Synchronization of non-identical chaotic systems: an exponential dichotomies approach

HOME | SEARCH | PACS & MSC | JOURNALS | ABOUT | CONTACT US Synchronization of non-identical chaotic systems: an exponential dichotomies approach This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2001 J. Phys. A: Math. Gen. 34 9143 (http://iopscience.iop.org/0305-4470/34/43/304) The Table of Contents and more related content is available Download details: IP Address: 190.169.254.95 The article was downloaded on 30/06/2009 at 15:27 Please note that terms and conditions apply. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 34 (2001) 9143–9151 PII: S0305-4470(01)26700-6 Synchronization of non-identical chaotic systems: an exponential dichotomies approach A Acosta1 and P Garcı́a2 1 Departamento de Matemática Aplicada, Facultad de Ingenierı́a, Universidad Central de Venezuela, AP 48110, Caracas, 1041-A, Venezuela 2 Departamento de Fı́sica Aplicada, Facultad de Ingenierı́a, Universidad Central de Venezuela, AP 48110, Caracas, 1041-A, Venezuela E-mail: pedro@apollo.ciens.ucv.ve Received 5 July 2001, in final form 29 August 2001 Published 19 October 2001 Online at stacks.iop.org/JPhysA/34/9143 Abstract In most applications, the synchronization of systems evolving under a chaotic regime requires the construction of identical systems or subsystems. In practical applications, systems should be created so that they match as closely as possible. Moreover, in real devices parameters can fluctuate resulting in loss of synchronization. In this paper, we consider a master–slave system of ordinary differential equations which are not identical. Considering bounded solutions of the master equation, we use those as an input in the slave equation. By using exponential dichotomies techniques we establish conditions that ensure synchronization. PACS numbers: 05.45.+b, 02.30.Hq 1. Introduction Synchronization between chaotic dynamical systems has been an active research topic since it was introduced by Fujisaka and Yamada [1]. The reason for this increasing interest is the wide range of applications which go from communication [2–5] to biology [6, 7]. The synchronization of chaotic systems can be achieved in several forms, for example, the work of Pecora and Caroll [8] shows that under suitable conditions, two chaotic systems S1 and S2 can be synchronized if S2 is formed copying a subsystem that is a replica of part of the system S1 . Another possibility consists in coupling S1 and S2 by a small linear term, in which the difference between the current state of the two systems is used as an inhibitory effect on the separation of the orbits [9]. A common feature of this and other methodologies is that the considered systems are identical. In practical applications, it is impossible to construct devices with identical 0305-4470/01/439143+09$30.00 © 2001 IOP Publishing Ltd Printed in the UK 9143 9144 A Acosta and P Garcı́a parameters. Therefore, if we want to synchronize real systems it seems to be more convenient to consider models with different parameters. In this paper, we study master–slave non-identical systems, where master–slave means that one of the systems (master) evolves freely while the other (slave) is driven by the master. The class of systems that we consider satisfies the conditions so that the parameters, which are not identical, appear to affect only linear terms. Systems illustrating this class are Lorenz and Rossler equations. In order to obtain synchronization between master–slave non-identical systems, we proceed as follows. First, a bounded solution of the master equation is considered and it is used as an input in the slave equation. Next, our attention is focused on the non-autonomous system that is obtained from the slave equation and, by using exponential dichotomies theory in an appropriate framework, we establish conditions that ensure synchronization of our master–slave system. The rest of this paper is organized as follows. In section 2, we set our problem and present what is needed regarding exponential dichotomies. Section 3 is devoted to proving the main result and in section 4, this result is applied to a particular example. Finally, in section 5 we give some concluding remarks. 2. Setting of the problem and exponential dichotomies We consider the system ẋ = f (µ̄, x) ẏ = f (µ, y) + ν(x − y) (1) (2) where ν is a real constant and f : ℜm × ℜn → ℜn is a continuous function that satisfies the following hypotheses: H1 f (µ, z) = B(µ)z + g(z), where B is a matrix of dimension n × n that depends on the vector parameter µ and g is a non-linear function. H2 f (µ, z + x) − f (µ, x) = f (µ, z) + C(x)z, where C is a matrix of dimension n × n that depends on x. H3 There exists K1 > 0 so that |f (µ, z) − f (µ̄, z)|
K1 |µ − µ̄| |z|. Also, we assume for the nonlinear function g that H4 |g(z) − g(w)|
η(ρ)|z − w| |z|, for all z, w ∈ ℜn such that |z|, |w|
ρ, where η is a continuous, non-decreasing, non-negative function on [0, ∞) with η(0) = 0. In addition to the previous hypothesis it is important to remark that throughout this work we assume µ̄ and µ as constant vectors. We define the set Bm as
 Bm := x0 ∈ ℜn : x(t, x0 , µ̄) is bounded on [0, ∞) . (3) Now let x(t, x0 , µ̄) denote a solution of equation (1) (master) satisfying x(0, x0 , µ̄) = x0 and consider it as an input in equation (2) (slave) for which y(t, x0 , y0 , µ, µ̄) denotes the solution satisfying y(0, x0 , y0 , µ, µ̄) = y0 . Definition 2.1. Let x0 ∈ Bm , we say that the system (1)–(2) synchronizes along the trajectory x(t, x0 , µ̄), t  0, if there exists a set V in ℜn such that: if given ǫ > 0, then δ > 0 exists such that if |µ − µ̄| < δ and y0 − x0 ∈ V , then lim sup|y(t, x0 , y0 , µ, µ̄) − x(t, x0 , µ̄)| < ǫ. t→∞ Synchronization of non-identical chaotic systems 9145 We now consider some properties of exponential dichotomies of linear systems of differential equations. We present some lemmas to be applied in the next two sections. Let A : J → ℜn×n be continuous, where J is some interval, and consider the differential equation: (4) ż = A(t)z Let (t, s), (t, t) = I , be the principal matrix solution of (4). Definition 2.2. We say that (4) has an exponential dichotomy on the interval J if there are projections P (t): ℜn → ℜn , t ∈ J , continuous in t, such that if Q(t) := I − P (t), where I is the identity matrix, then: (i) (t, s)P (s) = P (t)(t, s), t, s ∈ J. (ii) |(t, s)P (s)|
Ke−α(t−s) , t  s ∈ J. (iii) |(t, s)Q(s)|
Keα(t−s) , s  t ∈ J . where K and α are positive constants. The two cases of most interest are where J is the positive half-line [0, ∞) and the whole line ℜ. However, we are only interested in the first case. In the case of the autonomous equation ż = A0 z there is an exponential dichotomy on [0, ∞) if and only if no eigenvalue of the constant matrix A0 has zero real part. In this example associated with the trivial solution there are two sets called the stable manifold and the unstable manifold. The concept of exponential dichotomy provides the notions of those sets for the non-autonomous equations. Consider the inhomogeneous equation ż = A(t)z + f (t) (5) where f is in the Banach space of all bounded continuous functions with the supremun norm. Lemma 2.1. Suppose that (4) has an exponential dichotomy on [0, +∞). For any solution z(t) of (5) which exists and is bounded on [0, +∞), there is an z0 ∈ Range of P (0) such that z(t) satisfies  t  t z(t) = (t, 0)z0 + Q(t)(t, s)f (s) ds t  0. (6) P (t)(t, s)f (s) ds + 0 ∞ Conversely, any solution of (6) bounded on [0, +∞) is a solution of (6). Proof. See [10].  Now, we consider a perturbation of the differential equation (4). Let B : [0, +∞) → ℜn×n be a bounded, continuous matrix function. Lemma 2.2. Suppose that (4) has an exponential dichotomy on [0, +∞). If δ := sup |B(t)| < α/4K 2 , then the perturbed equation ż = (A(t) + B(t))z (7) also has an exponential dichotomy on [0, +∞) with constants K̃ and α̃ determined by K, α and δ. Moreover if P̃ (t) is the corresponding projection, then |P (t)− P̃ (t)| = O(δ) uniformly in t ∈ [0, +∞). Also |α̃ − α| = O(δ). 9146 A Acosta and P Garcı́a Proof. See [10, 11].  To set up the problem in a framework where exponential dichotomies can be applied, we consider, for x0 ∈ Bm , the following transformation of variables z = y − x(t, x0 , µ̄). (8) If y is a solution of the slave equation with input x(t, x0 , µ̄), then the transformation (8) applied to this equation yields the equation ż = A(ν, µ, x(t, x0 , µ̄))z + F (µ, z, x(t, x0 , µ̄)) (9) where A(ν, µ, x(t, x0 , µ̄)) := νI + B(µ) + C(x(t, x0 , µ̄)) (10) and F (µ, z, x(t, x0 , µ̄)) := g(z) + f (µ, z, x(t, x0 , µ̄)) − f (µ̄, z, x(t, x0 , µ̄)). (11) We will assume, in the next section, that the linear equation corresponding to (10), i.e. ż = A(ν, µ, x(t, x0 , µ̄))z (12) has an exponential dichotomy on [0, +∞). 3. Main result Two lemmas, one on the existence of solutions of equation (9) and another which is related to the Gronwall inequality, will be the key elements to establish our main result. From now on we assume that equation (12) has an exponential dichotomy on [0, +∞) with projections P (t) and where α and K are the corresponding constants. Also, we assume that x0 ∈ Bm . Let ρ > 0 and µ ∈ ℜm such that α η(ρ) < (13) 8K αρ . (14) |µ − µ̄| < 8KK1 supt
0 |x(t, x0 , µ̄)| With this choice of ρ and µ, and for any z0 in the range of P (0) with |z0 | < ρ/2K, we define G (z0 , ρ, ν, µ) as a set of continuous functions z : [0, +∞) → ℜn such that |z| := supt
0 |z(t)|
ρ and P (0)z(0) = z0 . G (z0 , ρ, ν, µ) is a closed bounded subset of the Banach space of all continuous functions taking [0, +∞) into ℜn with uniform topology. For any z ∈ G (z0 , ρ, ν, µ), we define T z by  t P (t)(t, s)F (µ, z(s), x(s, x0 , µ̄)) ds (T z)(t) = (t, 0)z0 + 0  t Q(t)(t, s)F (µ, z(s), x(s, x0 , µ̄)) ds t  0. + ∞ Lemma 3.1. T acts from G (z0 , ρ, ν, µ) into itself and also has a unique fixed point in G (z0 , ρ, ν, µ). Proof. Given G (z0 , ρ, ν, µ), it is easy to see that T z is defined and continuous for t  0 with P (0)(T z)(0) = z0 . Synchronization of non-identical chaotic systems 9147 The fact that (12) has an exponential dichotomy on [0, ∞) and the definition of F produce, for t  0, the estimation  t K e−α(t−s) |g(z(s))| ds |(T z)(t)|
K e−αt |z0 | + 0  ∞  t −α(t−s) K eα(t−s) |g(z(s))| ds Ke |f (µ, x(s)) − f (µ̄, x(s))| ds + + t 0 ∞ α(t−s) + Ke |f (µ, x(s)) − f (µ̄, x(s))| ds t where x(s) ≡ x(s, x0 , µ̄). Now, since g(0) = 0 and from H3, H4, we obtain |g(z(s))| + f (µ, x) − |f (µ̄, x)|
η(ρ)ρ + K1 |µ − µ̄| sup |x(s̃)|. s̃
0 Therefore,  |T z(t)|
K e−αt |z0 | + K η(ρ)ρ + K1 |µ − µ̄| sup |x(s̃)| s̃
0 ∞   t e−α(t−s) ds 0  eα(t−s) ds t   2K η(ρ)ρ + K1 |µ − µ̄| sup |x(s̃| .
K|z0 | + α s̃
0 +  Thus, from (14) and the condition |z0 | < ρ/2K, we obtain |T z| < ρ. Therefore, T acts from G (z0 , ρ, ν, µ) into itself. Furthermore, the same types of estimates yield, for z and w ∈ G (z0 , ρ, ν, µ), 2K 1 η(ρ)|z − w|
|z − w| for t  0. α 4 Thus, T is a contraction on G (z0 , ρ, ν, µ) and it has a unique fixed point. |(T z)(t) − (T w)(t)|
 Lemma 3.2. Suppose a > 0, b > 0, K, L, M are non-negative constants and u is a nonnegative bounded continuous solution of the inequality  ∞  t eb(t−s) u(s) ds t  0. e−a(t−s) u(s) ds + M u(t)
K e−at + L 0 t If β := L M + <1 a b then u(t)
(1 − β)−1 K e−[a−(1−β) Proof. See [12]. −1 L]t .  Let z∗ (·, z0 , ν, µ) denote the fixed point in lemma 3.1. An important remark is that using the same estimates as above, one shows that the function z∗ (·, z0 , ν, µ) is continuous on the variables z0 and µ and z∗ (·, 0, ν, µ) = 0. Our main result, which implies that the master–slave system synchronizes, is presented now. 9148 A Acosta and P Garcı́a Theorem 3.1. If the hypotheses H1, H2, H3 and H4 are satisfied and equation (12) has an exponential dichotomy on [0, +∞), then under the estimates (13) and (14), z∗ (·, z0 , ν, µ) satisfies the estimation 5 2KK1 1 4 |µ − µ̄||x(·, x0 , µ̄)| |z∗ (t, z0 , ν, µ)|
K|z0 | e− 6 αt + |z∗ (·, 0, ν, µ)| + 3 4 α t  0. (15) Proof. Let zz∗0 (t) denote the fixed point zz∗0 (t, z0 , ν, µ). First, estimations for zz∗0 and the difference between the fixed points zz∗0 and zz̃∗0 are obtained.  ∞  t eα(t−s) |g(z0∗ (s))| ds K e−α(t−s) |g(z0∗ (s))| ds + |z0∗ (t)|
t 0  t −α(t−s) Ke |f (µ, x(s)) − f (µ̄, x(s))| ds + 0  ∞ eα(t−s) |f (µ, x(s)) − f (µ̄, x(s))| ds + t  t  ∞ eα(t−s) ds
Kη(ρ)|z0∗ (·)| e−α(t−s) ds + t 0  t  ∞ eα(t−s) ds + KK1 |µ − µ̄||x(·)| e−α(t−s) ds + 0 t α ∗ 2 |z0 (·)| + KK1 |µ − µ̄||x(·)|
8 α 1 ∗ 2KK1 |µ − µ̄||x(·)| = |z0 (·)| + 4 α  α t zz∗0 (t) − zz̃∗0 (t)
K e−αt |z0 − z̃0 | + K e−α(t−s) |zz∗0 (s) − zz̃∗0 (s)| ds 8 0  α t + K eα(t−s) zz∗0 (s) − zz̃∗0 (s) ds. 8 0 If lemma 3.2 is applied with a = b = α, K = K|z0 − z̃0 |, L = M = α/8, then β = 1/4 and 5 4 K|z0 − z̃0 | e− 6 αt . 3 Finally, we apply the previous estimations, with z̃0 = 0, to the right-hand side of the following inequality: zz∗0 (t) − zz̃∗0 (t)
zz∗0 (t)
zz∗0 (t) − z0∗ (t) + |z0∗ (t)|. 4. Application In order to apply our main result we use the Lorenz equations: ẋ 1 = σ̄ (y1 − x1 ) ẏ 1 = r̄x1 − y1 − x1 z1 ż1 = x1 y1 − b̄z1 .  Synchronization of non-identical chaotic systems 9149 It satisfies hypotheses H1, H2, H3 and H4, and the master–slave system results in ẋ 1 = σ̄ (y1 − x1 ) ẏ 1 = r̄x1 − y1 − x1 z1 ż1 = x1 y1 − b̄z1 ẋ 2 = σ (y2 − x2 ) + ν(x2 − x1 ) ẏ 2 = rx2 − y2 − x2 z2 + ν(y2 − y1 ) ż2 = x2 y2 − bz2 + ν(z2 − z1 ). We concentrate our attention in the master–slave system with the usual parameters σ̄ = 10, r̄ = 28 and b̄ = 8/3. In this case let (x1 (t), y1 (t), z1 (t)) be a bounded solution of the master equation. In this particular case, the matrix A(ν, µ, x(t, x0 , µ̄)) := νI + B(µ) + C(x(t, x0 , µ̄)) where µ = (σ, r, b), µ̄ = (10, 28, 8/3), x0 = (x1 (0), y1 (0), z1 (0)) and x(t, x0 , µ̄) = (x1 (t, x0 , µ̄), y1 (t, x0 , µ̄), z1 (t, x0 , µ̄)), is given by       ν 0 0 −σ σ 0 0 0 0 0 ν 0  +  r 0 −x1 (t) . −1 0  + −z1 (t) 0 0 ν y1 (t) x1 (t) 0 0 0 −b are The eigenvalues of the matrix   ν −σ σ 0  r ν −1 0  0 0 ν −b λ1 = ν − b 2ν − σ − 1 + [(σ + 1)2 + 4σ (r − 1)]1/2 λ2 = 2 2ν − σ − 1 − [(σ + 1)2 + 4σ (r − 1)]1/2 λ3 = . 2 In particular, for σ = 10, r = 28 and b = 8/3, the eigenvalues are λ1 = ν − 8 3 √ 2ν − 11 ± 1201 . λ2,3 = 2 √ For ν < (11 − 1201)/2, these eigenvalues are negative and the system ż = (νI + B(µ̄))z has an exponential dichotomy on [0, ∞) with P = identity, K = 1 and α = (11 − 2ν − √ 1201)/2. For small deviations of the classical√parameters, i.e ‘µ − µ̄ small’, we have that for the interval (−∞, ν0 ) with ν0 close to (11 − 1201)/2 all the eigenvalues of the matrix νI + B(µ) are negative and the system ż = (νI + B(µ))z has an exponential dichotomy on [0, ∞) with P = identity, K = 1 and α close to √ (11 − 2ν − 1201)/2. 9150 A Acosta and P Garcı́a 50 40 z(t) 30 20 10 0 0 5 10 15 t 20 25 30 Figure 1. Synchronization of z-coordinates. Now if sup|C(x(t, x0 , µ̄))| < α/4, then from lemma 2.2, we obtain that the system ż = (νI + B(µ) + C(x(t, x0 , µ̄)))z has an exponential dichotomy on [0, ∞). In the simulation shown in figure 1 we have selected µ = (9.8, 28.2, 2.56) and ν = −23. We observe in the figure the evolution of the z-coordinate on the master equation and also in the slave equation. 5. Conclusions In order to establish synchronization of non-identical chaotic systems we have presented an approach based on the theory of exponential dichotomies. This approach also allows the following: • An estimation, in order to achieve control over a given chaotic system, of the intensity of the necessary perturbation to maintain the orbit of the slave system on the given orbit of the master system. • An estimation of the robustness of the synchronization against fluctuations in the parameter space around the given parameters in the master–slave system. Acknowledgment This research was partially supported by Consejo de Desarrollo Cientı́fico y Humanı́stico, Universidad Central de Venezuela. References [1] Fujisaka H and Yamada T 1983 Prog. Theor. Phys. 69 32 [2] Kokarev L and Parlitz U 1995 Phys. Rev. Lett. 74 528 Synchronization of non-identical chaotic systems [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] 9151 Gregory D, Van Wiggeren and Rajarshi Roy 1998 Phys. Rev. Lett. 87 3547 Rong He and Vaidya P G 1998 Phys. Rev. E 57 1532 Rodrigues H M 1996 Appl. Anal. 62 263 Glass L and Mackey M 1988 From Clock to Chaos: The Rhythms of Life (Princeton, NJ: Princeton University Press) Mirollo R and Strogatz S 1990 SIAM J. Appl. Math. 50 1645 Pecora L M and Caroll T L 1990 Phys. Rev. Lett. 64 821 Parmaranda P 1998 Phys. Lett. A 240 55 Palmer K J 1978 J. Diff. Eqns 55 225 Coppel W A 1978 Dichotomies in Stability Theory (Lecture Notes in Mathematics) vol 629 (New York: Springer) Hale J 1980 Ordinary Differential Equations (New York: Kriever)