International Journal of Bifurcation and Chaos, Vol. 16, No. 10 (2006) 2923–2933 c World Scientific Publishing Company
ADAPTIVE SYNCHRONIZATION OF COUPLED CHAOTIC DELAYED SYSTEMS BASED ON PARAMETER IDENTIFICATION AND ITS APPLICATIONS JIN ZHOU∗ Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China
[email protected] TIANPING CHEN of Nonlinear Science, Institute of Mathematics, Fudan University, Shanghai 200433, P. R. China
[email protected]
∗Laboratory
LAN XIANG Department of Physics, Shanghai University, Shanghai 200444, P. R. China
[email protected] Received February 2, 2005; Revised December 14, 2005 This paper investigates synchronization dynamics of a large class of chaotic delayed systems with all the parameters unknown. By a simple combination of adaptive control and linear feedback with the updated laws, some simple yet generic criteria for determining global synchronization based on parameter identification of uncertain chaotic delayed systems are derived by using the invariance principle of functional differential equations. It is shown that the approaches developed here further extend the ideas and techniques presented in recent literature. Furthermore, the theoretical results are applied to the well-known Chua’s circuit and a typical class of chaotic delayed Hopfied neural networks, and numerical simulations also demonstrate the effectiveness and feasibility of the proposed techniques. Keywords: Chaotic delayed systems; adaptive synchronization; parameters identification; Chua’s circuit; chaotic delayed Hopfied neural networks.
1. Introduction Since inspired by the pioneering work of Pecora and Carroll [1990], synchronization of coupled chaotic systems has attracted a great deal of attention in both theory and application. They have been successfully applied in chaos generator design, secure communication, chemical reactions, biological systems and so on [Chen & Dong, 1998; Pecora et al., 1998]. A wide variety of approaches have been proposed for synchronization of chaotic systems
which include linear and nonlinear feedback control, time-delay feedback control, adaptive design control, impulsive control method, and invariant manifold method, among many others (see [Chen & Dong, 1998; Chen et al., 2004b; Chen et al., 2005; Chua et al., 1993; Chua et al., 1996; Huang, 2004a, 2004b, 2004c, 2005; Jiang et al., 2003; Pecora et al., 1998, Zheng et al., 2002; Zhou et al., 2005a, 2005b, 2006; Wang et al., 1999] and references cited therein).
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However, to our best knowledge, the aforementioned methods and many other existing synchronization schemes mainly address synchronization of chaotic systems without delays, this feature limits obviously complexity of a large class of chaotic systems. It is believed that as many chaotic models developed in physics, chemistry and biology are formulated in terms of coupled nonlinear oscillators, time delay plays an important role in control and synchronization of such chaotic oscillators [Li et al., 2004; Mensour & Longtin, 1998; Peng & Liao, 2003; Pyragas, 1998]. In addition, literature dealing with chaos synchronization of coupled systems with time delays appears to be scarce due to the difficulties in mathematical analysis for the infinite-dimensional delayed dynamical systems [Chen & Zhou, 2004; Zhou et al., 2004a]. Moreover, most of the developed methods are valid only for chaotic systems whose parameters are precisely known. But in practical situation, the parameters of some systems cannot be exactly known a priori, the effect of these uncertainties will destroy synchronization and even break it [Chen & Dong, 1998; Chen et al., 2004a; Huang, 2004; Zhou et al., 2005a, 2005b]. Therefore, it is essential to investigate synchronization of coupled delayed chaotic systems in the presence of unknown parameters. Motivated by a simple adaptive technique of chaos synchronization for coupled dynamical systems without delays developed in recent works of Huang [2004a, 2004b, 2004c, 2005], we further investigate synchronization dynamics of a large class of chaotic delayed systems with all the parameters unknown, which includes almost all well-known chaotic systems with delays or without delays. In this research field, some results for particular unidirectionally coupled chaotic systems with delays are regarded [Li et al., 2004; Mensour & Longtin, 1998; Pyragas, 1998]. In addition, most of previous studies are predominantly concentrated on synchronization of some special chaotic systems with delays such as the Mackey–Glass system or the Ikeda equations, delayed Hopfied neural networks and delayed cellular neural networks (CNNs), and so on [Chen & Zhou, 2004; Li et al., 2004; Peng & Liao, 2003; Pyragas, 1998; Zhou et al., 2004b, 2005a, 2005b, 2006]. To the best of our knowledge, few (if any) further results for a more generic chaotic delayed system, in particular, with all the parameters unknown, have been published so far. In this paper, by a simple combination of adaptive control and linear feedback with the updated
laws, some simple yet generic criteria for determining global synchronization based on parameter identification of uncertain chaotic delayed systems are derived by using the invariance principle of functional differential equations [Hale, 1977; Kuang, 1993; Slotine, 2004]. These approaches provide a systematic and analytical procedure for adaptive synchronization and parameter identification of chaotic delayed systems in the presence of unknown parameters. It is shown that by using the techniques developed here, both adaptive synchronization and parameter identification are more rapidly achieved than by ones reported in recent literature [Huang, 2004a, 2004b, 2005]. To this end, the theoretical results will be applied to the well-known Chua’s circuit [Chua et al., 1993; Chua et al., 1996; Jiang et al., 2003] and a typical class of chaotic delayed Hopfied neural networks [Lu, 2002; Zhou et al., 2004b, 2005a, 2005b], and numerical simulations are also presented to demonstrate the effectiveness of the approaches. The rest of the paper is organized as follows. After giving some preliminaries, the problem formulations are presented for adaptive synchronization and parameter identification of uncertain chaotic delayed systems. Section 3 deals with adaptive synchronization problem of such coupled delayed systems, some simple yet generic criteria for determining global synchronization based on parameter identification of uncertain chaotic delayed systems are derived, where the approaches are based on the invariance principle of functional differential equations. Section 4 applies these results to study adaptive synchronization of the well-known Chua’s circuit and a typical class of chaotic delayed Hopfied neural networks, where numerical examples are given to verify and also visualize the theoretical results. Finally, some concluding remarks are given in Sec. 5.
2. Preliminaries and Problem Formulations First, we consider a class of n-dimensional delayed dynamical system, which is described by the following form of differential equations with delays x(t) ˙ = F (x(t), x(t − τ ), p, q),
(1)
in which x(t) = (x1 (t), . . . , xn (t)) ∈ Rn is the state of the system, and F (x(t), x(t − τ ), p, q) = (F1 (x(t), x(t − τ ), p, q), . . . , Fn (x(t), x(t − τ ), p, q))
Adaptive Synchronization of Coupled Chaotic Delayed Systems
with Fi (x(t), x(t − τ ), p, q) = ci (x(t)) +
m
(pij fij (x(t))
j=1
+ qij gij (x(t − τ ))),
i = 1, 2, . . . , n,
(2)
where ci (x), fij (x) and gij (x) are piecewise continuous nonlinear functions, the parameter matrixes p = (pij )n×m ∈ U, q = (qij )n×m ∈ U, U is a bounded subset of Rnm , and the time delay τ ≥ 0. The initial conditions of (1) are given by xi (t) = φi (t) ∈ C([−τ, 0], R), where C([−τ, 0], R) denotes the set of all continuous functions from [−τ, 0] to R. Next, we always assume that the vector-valued function F (x(t), x(t − τ ), p, q) satisfies uniform Lipschitz condition with respect to p, q ∈ U , i.e. for any p, q ∈ U and x(t) = (x1 (t), . . . , xn (t)) ∈ Rn , y(t) = (y1 (t), . . . , yn (t)) ∈ Rn , there exist constants ki > 0 satisfying n
(|xj (t) − yj (t)|
j=1
+|xj (t − τ ) − yj (t − τ )|),
i = 1, 2, . . . , n. (3)
Remark 1. Clearly, Eq. (1) is generalization of a
large class of n-dimensional systems without delays described in [Huang, 2004a]. It is easy to check that the class of systems in the form of Eqs. (1)–(3) includes almost all the well-known chaotic systems with delays or without delays such as the Lorenz system, R¨ossler system, Chen system, Chua’s circuit as well as the delayed Mackey–Glass system or delayed Ikeda equations, delayed Hopfied neural networks and delayed cellular neural networks (CNNs), and so on (see, for example, [Chen & Dong, 1998; Chen & Zhou, 2004; Huang, 2004a; Li et al., 2004; Lu, 2002; Peng & Liao, 2003; Pyragas, 1998; Zhou et al., 2004b, 2005a, 2005b, 2006] and the references therein). Now, we consider the master (or drive) system in the form of Eq. (1), which is assumed to be a chaotic delayed system. We also introduce an auxiliary variable y(t) = (y1 (t), . . . , yn (t)) ∈ Rn , the slave (or response) system is given by the following equation y(t) ˙ = F (y(t), y(t − τ ),p,q),
which has the same structure as the master system but all the parameters p = (pij )n×m and q = (q ij )n×m are completely unknown, or uncertain. In practical situation, the output signals of the master system (1) can be received by the slave system (4), but the parameter vectors of the master system (1) may not be known a priori, even awaits identification. To estimate all unknown parameters, by adding an adaptive controller to the slave system (4), we have the following controlled slave system y(t) ˙ = F (y(t), y(t − τ ),p,q) + U (t : x(t), y(t)).
(5)
Therefore, the goal of control is to design and implement an appropriate controller U for the slave system and the adaptive estimation laws of the parameters p and q, such that the controlled slave system could be synchronous with the master system (1), and all the parameters p → p and q → q as t → +∞.
3. Synchronization and Identification of Uncertain Chaotic Delayed Systems
|Fi (x(t), x(t − τ ), p, q) − Fi (y(t), y(t − τ ), p, q)| ≤ ki
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(4)
Based on the LaSalle invariance principle of functional differential equations [Hale, 1977; Kuang, 1993; Slotine, 2004], the following sufficient criteria for adaptive synchronization and parameter identification of chaotic delayed systems with all the parameters unknown are established. Theorem 1. Let the controller U (t : x(t), y(t)) =
ε(y(t) − x(t)), where the feedback strength ε = diag(ε1 , . . . , εn ) with the following update laws ε˙i = −δi e2i (t) exp(µt),
i = 1, 2, . . . , n.
(6)
and the parameter adaptive laws of p = (pij )n×m and q = (q ij )n×m are chosen as below p˙ ij = −αij ei (t)fij (y(t)) exp(µt), i = 1, 2, . . . , n, j = 1, 2, . . . , m. (7) ˙ij = −βij ei (t)gij (y(t − τ )) exp(µt), q i = 1, 2, . . . , n, j = 1, 2, . . . , m. in which µ ≥ 0 is a small enough real number properly selected, the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2, . . . , n), δi > 0 (i = 1, 2, . . . , n) and αij > 0, βij > 0 (i = 1, 2, . . . , n, j = 1, 2, . . . , m) are arbitrary constants, respectively. If there exist n positive number γ1 , . . . , γn and a positive number
J. Zhou et al.
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r ∈ [0, 1] such that n 2(1−r) (γi ki2r + γj kj ) < 0, −γi li +
i = 1, 2, . . . , n,
j=1
(8) where li ≥ −limt→+∞ εi (t) = −εi0 > 0 (i = 1, 2, . . . , n) are constants properly selected, then the controlled uncertain slave system (5) is globally synchronous with the master system (1) and satisfies 1 n 2 µ 2 (yi (t) − xi (t)) = O exp − t . (9) 2
where e(t) = (e1 (t), . . . , en (t)) . Now construct a Lyapunov functional of the form
n n 1 2(1−r) γi e2i (t) exp(µt) + ki V (t) = 2 i=1
×
t−τ
+
t→+∞
t→+∞
(10)
for all i = 1, 2, . . . , n, j = 1, 2, . . . , m. Let e(t) = y(t) − x(t) be the synchronization error between the master system (1) and the controlled slave system (5), one can obtain the error dynamical system as follows
Proof.
e(t) ˙ = (F (y(t), y(t − τ ), p, q) − F (x(t), x(t − τ ), p, q)) + (F (y(t), y(t − τ ),p,q) − F (y(t), y(t − τ ), p, q)) + εe(t),
V˙ (t) =
n
γi
e2j (s) exp(µ(s + τ ))ds
m m 1 1 (pij − pij )2 + (q − qij )2 αij βij ij j=1
i=1
Moreover lim (pij − pij ) = lim (q ij − qij ) = 0.
t
j=1
j=1
1 + (εi + li )2 . δi
(12)
From Condition (8), there exists a small enough real number µ ≥ 0, such that n µ 2(1−r) + (γi ki2r + γj kj exp(µτ )) −γi li − 2 j=1
< 0, (11)
i = 1, 2, . . . , n.
(13)
Differentiating V (t) with respect to time along the solution of (11), and by using the inequality xy ≤ (1/2)(x2 + y 2 ), one has
m ((pij − pij )fij (y(t)) ei (t) (Fi (y(t), y(t − τ ), p, q) − Fi (x(t), x(t − τ ), p, q)) +
i=1
j=1 n
1 2(1−r) 2 exp(µt) + ki (ej (t) exp(µτ) + (q ij − qij )gij (y(t − τ ))) + εi ei (t) exp(µt) + 2 2 j=1 m
m 1 1 1 2 (p − pij )p˙ ij + (q − qij )q˙ ij + (εi + li )ε˙i − ej (t − τ ))) exp(µt) + γij ij βij ij δi j=1 j=1
n n µ 2 e (t) + |ei (t)|ki γi (|ej (t)| + ej (t − τ )) exp(µt) −li + ≤ 2 i i=1 j=1 n 1 2(1−r) 2 2 ki (ej (t) exp(µτ ) − ej (t − τ )) exp(µt) + 2 j=1
n n µ 2 2(1−r) 2 e (t) + γi −li + (ki2r e2i (t) + ki ej (t) exp(µτ )) exp(µt) ≤ 2 i i=1 j=1
n n µ 2(1−r) + (γi ki2r +γj kj exp(µτ )) e2i (t) exp(µt) −γi li − ≤ 2 i=1
≤ 0.
µ e2i (t)
j=1
(14)
It is obvious that V˙ = 0 if and only if ei = 0 (i = 1, 2, . . . , n). Moreover, by the linear feedback updated laws (6) and the parameter adaptive laws (7), all the ei = 0 (i = 1, 2, . . . , n) implies that
Adaptive Synchronization of Coupled Chaotic Delayed Systems
ε˙i = 0 (i = 1, 2, . . . , n) and p˙ ij = 0, q˙ij = 0 (i = 1, 2, . . . , n, j = 1, 2, . . . , m). This leads to pij =
pij ,
q ij =
qij ,
εi = εi0 ,
i = 1, 2, . . . , n, j = 1, 2, . . . , m, (15) on which the set V˙ = 0. Note that the construction of the Lyapunov functional (12) and the inequality (14) implies the boundedness of all ei (t) (i = 1, 2, . . . , n). According to the well-known Liapunov– LaSalle-type theorem of functional differential equations (see Theorem 5.3 in Chapter 2 of [Kuang, 1993]), the trajectories of the error dynamical system (11), starting with arbitrary initial values, converge asymptotically to the largest invariant set E contained in V˙ = 0 as t → ∞, where the set )n×m , E = {e = 0 |p = (pij )n×m , q = (qij ε = diag(ε10 , . . . , εn0 )}.
(16)
Next, we need only to show that if the master system (1) is a chaotic system, then pij =
pij ,
, qij
qij = i = 1, 2, . . . , n, j = 1, 2, . . . , m.
(17)
Indeed, if x(t) is an arbitrary given chaotic solution trajectory of the master system (1), then y(t) is also a chaotic trajectory since limt→+∞ e(t) = 0, which is obviously bounded, namely, there exists a constant c > 0, such that y(t) ≤ c. It follows that y(t) ˙ is uniformly bounded in t, due to the uniform boundedness of the right-hand side of the system (5) on the compact set Sc = {y | y ≤ c } by assumptions of the functions ci , fij and gij , so y(t) is uniformly continuous in time t. Consequently, it is easy to see that e(t) ˙ is bounded and uniformly continuous in time t by the Lipschitz condition (3). It then follows from the Barbalat Lemma [Slotine, ˙ = 0. Thus, taking a limit 2004] that limt→+∞ e(t) on two sides of the system (11) gives 0 = lim (F (y(t), y(t − τ ),p,q) t→+∞
− F (y(t), y(t − τ ), p, q)).
(18)
Note that y(t) is a chaotic trajectory as t → +∞, so both limt→+∞ y(t) < +∞ and limt→+∞ y(t − τ ) < +∞, and these limits do not exist. Putting all these properties together, with the uniform continuity of the functions fij and gij , one obtains m ((pij − pij )fij (y(t)) 0 = lim t→+∞
j=1
+ (q ij − qij )gij (y(t − τ )))
=
m
(pij − pij )fij
j=1 − qij )gij + (qij
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lim y(t)
t→+∞
lim y(t − τ ) .
t→+∞
(19)
for all i = 1, . . . , n. This obviously implies that pij = = q (i = 1, 2, . . . , n, j = 1, 2, . . . , m), meanpij , qij ij ) ing that p = p = (pij )n×m and q = q = (qij n×m . As a consequence, the unknown parameters p and q with arbitrary initial values will approximate asymptotically the parameter identification values p and q of the master system as t → +∞, respectively. By the same arguments as in the proof of Theorem 2 in [Zhou, 2004a], one can also derive the following inequality n 1 2 (yi (t) − xi (t))2 |e(t)|2 = i=1
µ ≤ N φ2 exp − t , 2
N > 0,
(20)
which implies clearly (9). This completes the proof of Theorem 1. Theorem 2. Let the controller U (t : x(t), y(t)) =
ε(y(t) − x(t)), where the feedback strength ε = diag(ε1 , . . . , εn ) with the following update laws ε˙i = −δi |ei (t)| exp(µt),
i = 1, 2, . . . , n.
(21)
and the parameter adaptive laws of p = (pij )n×m and q = (q ij )n×m are chosen as below p˙ ij = −γij sgn ei (t) fij (y(t)) exp(µt), i = 1, . . . , n, j = 1, . . . , m. (22) ˙ q ij = −βij sgn ei (t) gij (y(t − τ )) exp(µt), i = 1, . . . , n, j = 1, . . . , m, in which µ ≥ 0 is a small enough real number properly selected, the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2, . . . , n), δi > 0 (i = 1, 2, . . . , n) and αij > 0, βij > 0 (i = 1, 2, . . . , n, j = 1, 2, . . . , m) are arbitrary constants, respectively. If there exist n positive numbers γ1 , . . . , γn such that −γi li + 2
n
γj kj < 0,
i = 1, 2, . . . , n,
(23)
j=1
where li ≥ − limt→+∞ εi (t) = −εi0 > 0 (i = 1, 2, . . . , n) are constants properly selected, then the controlled uncertain slave system (5) is globally synchronous with the master system (1) and
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satisfies n
≤ |yi (t) − xi (t)| = O(exp(−µt)),
n
γi −(li − µ)|ei (t)|
i=1
(24)
i=1
+
and (10).
n
ki |ej (t)|(1 + exp(µτ )) exp(µt)
j=1
Just as in the proof of Theorem 1, we construct a Lyapunov functional of the form
n n γi |ei (t)| exp(µt) + ki V (t) =
Proof.
i=1
j=1
×
t
t−τ
n n 1 1 1 2 (pij − pij ) + (q − qij )2 + 2 αij βij ij j=1 j=1 1 + (εi + li )2 . (25) δi It follows from (23) that there exist a small enough real number µ ≥ 0, such that n γj kj exp(µτ ) < 0, −γi (li − µ) + 2 j=1
i = 1, 2, . . . , n.
(21)
Calculating the upper Dini derivative of V with respect to time along the trajectories of (11), one has
n γi sgn ei (t) (Fi (y(t), y(t − τ ), p, q) V˙ (t) = i=1
− Fi (x(t), x(t − τ ), p, q)) m ((pij − pij )fij (y(t)) +
+ (q ij − qij )gij (y(t − τ ))) + εi ei (t) exp(µt) + |ei (t)|µ exp(µt) +
n
ki (|ej (t)| exp(µτ )
j=1
− |ej (t − τ )|) exp(µt) m 1 (p − pij )p˙ ij + αij ij j=1
n 1 1 + (q ij − qij )q˙ ij + (εi + li )ε˙i βij δi j=1
−γi (li − µ) + 2
i=1
n
γj kj exp(µτ )
j=1
× |ei (t)| exp(µt) ≤ 0.
|ej (s)| exp(µ(s + τ ))ds
j=1
≤
n
(26)
In the estimation of the inequality (26), sgn ei (t) × ei (t) = |ei (t)| has been used. It is obvious that V˙ = 0 if and only if ei = 0 (i = 1, 2, . . . , n). Thus, as in the proof of Theorem 1, Eq. (19) is easily derived. The rest of the proof is similar to that of Theorem 3 in [Zhou, 2004a], and hence it is omitted. The proof of Theorem 2 is thus completed. Remark 2. It can be seen that Theorems 1 and
2 supply a systematic and analytical procedure for determining global synchronization based on parameter identification of uncertain chaotic delayed systems. In particular, the Lyapunov functional techniques employed here can guarantee global exponential stability of the synchronization error system, and also give the corresponding exponential convergence rate. So just as stated in [Huang, 2004a, 2004b, 2005], these estimation approaches are not only robust against the effect of noise, but are also able to respond to changes in identifying parameters of the master system. It is useful to point out that the distinguished characteristics of our methods can be applied to almost all the chaotic dynamical systems without delays or with delays satisfying the uniform Lipschitz conditions. Therefore, the approaches developed here are very convenient to be implemented in practice. Remark 3. It is very important to point out that
Theorem 1 is actually a natural extension and generalization of an adaptive synchronization scheme proposed in [Huang, 2004a, 2004b, 2005] for uncertain chaotic systems without delays. From (9) and (24), we easily know that the exponential convergence rate of the synchronization error system in Theorem 2 is nearly twice as that in Theorem 1. Since both the feedback strength update laws and the parameter adaptive laws are related to the synchronization errors, by comparing Eqs. (6) and (7) with Eqs. (21) and (22) respectively, it is
Adaptive Synchronization of Coupled Chaotic Delayed Systems
easily verified that the linear feedback gains and all the unknown parameters presented in Theorem 2 will converge faster to their reference values respectively than those given in Theorem 1. It means that by selecting the control scheme proposed in Theorem 2, one can achieve global synchronization in a short time interval while identifying all the unknown parameters rapidly. Therefore, the approaches developed here extend the ideas and techniques presented in the literature [Huang, 2004a, 2004b, 2005]. This point will be further illustrated through the numerical simulations in the next section. Remark 4. Finally, from the proof of Theorems 1
and 2, we point out that Conditions (8) and (23) can be relaxed to n 1 2(1−r) (γi ki2r + γj kj ) < 0, −γi li + 2 j=1
i = 1, 2, . . . , n.
(27)
i = 1, 2, . . . , n.
(28)
and −γi li +
n
γj kj < 0,
j=1
respectively, the consequences of Theorems 1 and 2 still hold for uncertain chaotic systems without delays.
4. Application Examples and Simulation Results As an application of the above-derived theoretical criteria, the synchronization and identification problem of some typical examples of chaotic systems is worked out in this section. Example 1. Consider the original Chua’s circuit
described by [Chua et al., 1993; Chua et al., 1996] x˙ 1 = p1 (x2 − x1 − f (x1 )), x˙ 2 = x1 − x2 + x3 , (29) x˙ 3 = −p2 x2 − p3 x3 , where the parameters p1 > 0, p2 > 0, p3 > 0, a < b < 0, and the piecewise linear function f (x) is described by 1 (30) f (x) = bx + (a − b)(|x + 1| − |x − 1|). 2 As is well known, the Chua circuit plays an important role since it is an extremely simple system yet exhibits complex dynamics of bifurcation
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and chaos [Chua et al., 1993; Chua et al., 1996; Jiang et al., 2003; Zheng et al., 2002; Wang et al., 1999]. It is known that with parameters p1 = 10, p2 = 15, p3 = 0.0385, a = −1.27 and b = −0.68, the solution trajectory of (29) approaches a chaotic attractor, as shown in Fig. 1. In order to verify the effectiveness of the proposed methods, let the master output signals be from the Chua circuit (29). We assume that the parameters p1 , p2 and p3 will be identified, then the controlled slave system is given for the drive system (29) with a linear unidirectional coupling y˙ 1 = p1 (y2 − y1 − f (y1 )) + ε1 (y1 − x1 ), y˙ 2 = y1 − y2 + y3 + ε2 (y2 − x2 ), (31) y˙ = −p y −p y + ε (y − x ). 3 3 3 3 2 2 3 3 Next, one chooses the feedback strength update laws and the parameter adaptive laws ε˙i = −(yi − xi )2 exp(0.0035t), i = 1, 2, 3. ˙ p1 = −13(y1 − x1 )(y2 − y1 − f (y1 )) × exp(0.0035t), (32) p˙ 2 = 10.15(y3 − x3 )y2 exp(0.0035t), ˙ p3 = 15(y3 − x3 )y3 exp(0.0035t), and ε˙i = −|yi − xi | exp(0.0035t), i = 1, 2, 3. p˙ = −13 sgn(y1 − x1 )(y2 − y1 − f (y1 )) 1 × exp(0.0035t), p˙ 2 = −10.15 sgn(y3 − x3 )y2 exp(0.0035t), ˙ p3 = −15 sgn(y3 − x3 )y2 exp(0.0035t),
(33)
respectively.
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −4
−3
−2
−1
0
1
2
3
4
Fig. 1. A double scroll chaotic attractor of the Chua’s circuit (29) with parameters p1 = 10, p2 = 15 and p3 = 0.0385.
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Let li > 68.1 (i = 1, 2, 3), it is easy to verity that both Conditions (27) and (28) are satisfied with γi = 1, ki = 22.7 (i = 1, 2, 3) and r = 1/2. It follows from Remark 3, Theorems 1 and 2 that the controlled uncertain slave system (31) is globally synchronous with the master system (29) and satisfies lim (pi − pi ) = 0.
i = 1, 2, 3,
t→+∞
(34)
based on the feedback strength update laws and the parameter adaptive laws (32) and (33), respectively. Figures 2 and 3 display the simulation results, which show clearly the parameter identification and adaptive synchronization of the uncertain Chua’s circuits. In the simulations, the initial conditions of the feedback strength and the unknown parameters of the controlled slave system are chosen as follows: (ε1 (0), ε2 (0), ε2 (0)) = (−6, −6, −6) ,
20
2
p2 e2(t)
15
3
1
e (t)
p ,p ,p
1
2
0.5 0 −0.5
10
p1
1
e1(t), e2(t), e3(t),
1.5
5
e (t)
p3
3
−1
0.0385
−1.5 −2 0
50
100
−5 0
150
50
100
150
time t
time t (a)
(b)
Fig. 2. Adaptive synchronization and parameter identification of the coupled Chua’s circuits (31) and (29) based on the feedback strength update laws and the parameter adaptive laws (32) in time interval [0, 120], where the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2, 3).
20
2 1
p
2
15
e2(t)
3
1
e (t)
0.5
10
2
p ,p ,p
0
1
e1(t), e2(t), e3(t)
1.5
−0.5
p
1
5
p
3
−1
e3(t)
0
−1.5 −2 0
1
2
3
time t (a)
4
5
6
−5 0
1
2
3
4
5
6
time t (b)
Fig. 3. Adaptive synchronization and parameter identification of the coupled Chua’s circuits (31) and (29) based on the feedback strength update laws and the parameter adaptive laws (33) in time interval [0, 6], where the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2, 3).
Adaptive Synchronization of Coupled Chaotic Delayed Systems
(p1 (0),p2 (0),p3 (0)) = (13, 11.25, 1) (see Figs. 2 and 3). Example 2. Consider a typical delayed Hopfied neural network with two neurons [Lu, 2002; Zhou et al., 2004b, 2005a, 2005b, 2006]
x(t) ˙ = −Cx(t) + Af (x(t)) + Bf (x(t − τ )),
(35)
(t)) ,
f (x(t)) = (tanh(x1 (t)), where x(t) = (x1 (t), x2 tanh(x2 (t))) , τ = 1, and 1 0 3.0 5.0 C= , A= with 0 1 0.1 2.0 −2.5 0.2 B= . 0.1 −1.5 It should be noted that the network is actually a chaotic delayed Hopfied neural network [Lu, 2002] (see Fig. 4). Let the master output signals be from the delayed neural network (35). For simplicity, we assume only that the four parameters a11 = 3.0, a22 = 2.0, b11 = −2.5 and b22 = −1.5 will be identified, then the controlled slave system is given by the following equation y(t) ˙ = −Cy(t) +Af (y(t)) +Bf (y(t − τ )) + ε(y(t) − x(t)),
(36)
(t))
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −2
−1
0
1
and ε˙i = −3|yi − xi |, a˙ ii = −3 sgn(yi − xi ) tanh yi (t), ˙ bii = −10 sgn(yi − xi ) tanh yi (t − 1),
i = 1, 2. i = 1, 2. i = 1, 2, (38)
respectively. It is easy to show that both Conditions (8) and (23) are also satisfied with γi = 1, ki = 6 (i = 1, 2) and r = 1/2 if li > 24 (i = 1, 2). From Theorems 1 and 2, one can conclude that the controlled uncertain slave system (36) is globally synchronous with the master system (35) and satisfies lim (aii − aii ) = lim (bii − bii ) = 0, t→+∞
i = 1, 2, (33)
0.8
−3
(37)
with
1
−1 −4
and the feedback strength ε = diag(ε1 , ε2 ). According to Theorems 1 and 2, one can easily construct the feedback strength update laws and the parameter adaptive laws as follows 2 i = 1, 2. ε˙i = −3(yi − xi ) , ˙aii = −2(yi − xi ) tanh yi (t), i = 1, 2. ˙ bii = −10(yi − xi )(tanh yi (t − 1)), i = 1, 2,
t→+∞
and where y(t) = (y1 (t), y2 1 0 a11 5.0 C= , A= 0 1 0.1 a22 b11 0.2 B= , 0.1 b22
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2
3
4
Fig. 4. A fully developed double-scroll-like chaotic attractor of the delayed Hopfied neural networks (35).
based on the feedback strength update laws and the parameter adaptive laws (37) and (38), respectively. Let the initial conditions of the feedback strength and the unknown parameters of the controlled slave system be taken as follows: (ε1 (0), ε2 (0)) = (−6, −6) , (a11 (0), a22 (0), b11 (0), b22 (0)) = (0, 2.8, −1.5, −3) , respectively. The numerical simulations show that parameter identification and adaptive synchronization are achieved successfully (see Figs. 5 and 6). The above numerical simulations are done by using the Delay Differential Equations (DDEs) Solvers in Matlab Simulink Toolbox. These simulations in Examples 1 and 2 have clearly shown that by using the schemes proposed in Theorem 2, both adaptive synchronization and parameter identification are more rapidly achieved for the uncertain slave system (5) with the master system (1) than by those presented in Theorem 1. This conclusion is obviously consistent with the theoretical results of this paper. Therefore, the approaches developed here further extend the ideas and techniques presented so far in the literature.
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J. Zhou et al. 2
4
a 22
1
0
1 0
11
e2(t) −1
100
11
e (t)
−2 0
2
a ,a ,b ,b
e1(t), e2(t)
1
22
3
200
300
400
500
−1 −1.5 −2 −2.5 −3 −4 0
600
11
a22
b
22
b
11
100
200
300
time t
400
500
600
time t
(a)
(b)
Fig. 5. Adaptive synchronization and parameter identification of the coupled delayed Hopfied neural networks (36) and (35) based on the feedback strength update laws and the parameter adaptive laws (37) in time interval [0, 120], where the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2).
2
4
a11
1
22
0
1 0
11
e2(t) −1
1
11
1
−2 0
2
a ,a ,b ,b
e (t)
2
e (t), e (t)
1
22
3
2
3
4
5
6
time t (a)
a
22
b
22
−1 −1.5 −2 −2.5 −3 −4 0
b11 1
2
3
4
5
6
time t (b)
Fig. 6. Adaptive synchronization and parameter identification of the coupled delayed Hopfied neural networks (36) and (35) based on the feedback strength update laws and the parameter adaptive laws (37) in time interval [0, 6], where the synchronization errors ei (t) = yi (t) − xi (t) (i = 1, 2).
5. Conclusions In this paper, we introduce two methods for adaptive synchronization and parameter identification of coupled chaotic delayed systems with all the parameters unknown. These approaches supply a systematic and analytical procedure for determining global synchronization based on parameter identification of uncertain chaotic delayed systems, and even they are very convenient to be implemented in
practice. It is shown that the approaches developed here further extend the ideas and techniques presented in recent literature. To this end, the theoretical results can be applied to the well-known Chua’s circuit and a typical class of chaotic delayed Hopfied neural networks. Numerical experiment shows the effectiveness of the proposed methods. A possible application of the proposed methods is to secure message transmission using parameter modulation.
Adaptive Synchronization of Coupled Chaotic Delayed Systems
It is believed that these techniques are essential for chaos communication in practical designs and engineering applications.
Acknowledgments This work was supported by the National Science Foundation of China (Grant nos. 60474071 and 10672094), the China Postdoctoral Science Foundation (Grant no. 20040350121) and the Science Foundation of Education Commission of Hebei Province (Grant no. 2003013). The authors wish to acknowledge the benefit received from valuable discussions with Prof. Debin Huang.
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