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Synchronization of non-identical chaotic systems: an exponential dichotomies approach
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2001 J. Phys. A: Math. Gen. 34 9143
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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
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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 supt0 |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| := supt0 |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)|.
s̃0
Therefore,
|T z(t)| K e−αt |z0 | + K η(ρ)ρ + K1 |µ − µ̄| sup |x(s̃)|
s̃0
∞
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
s̃0
+
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.
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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.
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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.
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Synchronization of non-identical chaotic systems
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