Synchronization of two Lorenz systems using active control

Chaos.

Solrrons

& Frarmls Vol. 8. No I, pp 51--5X. 1997 Copyright 0 1997 Elsmer Science Ltd Printed m Great Britain. All rights reserved 096@0779/97 $17 on + 0.00

PII: SO960-0779(96)00060-4

Synchronization of Two Lorenz Systems Using Active Control

Department

of Electrical

ER-WE1

BAI and KARL

and Computer

Engineering,

E. LONNGREN University

of Iowa,

Iowa

City,

IA 52242,

USA

(Accepted I6 May 1996)

Abstract-Using techniques from active control system can be synchronized. The synchronization Copyright 0 1997 Elsevier Science Ltd

theory, is verified

we demonstrate that a coupled Lorenz using the Simulink feature in MATLAB.

1. INTRODUCTION

Recently, Liu and Barbosa examined a coupled system with the goal of controlling a 15NH3 laser system with a weaker one [l]. They modeled each laser with a Lorenz system of equations. The Lorenz system of ordinary differential equations that has been used to model the dynamic movement of an atomospheric fluid is written as [2] dx = a(y - x) dt dy -=rx-y-xz

.

dt

(1)

I

dz =xy- bz I dt This set is known to lead to chaotic behavior. They noted that the two systems would become locked to each other if a particular weak coupling was introduced between the two systems. This coupling property is fairly general and is of current interest [3]. The purpose of the present paper is to show that this locking or synchronization of the two Lorenz systems can be achieved with a fairly general coupling that is based on an application of techniques found in active control theory. We describe this active control mechanism in Section 2. In Section 3, we summarize the results of a numerical simulation that verifies the technique. Section 4 is the conclusion.

2. ACTIVE

CONTROL

We assume that we have two Lorenz systems and that the system with the subscript 1 is to control the system with the subscript 2. The systems are: 51

ER-WEI BAI and K. E. LONNGREN

52

dxl - =

4Yl

dt

- x1)

-dy, = IX 1 - y, - XlZl dt dzl = xly, - bz, dt

and dx, = 4Y2 dt

-

dyz

-

- x2) + &l(t)

= rx2 - y2

dt

-

x27-2

+

Pbub(f)

(3)

.

I

dzz

-

=

X2Y2

-

bz2

+

IL,(~)

dt three control functions p,(t),

J pb( t), and pC(t). These functions

In (3), we have introduced are to be determined. In order to ascertain the control functions, we subtract (2) from (3). It is convenient to define the differences between the Lorenz system that is to be controlled and the controlling system using x3

Using this notation,

=

x2

-

Y3 = Y2 -

x1;

z3

Yli

=

z2

-

11.

(4)

we obtain 3

dx3 = 4Y3 dt -

dy3

=

dz3

dt

x3)

rx 3 -

dt -

-

y,

+

PAf>

-

x2.22

f

XlZ1

+

/lb(t)

I.

(5)

= X2Y2 - X~YI - bz3 + cl,(t)

i We define the active control functions vu.(t), pb(t), and p,(t) as Pa(t) = V.?(t) L%(t)

=

x2z2

CL,(t)

=

-X2Y2

-

XlZl

+

+ XlYl

Vblb(l)

.

(6)

+ VAtI

This leads to

dx3

-

-

dt dy3

dt

= 4Y3

=

- z3) + Va(t)

rx 3 -

y3

+

-dZ3 = -bz3 + VJt) dt

vb(t)

=,

(7)

53

Coupled Lorenz system

Equation (7) describes the error dynamics and can be considered problem where the system to be controlled is a linear system with V,(t) and V,(t) as functions of x3, y, and z3 [4]. As long as these system, xg, y3 and z3 converge to zero as time t goes to infinity. Lorenz systems are synchronized with feedback control. There are for the control V,(t), V,(t) and Vc( t). We choose

in terms of a control a control input V,(t), feedbacks stabilize the This implies that two many possible choices

where A is a 3 x 3 constant matrix. For proper choice of the elements of the matrix A, the feedback system must have all of the eigenvalues with negative real parts. In this case, the closed loop system will be stable. Let us choose a particular form of the matrix A that is given by A=

u-l -r

-0 0

0

0

t

0 0 b-l

.

(9)

i

For this particular choice, the closed loop system has eigenvalues that are found to be -1, - 1 and -1. This choice will lead to a stable system and as we will observe in a numerical investigation, lead to the synchronization of two Lorenz systems.

3. NUMERICAL

VERIFICATION

The numerical simulation procedure that is employed here is to use the software Simulink that is a graphical interface software that has recently been incorporated into MATLAB. The final results using this tool are displayed on oscilloscopes and XY recorders that are included in the Simulink toolbox. Using this tool, we have essentially converted the digital computer with its number crunching capabilities into an analog computer with its ease in modeling circuits, systems, and equations. It is widely used in control systems investigations [4]. In order to demonstrate the veracity of this tool in investigating chaotic systems, we first examined the characteristics of a single Lorenz system of equations. The Lorenz system is defined by the set of coupled ordinary differential equations in (1). We have to specify numerical values for the constants and we will select them to have the same values that were in the original study [2] u = 10

r = 28

b = ;.

(10)

A solution of this set of equations can be given as a graphical presentation of each of the dependent variables x, y , and z as a function of the independent variable time t. An alternative method is to present a graphical display of x vs y, x vs z and y vs z. It is this latter phase space display of the results that will be obtained in this simulation. The Simulink circuit that represents the set of differential equations given in (1) is shown in Fig. 1. In this circuit, there are three separate branches that represent the three equations given in (1). The terms x, y, and z are just the time integrals of the derivatives (dx/‘dt), (dybt), and Cd.&), respectively. Having identified the locations of the terms x,

ER-WEI BAI and K. E. LONNGREN

54

r*x 4jizJ Y

)
-lzl m

r*x-y-xz

Y xy-b

,bL

h

I

’

b +wux

b bus bar

chaosmat to file

Fig. 1. Simulink circuit for the Lorenz system of equations (1) with the constants given in (IO).

y , and z, it is possible to perform the rest of the operations indicated in (1). The numerical constants in (10) are introduced by changing the gain of the ‘op amps’ in the Simulink terminology. All numerical values are menu specified. The addition, subtraction, and multiplication operations are standard tools in the software. Initial values for the parameters x, y, and z also have to be specified. These can be arbitrarily specified to have a nonzero value at the start of a simulation. There will be an initial transient that will approach the final asymptotic values after a certain time. These asymptotic values for the variables x, y, and z can be collected with the ‘MUX’ and ‘to file’ operators in Simulink and then ‘loaded’ into and read from MATLAB. Using these asymptotic values in the next simulation as the initial conditions removes any undesired transient effects. Since our goal in this initial simulation is to obtain a graphical display of the three components of phase space: x vs y, x vs z and y vs z; we use the ‘XY recorder’ output. Each of these outputs can be individually displayed. We, however, will employ the ‘MUX’ operator and store all the data in a FILE that can be directly read by MATLAB. This data can then be displayed as a three-dimensional object that is shown in Fig. 2. In this case, the three perspective x vs y, x vs z and y vs z graphs are displayed on the three walls that appear to surround the three-dimensional object. Note that the graph shown in Fig. 3.2.2 of a recent book by Matsumoto et al. appears on one of these walls [5]. The Simulink circuit shown in Fig. 1 was duplicated along with additional items that represented the terms in the active control that are specified in (6). The initial condition of one of the integration elements was changed in order to create a difference in the two

55

Coupled Lorenz system

N

‘30

Fig. 2. Solution of the Lorenz system of equations.

Lorenz systems. We also included a switch that allowed us to activate or deactivate the active control. Oscilloscopes are included at appropriate points to display the temporal evolution of various signals from the time of the start of the simulation at t = 0. The results of the simulation of the two identical Lorenz systems with the active controller disconnected from the circuit are shown in Fig. 3. Figure 3(a) displays the signals x1 and x2. Figure 3(b) displays the signals y, and y2. Figure 3(c) displays the signals z1 and z2. Figure 4 displays the same sequence of signals with the active controller connected into the circuit. We note that the active controller has synchronized the two Lorenz systems. The synchronizing of the two Lorenz systems can also be observed by monitoring the difference of the two signals: x3, y3 and z3. Figure 5(a) displays the difference signals with the active controller disconnected from the circuit. Figure 5(b) displays the same signals with the active controller connected into the circuit. As expected, the signals shown in Fig. 5(b) decay to zero at the time of synchronization observed in Fig. 4.

4. CONCLUSION

This paper demonstrates that chaos in a Lorenz system of equations can be easily controlled using techniques found in active controls. We believe that the technique can be generalized.

56

ER-WE1 BAI and K. E. LONNGREN

(a)

-20 '

/I

3

04

4

5

6

7

a

“V

40 '1 1 20 '\/. ', :h----OP 0

I 1

-.

:\ .' '\

11 \

'\. I 2

\.'

r

.T\ '\ I 3

\

;\.I 'd

'. 1' I 4

I\ '\,/

.\ '1.1. I 5

* '1 .\ \ r I'., \ I. I, /\ I '\ ; '\ ' \ '._I I d " J .\ I 6

I 7

I

a

I 9

I 10

Fig. 3. Solution of the coupled Lorenz system of equations with the active control deactivated. (a) Signals x1 and x2; (b) signals y, and yz; (c) signals z1 and z2 (x1, y1 and z1 -‘-; x2. y2 and z2 -).

Coupled Lorenz system

(4

-20'

-50’

Olb)

I 1

I 2

I 3

L 4

I

I

I

I

I

I

1

,

I

0

1

2

3

4

5

6

7

8

9

I 5

I 6

I 7

I 8

1 9

I 10

10

Fig. 4. Solution of the coupled Lorenz system of equations with the active control activated. (a) Signals x1 and x2; (b) signals yl and ~2; (c) signals z1 and z2 (XI, y1 and ZI -. -; x2, y2 and z2 -).

ER-WE1 BAI and K. E. LONNGREN

58

-4o-40' 0

I 1

I 2

I 3

I 4

I 5

t 6

, 7

, a

, 9

I 10

Fig. 5. Difference signals: x3, y3 ys and z3. zs. (a) Active controller deactivated; (b) active controller activated (xs -. y3 -.-and z3 ...).

REFERENCES 1. Y. Liu and L. C. Barbosa, Periodic locking and coupled Lorenz systems, Phys. Left. A 197, 13-18 (1995). 2. E. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Science, 20 130-141 (1963). 3. e.g. P. Ashwin, J. Buescu and I. Stewart, Bubbling of attractors and synchronisation of chaotic oscillators. Phys. Len. A. 193, 126-139 (1994); J. F. Heagy, T. L. Carroll and L. M. Pecora, Synchronous chaos in coupled oscillator systems, Phys. Rev. E. 50, 1874-1885 (1994); J. F. Heagy, T. L. Carroll and L. M. Pecora, Experimental and numerical evidence for riddled basins in coupled chaotic systems, Phys. Rev. Left. 73, 3528-3531 (1994). 4. e.g. R. H. Bishop, Modern Controt Systems Analysis and Design using MATLAB. Addison-Wesley, Reading, MA (1993). 5. T. Matsumoto, M. Komuro, H. Kokubu and R. Tokunaga, Bifurcations. Springer, New York (1991).