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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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