figures 6(a)–6(d). The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 7(a)–7(d). The synchronization errors are shown in Figures 8(a)–8(d). Figures 9(a)–9(d) depicts the projection of the synchronized attractors. Figures 10(a)–10(f) depicts the phase portraits of the synchronized attractors.
The solutions of the master and slave systems without active control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems without active control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems without active control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems without active control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the drive and response systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the drive and response systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the drive and response systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the drive and response systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.
The projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.
The projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.
The projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
Case 2 (generalized projective synchronization control).
Theorem II. When [a11, a12, a21, a22, a31, a32, a41, a42] = [0, m1, 0, m2, 0, m3, 0, m4], mi’s are different, a > 0 and b > 0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i = 1,2, 3,4) is given for the slave system (36):
()
there exist many possible choices for v1, v2, v3, and v4. Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential generalized projective synchronization.
The concrete proof of Theorem II can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The parameters are chosen to be m1 = 2, m2 = 3, m3 = 3, and m4 = 5. The initial values and others parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 11(a)–11(d). The synchronization errors are shown in Figures 12(a)–12(d).
The solutions of the master and slave systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The solutions of the master and slave systems with control law. (a) Signals x1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
The dynamics of synchronization errors. (a) Signal e1, (b) signal e2, (c) signal e3, and (d) signal e4.
Case 3 (function synchronization control [45–58]).
Theorem III. When f1 = a11x2 + a12, f2 = a21y2 + a22, f3 = a31z2 + a32, f4 = a41w2 + a42, ai1 ≠ 0, a > 0, b > 0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i = 1,2, 3,4) is given for the slave system (36):
()
where k1 > 0, k2 > 0, k3 > 0, and k4 > 0, then the zero solution of the error dynamical system (39) is globally stable, and thus global function, projective synchronization occurs between the master systems (36) and (37).
The concrete Proof of Theorem III can be demonstrated by Appendix B. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The initial values and the parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with active control law is presented in Figures 13(a)–13(d). The synchronization errors are shown in Figures 14(a)–14(d).
The function projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes the drive system; the dashed line denotes respond system synchronized.
The function projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes the drive system; the dashed line denotes respond system synchronized.
The function projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes the drive system; the dashed line denotes respond system synchronized.
The function projection of the synchronized attractors in different spaces: (a) (x, y, z), (b) (x, y, w), (c) (x, w, z), and (d) (y, w, z). The solid line denotes the drive system; the dashed line denotes respond system synchronized.
The dynamics of synchronization errors. (a) Signal e1, (b) signal e2, (c) signal e3 and (d) signal e4.
5. Summary and Conclusions
In this paper, we have introduced the tracking control and generalized synchronization of the hyperchaotic system which is different from the Lorenz-Stenflo attractor. We suppress the chaos to unstabilize equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls of two Lorenz-Stenflo systems. As a result, some powerful controllers are obtained. Then, we investigate the hyperchaotic system applying the complex system calculus technique. Moreover, numerical simulations are used to verify the effectiveness of our results, through the contrast between the orbits before being stabilized and the ones after being stabilized.
Acknowledgments
This work is supported by the NSF of China (no. 11171355), the Ph.D. Programs Foundation of Ministry of Education of China (no. 20100171110040), and Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University. The authors thank the handling editor and the reviewers for their valuable comments and suggestions, which improved the completeness of the paper.
Appendices
A. The Maple Program
>restart: with(student): with(PDEtools): with(linalg): with (LinearAlgebra):
Let us consider the Lyapunov function for the system (B.2) as follows:
()
In addition, the derivative of V has the form
()
which is negatively defined. So (B.2) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective and synchronized.
Proof of Theorem II. Consider the controller (41) and choose vi as follows:
()
Then the following steps are the same (B.2), (B.3), and (B.4). So we know that the two Lorenz-Stenflo hyperchaotic systems are modified, projective, and synchronized.
Proof of Theorem III. Consider the controller (42) and choose the following positive definite, quadratic form of Lyapunov function:
()
We differentiate V(t) and substitute the trajectory of system (39) which yields
()
where P = diag (k1, k2, k3, k4), which implies that the conclusion of Theorem III is true. So (39) is asymptotically stable. This implies that the two Lorenz-Stenflo hyperchaotic systems are projective synchronized functions.
3Pecora L. M. and
Carroll T. L., Synchronization in chaotic systems, Physical Review Letters. (1990) 64, no. 8, 821–824, https://doi.org/10.1103/PhysRevLett.64.821, MR1038263, ZBL0938.37019.
6Liao X. and
Yu P., Analysis on the globally exponent synchronization of Chua′s circuit using absolute stability theory, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. (2005) 15, no. 12, 3867–3881, https://doi.org/10.1142/S0218127405014350, MR2205622, ZBL1094.37502.
7Li Y.,
Tang W. K. S., and
Chen G., Generating hyperchaos via state feedback control, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. (2005) 15, no. 10, 3367–3375, 2-s2.0-29244467721, https://doi.org/10.1142/S0218127405013988.
9Boutayeb M.,
Darouach M., and
Rafaralahy H., Generalized state-space observers for chaotic synchronization and secure communication, IEEE Transactions on Circuits and Systems. (2002) 49, no. 3, 345–349, https://doi.org/10.1109/81.989169, MR1890971.
10Guan X. P.,
Fan Z. P., and
Chen C. L., Chaos Control and Its Application in Secure Communication, 2002, National Defence Industry Press, Beijing, China.
11Sundar S. and
Minai A. A., Synchronization of randomly multiplexed chaotic systems with application to communication, Physical Review Letters. (2000) 85, no. 25, 5456–5459, 2-s2.0-0034499813, https://doi.org/10.1103/PhysRevLett.85.5456.
15Ho M.-C.,
Hung Y.-C., and
Chou C.-H., Phase and anti-phase synchronization of two chaotic systems by using active control, Physics Letters A. (2002) 296, no. 1, 43–48, 2-s2.0-0037041689, https://doi.org/10.1016/S0375-9601(02)00074-9.
16Pan L.,
Zhou W.,
Fang J., and
Li D., A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems, Nonlinear Dynamics. (2010) 62, no. 1-2, 417–425, https://doi.org/10.1007/s11071-010-9728-0, MR2737004, ZBL1209.93071.
17Salarieh H. and
Alasty A., Adaptive synchronization of two chaotic systems with stochastic unknown parameters, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 2, 508–519, https://doi.org/10.1016/j.cnsns.2007.09.002, MR2458824, ZBL1221.93246.
18Hu M. and
Xu Z., Adaptive feedback controller for projective synchronization, Nonlinear Analysis. Real World Applications. (2008) 9, no. 3, 1253–1260, https://doi.org/10.1016/j.nonrwa.2007.03.005, MR2392409, ZBL1144.93364.
19Liu Y.-J.,
Wang W.,
Tong S.-C., and
Liu Y.-S., Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters, IEEE Transactions on Systems, Man, and Cybernetics A. (2010) 40, no. 1, 170–184, 2-s2.0-72949089051, https://doi.org/10.1109/TSMCA.2009.2030164.
20Liu Y.-J. and
Zheng Y.-Q., Adaptive robust fuzzy control for a class of uncertain chaotic systems, Nonlinear Dynamics. (2009) 57, no. 3, 431–439, https://doi.org/10.1007/s11071-008-9453-0, MR2520222, ZBL1176.93040.
21Liu Y.-J. and
Wang W., Adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems, Information Sciences. (2007) 177, no. 18, 3901–3917, https://doi.org/10.1016/j.ins.2007.03.005, MR2337220, ZBL1121.93037.
22Elabbasy E. M.,
Agiza H. N., and
El-Dessoky M. M., Adaptive synchronization of a hyperchaotic system with uncertain parameter, Chaos, Solitons & Fractals. (2006) 30, no. 5, 1133–1142, https://doi.org/10.1016/j.chaos.2005.09.047, MR2249221, ZBL1142.37325.
23Huang J., Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters, Physics Letters A. (2008) 372, no. 27-28, 4799–4804, 2-s2.0-51749111833, https://doi.org/10.1016/j.physleta.2008.05.025.
24Liu Y.-J.,
Tong S.-C.,
Wang W., and
Li Y.-M., Observer-based direct adaptive fuzzy control of uncertain nonlinear systems and its applications, International Journal of Control, Automation and Systems. (2009) 7, no. 4, 681–690, 2-s2.0-69749095269, https://doi.org/10.1007/s12555-009-0420-4.
25Li C.,
Liao X., and
Wong K.-w., Lag synchronization of hyperchaos with application to secure communications, Chaos, Solitons & Fractals. (2005) 23, no. 1, 183–193, https://doi.org/10.1016/j.chaos.2004.04.025, MR2091435, ZBL1068.94004.
27Yan Z. and
Yu P., Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems, Chaos, Solitons and Fractals. (2007) 33, no. 2, 419–435, https://doi.org/10.1016/j.chaos.2005.12.059, MR2313551, ZBL1136.93430.
28Liao X. and
Yu P., Study of globally exponential synchronization for the family of Rössler systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. (2006) 16, no. 8, 2395–2406, https://doi.org/10.1142/S0218127406016148, MR2266022, ZBL1192.37042.
29Ghosh D. and
Bhattacharya S., Projective synchronization of new hyperchaotic system with fully unknown parameters, Nonlinear Dynamics. (2010) 61, no. 1-2, 11–21, https://doi.org/10.1007/s11071-009-9627-4, MR2661780, ZBL1204.93101.
30Ma D.-Z.,
Zhang H.-G.,
Wang Z.-S., and
Feng J., Fault tolerant synchronization of chaotic systems based on T-S fuzzy model with fuzzy sampled-data controller, Chinese Physics B. (2010) 19, no. 5, 050506, 2-s2.0-77952840601, https://doi.org/10.1088/1674-1056/19/5/050506.
31Yang S. S. and
Duan C. K., Generalized synchronization in chaotic systems, Chaos, Solitons and Fractals. (1998) 9, no. 10, 1703–1707, https://doi.org/10.1016/S0960-0779(97)00149-5, MR1663088, ZBL0946.34040.
34Zhang Y. and
Sun J., Chaotic synchronization and anti-synchronization based on suitable separation, Physics Letters A. (2004) 330, no. 6, 442–447, https://doi.org/10.1016/j.physleta.2004.08.023, MR2094455, ZBL1209.37039.
35Aguilar-López R. and
Martínez-Guerra R., Synchronization of a class of chaotic signals via robust observer design, Chaos, Solitons and Fractals. (2008) 37, no. 2, 581–587, https://doi.org/10.1016/j.chaos.2006.08.026, MR2402651, ZBL1141.93314.
36Chen J.-H., Controlling chaos and chaotification in the Chen-Lee system by multiple time delays, Chaos, Solitons and Fractals. (2008) 36, no. 4, 843–852, 2-s2.0-38349106246, https://doi.org/10.1016/j.chaos.2006.10.049.
38Li Y.,
Chen Y., and
Li B., Adaptive control and function projective synchronization in 2d discrete-time chaotic systems, Communications in Theoretical Physics. (2009) 51, no. 2, 270–278, 2-s2.0-65749111918, https://doi.org/10.1088/0253-6102/51/2/17.
40Li Y.,
Li B., and
Chen Y., Anticipated function synchronization with unknown parameters of discrete-time chaotic systems, International Journal of Modern Physics C. (2009) 20, no. 4, 597–608, 2-s2.0-67649408913, https://doi.org/10.1142/S0129183109013820.
41Li Y. and
Li B., Chaos control and projective synchronization of a chaotic Chen-Lee system, Chinese Journal of Physics. (2009) 47, no. 3, 261–270, 2-s2.0-77349095474.
42Lü J.,
Yu X.,
Chen G., and
Cheng D., Characterizing the synchronizability of small-world dynamical networks, IEEE Transactions on Circuits and Systems I. (2004) 51, no. 4, 787–796, https://doi.org/10.1109/TCSI.2004.823672, MR2066214.
44Lü J. and
Chen G., A time-varying complex dynamical network model and its controlled synchronization criteria, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control. (2005) 50, no. 6, 841–846, https://doi.org/10.1109/TAC.2005.849233, MR2142000.
46Chen Y. and
Li X., Function projective synchronization between two identical chaotic systems, International Journal of Modern Physics C. (2007) 18, no. 5, 883–888, 2-s2.0-34547747872, https://doi.org/10.1142/S0129183107010607.
47Du H.,
Zeng Q.,
Wang C., and
Ling M., Function projective synchronization in coupled chaotic systems, Nonlinear Analysis. Real World Applications. (2010) 11, no. 2, 705–712, https://doi.org/10.1016/j.nonrwa.2009.01.016, MR2571243, ZBL1181.37039.
48Wu X.-J.,
Wang H., and
Lu H.-T., Hyperchaotic secure communication via generalized function projective synchronization, Nonlinear Analysis. Real World Applications. (2011) 12, no. 2, 1288–1299, https://doi.org/10.1016/j.nonrwa.2010.09.026, MR2736310, ZBL1203.94128.
49Du H.,
Zeng Q., and
Wang C., A General method for function projective synchronization, International Journal of Innovative Computing, Information and Control. (2009) 5, no. 8, 2239–2248, 2-s2.0-67649641041.
50Du H.,
Zeng Q., and
Wang C., Function projective synchronization of different chaotic systems with uncertain parameters, Physics Letters A. (2008) 372, no. 33, 5402–5410, 2-s2.0-47349112935, https://doi.org/10.1016/j.physleta.2008.06.036.
51Wu Z. and
Fu X., Adaptive function projective synchronization of discrete chaotic systems with unknown parameters, Chinese Physics Letters. (2010) 27, 050502.
52Du H.,
Li F., and
Meng G., Robust function projective synchronization of two different chaotic systems with unknown parameters, Journal of the Franklin Instituts. (2011) 348, no. 10, 2782–2794, https://doi.org/10.1016/j.jfranklin.2011.08.016, MR2847571, ZBL1254.93076.
53Park J. H., Further results on functional projective synchronization of genesiotesi chaotic system, Modern Physics Letters B. (2009) 23, no. 15, 1889–1895, 2-s2.0-67749099501, https://doi.org/10.1142/S0217984909020059.
54Zhang R.,
Yang Y.,
Xu Z., and
Hu M., Function projective synchronization in drive-response dynamical network, Physics Letters A. (2010) 374, no. 30, 3025–3028, 2-s2.0-77955423005, https://doi.org/10.1016/j.physleta.2010.05.041.
55An H. and
Chen Y., The function cascade synchronization method and applications, Communications in Nonlinear Science and Numerical Simulation. (2008) 13, no. 10, 2246–2255, https://doi.org/10.1016/j.cnsns.2007.05.029, MR2417585, ZBL1221.93124.
56Chen Y.,
An H., and
Li Z., The function cascade synchronization approach with uncertain parameters or not for hyperchaotic systems, Applied Mathematics and Computation. (2008) 197, no. 1, 96–110, https://doi.org/10.1016/j.amc.2007.07.036, MR2396294, ZBL1136.93410.
57Li D.-J., Adaptive output feedback control of uncertain nonlinear chaotic systems based on dynamic surface control technique, Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems. (2012) 68, no. 1-2, 235–243, https://doi.org/10.1007/s11071-011-0222-0, MR2904171, ZBL1243.93057.
58Li Y. and
Zheng C.-l., The complex network synchronization via chaos control nodes, Journal of Applied Mathematics. (2013) 11, 823863, MR3035181, ZBL1266.93121.
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