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Preservation of Synchronization in Dynamical Systems via
Lyapunov Methods
G. FERNÁNDEZ ANAYA
Universidad Iberoamericana
Departamento de Física
y Matemáticas
Prol. Paseo de la Reforma 880,
Lomas de Santa Fe, México,
D. F., C.P. 01219
MÉXICO
guillermo.fernandez@uia.mx
C. RODRÍGUEZ LUCATERO
Universidad Autónoma,
Metropolitana Cuajimalpa
Departamento de Tecnologías
de la Información
Av. Constituyentes 1056,
Col. Lomas Altas, México,
D.F.,C.P. 11950
MÉXICO
profesor.lucatero@gmail.com
J.J. FLORES-GODOY &
C. MIRANDA-REYES
Universidad Iberoamericana
Departamento de Física
y Matemáticas
Prol. Paseo de la Reforma 880,
Lomas de Santa Fe,
México, D. F., C.P. 01219
MÉXICO
job.flores@uia.mx,
mirandare@gmail.com
Abstract: In this paper, we use, extend and apply some classic results of the theory of dynamical systems
to study the preservation of synchronization in chaotical dynamical systems via Lyapunov method. The
obtained results show that synchronization can be preserved after a particular class of changes are made
to the linear part of the dynamical system. For illustrative purposes we apply a compound control law
to achieve synchronization in a master-slave system. We also show that it is possible to preserve partial
synchronization when an additive perturbation is included in the control law. We present numerical
simulations to show the effectiveness of our method.
Key–Words: Chaotic Systems, control theory, convergence, stability.
1 Introduction
Chaotic dynamical systems and chaos control is
a theme that has been widely developed in the
last years. The study of systems with this kind
of behavior is well documented and there are sev-
eral papers where applications are presented, as
described in [1]. Also, chaos control and synchro-
nization of systems have been studied in [5, 7].
Studies on preservation of synchronization and
chaos structure can be found in [3, 4, 2].
In [4], mathematical tools to ensure synchro-
nization and preservation of hyperbolic points are
developed, which in this paper are extended, in
particular the stable-unstable manifold theorem.
In [3, 4, 2], local stability is preserved, in the
present work, it may be possible to achieve global
stability if certain conditions are met.
An extension of the stable-unstable manifold
theorem is presented. This extension is based
on the modification through matrix multiplication
and group action over the Jacobian matrix of the
dynamical systems used. In general, we will de-
fine a nonlinear function that acts over the linear
part of the dynamical system, changing it in such
a way that this modification preserves hyperbolic
equilibrium points.
It was proved in [9], [8], that using a compos-
ite control law formed by a linear and a nonlinear
part, it is possible to synchronize a master-slave
system. They also show that the control law en-
sures ultimate boundedness in the presence of ad-
ditive perturbations. In this paper we will extend
this work and show that the same design proce-
dure can be used to preserved synchronization in
a modified master-slave system.
We provide two simulation examples of
master-slave systems to illustrate the application
of our methodology. The benchmark dynamical
systems are Chua’s system and the Sprott’s Q
system. The simulations show how the modified
systems still preserve its synchronization proper-
ties.
2 Preliminaries
In this section, we present preliminary results
needed for the development of our main result.
As we know, the local stability of a dynamical
system around an equilibrium point is completely
related to the sign of the real part of the eigenval-
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ues of the matrix that represents the linear part
of the system. The main idea of this work is to
preserve the stability of a dynamical system af-
ter applying a class of transformations. The class
of transformations that we study are applied only
on the linear part of the dynamical system, these
transformations are designed to preserve the char-
acteristics of the original system.
Let Cand Rrepresent the field of complex
and real numbers, respectively. First, we state
the following definitions.
Definition 1 ([6]).Given a function h:C→C,
we say that h(·) is defined on the spectrum of a
matrix A∈Rn×n, if there exist the numbers
h(λk), h′(λk), ..., h(mk−1) (λk), k =1,2, ..., s,
(1)
where
h(mk−1) (λk) = dmk−1h(x)
dxmk−1x=λk
.(2)
with λkthe eigenvalues of Awith multiplicity mk,
Definition 2. An equilibrium point x0of a dy-
namical system
˙x=f(x) (3)
is said to be hyperbolic, if the Jacobian matrix
evaluated at x0,∂f
∂x x0
=Df (x0), has eigenval-
ues with strictly positive and/or strictly negative
real part, i. e., the eigenvalues do not lie on the
imaginary axis.
The following lemma characterizes some prop-
erties that will be used along this work.
Lemma 3. Let h1(s),..., hn(s)be a set of func-
tions, where hk:C→C, for k= 1, ..., n. If
hk(C−)⊂C−and hkC+⊂C+then
1. n
X
k=1
hkC−!⊂C−.
2. n
X
k=1
hkC+!⊂C+.
3. hkhlC−⊂C−,with k, l = 1, ..., n.
4. hkhlC+⊂C+, with k, l = 1, ..., n.
3 Basic Result
In this section we extend a classical result on
properties of dynamical systems, in particular we
present the next proposition, which is an exten-
sion of the Local Stable-Unstable Manifold Theo-
rem.
Proposition 4. Let Ebe an open subset of Rn×n
containing the origin, let fbe a continuous differ-
entiable function on E, and let φA,t be the flow of
the nonlinear system ˙x=f(x) = Ax +g(x). Sup-
pose that f(0) = 0 and that A=Df (0) (where
Df (0) is the Jacobian matrix of fevaluated at
0), has k eigenvalues with negative real part and
n−keigenvalues with positive real part. Given a
function h:C→C, defined on the spectrum of
A; if h(C−)⊂C−and, hC+⊂C+,h(0) = 0
and his analytic in an open set containing the
origin. There exist a k-dimensional differentiable
manifold Shtangent to the stable subspace ES
hof
the linear system ˙x=h(A)xat 0 such that for al l
t≥0,φh(A),t(Sh)⊂Shand for all x0∈Sh
lim
t→∞ φh(A),t(x0) = 0,(4)
where φh(A),t is the flow of the nonlinear system
˙x=h(A) + g(x). There also exists a (n−k)-
dimensional differential manifold Whtangent to
the unstable subspace EW
hof the linear system ˙x=
h(A)xat 0, such that for all t≤0,φh(A),t(Wh)⊂
Whand for all x0∈Wh
lim
t→−∞ φh(A),t(x0)= 0.(5)
Proof. Let Abe a matrix with keigenvalues with
negative real part and n−keigenvalues with pos-
itive real part. We take the Schur decomposition
of the matrix A, [6],
A=UTAU⊤,(6)
where TAis an upper triangular matrix containing
the eigenvalues of Ain the diagonal, i. e.,
TA=
λ1⋆··· ⋆
0λ2··· ⋆
.
.
..
.
.....
.
.
0 0 ··· λn
,(7)
with λ1, ..., λnthe eigenvalues of Aand ⋆elements
of Athat may or may not be zero.
We take the power series expansion of h(x)
h(x) =
∞
X
k=0
ckxk,(8)
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substituting (6) in (8) gives
h(A)=
∞
X
k=0
ck(UTAU⊤)k
=U ∞
X
k=0
ckTk
A!U⊤.
(9)
Since the multiplication of triangular matrices is
a triangular matrix we have
h(A) = U
P∞
k=0ckλk
1⋆· · · ⋆
0P∞
k=0 ckλk
2· · · ⋆
.
.
..
.
.....
.
.
0 0 · · · P∞
k=0 ckλk
n
U⊤,
(10)
which is equivalent to
h(A) = U
h(λ1)⋆·· · ⋆
0h(λ2)··· ⋆
.
.
..
.
.....
.
.
0 0 ··· h(λn)
U⊤.(11)
Since the function h(·) maps the right-half com-
plex plane and left-half complex plane into them-
selves, the resultant matrix h(A), has keigenval-
ues with negative real part and n−keigenvalues
with positive real part. Now, the result is a con-
sequence of the stable-unstable manifold theorem
and Lemma 3.
This result is an extension of Theorem 3.2
from [3], which implies that if we can keep the
sign structure of the jacobian matrix, after we ap-
ply a certain function to the linear part of a dy-
namical system, there will still exists a stable and
unstable manifolds with the original dimensions.
Based on Lemma 3, there exist an infinite family
of functions h(·) that can be applied to dynam-
ical systems and preserve hyperbolic equilibrium
points and its dynamical properties.
4 Preservation of Control Law
and Synchronization
As we mentioned above, one goal of this paper is
to preserve synchronization in dynamical systems.
To achieve this we applied a function to the linear
part of the system and then we use the control
law developed in [9], where the control vector uis
formed by a linear and a nonlinear part,
u=uL+uN.(12)
Consider the master-slave system
˙x=Ax +g(x),(13)
˙
˜x=A˜x+g(˜x) + Bu (14)
where u∈Rmand x, ˜x∈Rnare the master and
slave state vectors, respectively. We also suppose
that g(x) is such that for all x, ˜x∈χ⊂Rn
kg(x)−g(˜x)k ≤ lkx−˜xk, l > 0.(15)
Now, it is proved in [9], that using as control (12)
with
uL=Ke
uN=τ(e)B⊤P e (16)
where e=x−˜xis the error between the state
variables, Kis such that A−BK is Hurwitz and
P=P⊤>0 is the solution of the Lyapunov
equation
(A−BK )⊤P+P(A−BK) = −Q, Q =Q⊤>0.
(17)
we can ensure the error between the master-slave
system becomes asymptotically zero. If we choose
τ(e) as a constant in equation (16), this control
is reduced to a proportional control law, but in
general it is a function of the error.
Theorem 5. Consider a linear function h(·)as
in Proposition 4 such that h(A−BK) = h(A)−
h(BK), if we have the system
˙x=h(A)x+g(x),
˙
˜x=h(A)˜x+g(˜x) + ˆu, (18)
where g(x)is locally Lipschitz in a domain x∈
χ⊂Rnand h(C−)⊂C−. Choosing a control
law such that
ˆu=h(BK)e+τ(e)BB⊤ˆ
P e, (19)
with the pair (A, B)stabilizable and τ:Rn→
[0,∞), locally Lipschitz and positive. Then, the
error of the system is asymptotically stable if
λmin ˆ
Q>2lλmax ˆ
P,(20)
where
(h(A)−h(BK))⊤ˆ
P+ˆ
P(h(A)−h(BK)) = −ˆ
Q,
(21)
is satisfied with ˆ
Q > 0.
Proof. By Proposition 4, h(A−BK ) = h(A)−
h(BK) is Hurwitz. Therefore, there exists a ma-
trix ˆ
P=ˆ
P⊤>0, such that (21) is satisfied.
Forming the error system e=x−˜x
˙e= (h(A)−h(BK )) e+g(x)−g(˜x)−τ(e)BB⊤ˆ
P e,
(22)
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choosing v(e)=e⊤ˆ
P e as a Lyapunov candidate
function and evaluating this along the trajectories
of the error system we have
˙v(e) = ˙e⊤ˆ
P e +e⊤ˆ
P˙e, (23)
taking the transpose of ˙e, substituting in ˙v(e) and
using Lemma 3,
˙v(e) =e⊤[(h(A−BK))⊤ˆ
P+ˆ
P h(A−BK )]e
+ 2e⊤ˆ
P(g(x)−g(˜x)) −2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
≤ − e⊤ˆ
Qe + 2 kekλmax (ˆ
P)lkek − 2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
≤−kek2(λmin(ˆ
Q)−2lλmax(ˆ
P)) −2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
(24)
If λmin(Q)>2lλmax (P) we ensure that ˙v(e)<0
regardless of the values of τ(e) (providing it is
positive semidefinite). This implies that the ori-
gin of the error system is locally asymptotically
stable. If we choose χarbitrary large, then the
error system is semi globally stable. If χ=Rn
the error system is globally stable.
This theorem is a result for preservation of
synchronization in nonlinear dynamical systems,
it ensures local asymptotical stability or global
asymptotical stability depending of the size of χ.
We will now prove, that this result can be ex-
tended to achieve synchronization when we intro-
duce a perturbation to the system in the control
law.
A stronger result similar to the proposition 1
in [9] is the following
Theorem 6. Suppose that the pair (A, B)is sta-
bilizable with Bfull column rank and also that
Im(B) = Im(g(., t)), then under the conditions
of Theorem 5, there exist a control law of the
form (19) such that the error of the modified sys-
tem equation (22) is locally asymptotically stable,
where h(A) = M A and Mbeing a nonsingular
matrix . Furthermore if g(x)is semi-globally Lip-
schitz (resp. globally Lipschitz), then there exists
a control law such that the error of the modified
system is semi-globally (resp. globally) asymptot-
ically stable.
Proof. Notice that Bhas full column rank and if
Im(B) = Im(g(., t)), there exists a state-similarity
transformation such that we can write the system
as
h˙x1
˙x2i=hA11 A12
A21 A22 ih x1
x2i+h0
Biu+h0
Iig2(x, t)
(25)
where Bis nonsingular and g2(x, t) is locally
Lipschitz with Lipschitz constant l. In an explicit
way we are going to make the state-similarity
transformation as follows; consider x=T z and
e
x=Te
z
e
e=z−e
z
˙
e
e=e
Ae
e−g
BK e
e
[T(e
A−g
BK)T−1]Tˆ
P+ˆ
P T (e
A−g
BK)T−1
⇒
(e
A−g
BK)TTTˆ
P T +TTˆ
P T (e
A−g
BK) = −TTˆ
QT
ˆ
¯
P=TTˆ
P T
ˆ
¯
Q=TTˆ
QT
e
A=T−1h(A)T=T−1MAT
"0
MBK #=g
BK
g
BK =h(BK) = M BK
Observe that the Im(MB) = Im(B) because M
is an isomorphism, then Im(M B) = Im(g(., t)).
Now the new error equation is:
˙
˜e= ( e
A−g
BK)˜e+"0
I#g2(x, t).(26)
Replacing the matrices A, BK, ˆ
P , ˆ
Qand vari-
ables u, x, e
xand ein the proof of Proposition 1
in [9] by the matrices e
A, g
BK, ˆ
¯
P, ˆ
¯
Qand variables
e
u, z, e
zand e
ethe demonstration follows the same
procedure.
In practical situations there are always per-
turbations present. We introduce an additive per-
turbation at the control input such that the ap-
plied control signal ˜uto the slave system is dis-
torted by the perturbation, i. e.,
˜u= ˆu(t) + Bd(t).(27)
In this case d(t) represents the additive perturba-
tion that we introduce to the system, which we
assume to be bounded,
kd(t)k ≤ δ0.(28)
If this perturbation is present we can not expect
to achieve local asymptotic stability, but we will
show that it is possible to reach ultimate bounded
stability.
Theorem 7. Consider the dynamical system
˙x=h(A)x+g(x),
˙
˜x=h(A)˜x+g(˜x) + ˜u, (29)
and h(·)as in Theorem 5, where (A, B)is stabi-
lizable, g(x)is locally Lipschitz in xin a domain
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x∈χand h(C−)⊂C−. Then, with the noisy
control (27) with a bounded perturbation given by
(28), and
ˆu=h(BK )e+τ(e)BB⊤ˆ
P e, (30)
with τ(·) : Rn→[0,∞)locally Lipschitz and pos-
itive, and e=x−˜x. Then, the error is locally
ultimately bounded within the ball
G(e) = (e:kek ≤ 2κ(ˆ
P1
2)λmax(ˆ
P)kBkδ0
λmin(Q)−2lλmax (P)),
(31)
providing that
λmin(ˆ
Q)>2lλmax(ˆ
P),(32)
where
(h(A)−h(BK))⊤ˆ
P+ˆ
P(h(A)−h(BK)) = −ˆ
Q,
(33)
with ˆ
Q > 0.
If this holds for an arbitrary large set χ,
then the error system is semi globally ultimately
bounded. If this holds ∀x, ˜x∈Rn, the error sys-
tem is globally ultimately bounded.
Proof. Since the pair (A, B) is stabilizable, there
exists a matrix Ksuch that A−BK is Hur-
witz. By Proposition 4, the matrix h(A−BK) =
h(A)−h(BK) is also Hurwitz. In consequence,
there exists a matrix ˆ
P > 0 that satisfies the Lya-
punov equation (33).
Now, we form the error system e=x−˜x
˙e=h(A)−h(BK)e+g(x)−g(˜x)−τ(e)BB⊤ˆ
P e−B d, (34)
choosing v(e) = e⊤ˆ
P e as a Lyapunov candidate
function, and evaluating its derivative along the
trajectories of the error system we have
˙v(e) = ˙e⊤ˆ
P e +e⊤ˆ
P˙e, (35)
substituing into ˙v(e) and using Lemma 3,
˙v(e) =e⊤h(A−BK)⊤ˆ
P+ˆ
P h(A−BK )e
+ 2e⊤ˆ
P(g(x)−g(˜x)) −2τ(e)e⊤ˆ
P BB ⊤ˆ
P e −2e⊤ˆ
P Bd
≤ − e⊤ˆ
Qe + 2 kekλmax (ˆ
P)lkek − 2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
−2kekλmax(ˆ
P)kBkδ0−2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
≤kek−kekλmin(ˆ
Q) + 2lλmax(ˆ
P)kek+ 2λmax(ˆ
P)kBkδ0
−2τ(e)e⊤ˆ
P BB ′ˆ
P e.
(36)
Assuming that λmin(ˆ
Q)>2lλmax(ˆ
P) and defining
−λmin(ˆ
Q) + 2lλmax(ˆ
P) = −α, (37)
yields to
˙v(e)≤ kek−αkek+2λmax (ˆ
P)kBkδ0−2τ(e)e⊤ˆ
P BB ⊤ˆ
P e
≤ kek(−α√v
pλmax(ˆ
P)
+ 2λmax(ˆ
P)kBkδ0)
−2τ(e)e⊤ˆ
P BB ⊤ˆ
P e.
(38)
Then, for
√v > 2λmax (ˆ
P)qλmax(ˆ
P)kBkδ0
α,(39)
˙v(e)<0, defining a set of ultimately boundedness
in v. If
e∈G(e) := ne:kek ≤ 2λmax(ˆ
P)κ(ˆ
P1
2)kBkδ0
αo,(40)
guarantees, for a condition number κ(ˆ
P1
2)of ˆ
P1
2
that ˙v(e)<0, which implies that G(e) is ulti-
mately entered. If χis chosen arbitrarily large,
the ultimately boundedness holds for all x, ˜x∈χ,
providing that G(e)⊂χ.
This theorem ensures ultimately bounded sta-
bility inside a region defined by G(e), which is
function of δ0,λmax(ˆ
P) and λmin(ˆ
Q). The size
of G(e) depends on how small λmax(ˆ
P) is and
how large λmin(ˆ
Q) is, the smaller λmax (ˆ
P) and
the larger λmin(ˆ
Q) are, the smaller the size of G(e)
will be.
As it is mention in [9] the condition λmin(ˆ
Q)>
2lλmax(ˆ
P) is restrictive, nevertheless it is possible
to apply the same technique use in Theorem 6 to
relax this condition.
5 Examples of Synchronicity
Preservation
In this section, we present two examples of a
master-slave system driven by a control law de-
signed according to the last section.
5.1 Sprott’s Q system
The dynamical systems used are known as
Sprott’s Q systems. Let us consider the master
system
˙x1=−x3,(41)
˙x2=x1−x2,(42)
˙x3= 3.1x1+x2
2+ 0.5x3,(43)
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and the slave system
˙
˜x1=−˜x3,
˙
˜x2=˜x1−˜x2,
˙
˜x3= 3.1˜x1+ ˜x2
2+ 0.5˜x3+u3.
(44)
The matrix A that represents the linear part
around the origin is
A=
0 0 −1
1−1 0
3.1 0 0.5
(45)
andB= [0,0,1]. To place the poles of the sys-
tem at [−2,−2.5,−3] the state feedback matrix
K= [−8.9,−3,7]. Choosing Q=I3and solving
equation (17) gives
P=
1.4606 0.2758 −0.0646
0.2758 0.4939 −0.0020
−0.0646 −0.0020 0.0869
.(46)
Setting the value of τ(e) = e2, the initial con-
ditions of the Master System to x1(0) = x2(0) =
x3(0) = 0.05 and the Slave system initial condi-
tions e
x1= 0.1,e
x2=e
x3= 0. We let the mas-
ter and slave systems evolve without control until
t= 100 when the control law is engage.
Figure 1 shows the phase portrait of the mas-
ter and slave systems which exhibit a chaotic be-
havior. In Fig. 2 the absolute value of the error
between the master and slave systems are plotted
in semi-logarithmic scale. It can be seen in Fig. 2
that the absolute value of the error between the
two systems converges to zero after the control
law is activated at time t= 100.
−10 −5 05
−5
0
5
−10
−5
0
5
10
x1
x2
x3
Master
Slave
Figure 1: Original Sprott Q attractor showing
synchronization between master and slave sys-
tems (initial conditions x1(0) = x2(0) = x3(0) =
0.05, ˜x1(0) = 0.1, ˜x2(0) = ˜x3(0) = 0).
0 50 100 150 200 250 300
10−20
10−15
10−10
10−5
100
105
time
Log|e|
e1
e2
e3
Figure 2: Magnitude of error between master and
slave systems (|e|=|x−˜x|).
Now we modify the system applying the linear
function
h(A) = MA, (47)
to the linear parts of the system and the control
law. We choose M to be positive definite, so it
preserves the sign characteristic of the eigenvalues
of (A−BK). Since (A−BK) = U T(A−BK )U−1,
where
U=
0.3123 0.4827 −0.8182
−0.1562 −0.8235 −0.5455
0.9370 −0.2981 0.1818
.(48)
If we choose M=UTMU−1and we set TMas
TM=
0.5−0.05 −0.05
0 0.5−0.05
0 0 1
,(49)
M will be given by
M=
0.8597 0.2577 −0.0770
0.1868 0.6156 −0.0430
−0.0709 0.0064 0.5247
.(50)
Choosing again ˜
Q=I3and solving (21) gives
ˆ
P=
1.5489 −0.0677 0.0853
−0.0677 0.8634 0.0700
0.0853 0.0700 0.1425
.(51)
Using τ= 30 + 500 exp(−0.001(|e1|+|e2|+|e3|))
and the same initial conditions used for the orig-
inal master-slave system, we solve the modified
systems.
Figure 3 shows the synchronization between
the master-slave modified system, therefore the
original system definitely changes, the new sys-
tem preserves the synchronization. This can be
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−5
0
5
−4
−2
0
2
−4
−2
0
2
4
6
x1
x2
x3
Master
Slave
Figure 3: Master-Slave modified system, initial
conditions x1=x2=x3= 0.05, e
x1= 0.1,e
x2=
e
x3= 0.
0 50 100 150 200 250 300
10−20
10−15
10−10
10−5
100
105
time
Log|e|
e1
e2
e3
Figure 4: Magnitude of error between master and
slave modified systems (|e|=|˜x−x|).
seen in Fig. 4 where the errors between the modi-
fied master-slave system are presented, in a semi-
logarithmic plot, to empathize the convergence to
zero. Here again the control law is activated at
t= 100.
5.2 Chua’s system
Now, we use the well known chaotic system,
Chua’s circuit, to show in a simulation how syn-
chronization is preserved. Let the Master system
be given by
˙x1=10
7x1+ 10x2−20
7x3
1,
˙x2=x1−x2+x3,
˙x3=−100
7x2.
(52)
−1 012
−0.2
0
0.2
−2
−1
0
1
2
x1
x2
x3
Master
Slave
Figure 5: Chua’s Master-Slave system, initial con-
ditions x1= 0.02, x2= 0.05, x3= 0.04, e
x1=e
x2=
e
x3= 0.
and the slave system
˙
˜x1=10
7˜x1+10˜x2−20
7˜x3
1+u1,
˙
˜x2=˜x1−˜x2+ ˜x3,
˙
˜x3=−100
7˜x2.
(53)
The matrix A that represents this the linear part
of the system is
A=
10/7 10 0
1−1 1
0−100/7 0
.(54)
and the matrix B is [1,0,0]. We place the poles of
the system in [−1,−1.5,−2] thus, the state feed-
back matrix is K= [4.9286,−1.2857,3.29]. We
choose Q=I3and solving equation (17) gives
P=
11.2795 38.9782 6.5176
38.9782 141.2133 20.9429
6.5176 20.9429 4.5911
.(55)
Now, we set τ(e) = 3.9e3 + 500 exp(−0.001(|e1|+
|e2|+|e3|)), as suggested by Theorem 6; the ini-
tial conditions for the Master system as x1=
0.02, x2= 0.05, x3= 0.04 and the initial condi-
tions for the slave system e
x1= 0.01,e
x2=e
x3= 0.
Figure 5 shows a phase portrait of the origi-
nal Chua’s Master-Slave system, where we can see
this system reaches synchronization. In Fig. 6 the
absolute value for the error between the Master
and slave system are shown in a semi-logarithmic
plot, to empathize the fact that the error con-
verges to zero.
As shown before, we design a matrix M, that
is simultaneously triangularizable with A−BK
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0 50 100 150 200 250 300
10−15
10−10
10−5
100
105
time
Log|e|
e1
e2
e3
Figure 6: Magnitude of error between master and
slave systems in semi-logarithmic scale (log|e|=
log |˜x−x|).
and that preserves the sign structure of the eigen-
values of A−BK. We know that A−BK =
UTA−B K U′, where Uis an unitary matrix
U=
−0.7486 −0.5591 −0.3564
0.0919 0.4448 −0.8909
0.6566 −0.6997 −0.2815
,(56)
If we let M be given by M=UTMU′and we
choose the upper triangular matrix TMas
TM=
0.5000 0 0.0010
0 0.5000 −0.0010
0 0 1.1000
,(57)
M will be given by
M=
0.5763 0.1907 0.0603
0.1906 0.9766 0.1506
0.0597 0.1493 0.5472
.(58)
Using these values, we solve equation (21)
ˆ
P=
2.9698 15.9383 0.1208
15.9383 94.7681 −0.7060
0.1208 −0.7060 1.0032
.(59)
The initial conditions for the modified system are
the same that we used for the original system,
x1= 0.02, x2= 0.05, x3= 0.04 for the Master
system and e
x1= 0.1,e
x2=e
x3= 0 for the slave
system.
Figure 7 shows the trajectories for the solu-
tions of the master and slave systems. In Fig. 8
the error between the master and slave system is
plotted in a semi-logarithmic scale so it can be
seen that the error converges to zero and the syn-
chronization is achieved.
−4 −2 02
−2
0
2
−5
0
5
x1
x2
x3
Master
Slave
Figure 7: Chua’s modified Master-Slave system,
initial conditions x1= 0.02, x2= 0.05, x3= 0.04,
e
x1= 0.1,e
x2=e
x3= 0.
0 50 100 150 200 250 300
10−15
10−10
10−5
100
105
time
Log|e|
e1
e2
e3
Figure 8: Magnitude of error between master and
slave modified system in semi-logarithmic scale
(log |e|= log |˜x−x|).
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−6 −4 −2 02
−2
0
2
−5
0
5
10
x1
x2
x3
Master
Slave
Figure 9: Chua’s modified Master-Slave sys-
tem with perturbation, initial conditions x1=
0.02, x2= 0.05, x3= 0.04, e
x1= 0.01,e
x2=e
x3= 0.
−5
0
5
−4
−2
0
2
−4
−2
0
2
4
6
x1
x2
x3
Master
Slave
Figure 10: Sprott’s Q modified Master-Slave sys-
tem with perturbations, initial conditions x1=
0.02,x2= 0.05, x3= 0.04, e
x1= 0.1,e
x2=e
x3= 0.
6 Partial Synchronization with
Perturbations
In this section we use the two modified systems,
the modified Sprott’s Q and the modified Chua’s
system, to prove that we can achieve partial syn-
chronization when a sinusoidal function, which
represents the perturbation, is introduced to the
feedback control of the system. Therefore, the
new control law has the extra term d(t) and it
will be given by
˜u=h(BK)e+τ(e)BB′ˆ
P e +Bd(t).(60)
For the Sprott’s Q system we used d(t) =
0.1 sin(t) and for Chua’s system we used d(t) =
0.2 sin(5t). Now we simulate the systems with
this new control and using the same parameters
that we use for the original systems.
Figure (9) and Fig.(10) show the phase por-
trait diagrams for the error systems of the mod-
ified Chua’s system and Sprott’s Q systems. In
0 50 100 150 200 250 300
10−15
10−10
10−5
100
105
time
Log|e|
e1
e2
e3
Figure 11: Magnitude of error between master
and slave modified system in semi-logarithmic
scale (log |e|= log |˜x−x|).
0 50 100 150 200 250 300
10−8
10−6
10−4
10−2
100
102
time
Log|e|
e1
e2
e3
Figure 12: Magnitude of error between master
and slave modified system in semi-logarithmic
scale (log |e|= log |˜x−x|).
this simulations we can see that the error between
the state variables do not converges to zero but
it remains inside a region ultimately bounded, as
was predicted by the theorem.
In Fig. (11) and Fig. (12) is easier to see that
the error do not converges asymptotically to zero
and it remains bounded as time increases. These
plots are in a logarithmic scale and can be com-
pared with Fig. 8 for Chua’s modified system and
Fig. 4 for Sprott’s system, where it can be seen
that the error converges to zero when the pertur-
bation is not present.
7 Conclusions
If a dynamical system is changed by applying a
function of the kind described in this work, when
the structure of the sign of the eigenvalues of
the jacobian matrix, evaluated in the hyperbolic
points is preserved, the new pair master-slave sys-
tem also achieves synchronization. We suppose
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that the chaotic dynamics that the original sys-
tem exhibits is also preserved. Although the prin-
cipal interest of this work was to preserve synchro-
nization after modifying a dynamical system, we
have proved that the control law we used along
this work, which was proved in other paper to
work correctly synchronizing a master-slave sys-
tem, can be modified in its linear part to make
the modified master slave system achieve synchro-
nization via Lyapunov method. We have to point
out that the value of the parameter τis important
since the system is really sensible to it.
When a perturbation was introduced to the
system, we showed that it is possible to preserve
the stability of the error system, but the kind of
stability we can ensure is ultimately bounded in
contrast of the local asymptotic stability that is
obtained when the perturbation in the control law
is not present. This ultimate bounded stability
means that the error between the state variables,
is inside a region defined by G(e), defined in the
last theorem.
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