Optically injected laser system: Characterization of chaos, bifurcation, and control

8/9/2019 Optically injected laser system: Characterization of chaos, bifurcation, and control

1/12

Optically injected laser system: Characterization of chaos bifurcation and

control

Santo Banerjee, Papri Saha, and A. Roy Chowdhury

Citation: Chaos 14, 347 (2004); doi: 10.1063/1.1755179

View online: http://dx.doi.org/10.1063/1.1755179

View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v14/i2

Published by the American Institute of Physics.

Related Articles

Effect of uniaxial stress on electroluminescence, valence band modification, optical gain, and polarization modesin tensile strained p-AlGaAs/GaAsP/n-AlGaAs laser diode structures: Numerical calculations and experimentalresults J. Appl. Phys. 112, 093113 (2012)

A bi-functional quantum cascade device for same-frequency lasing and detection

Appl. Phys. Lett. 101, 191109 (2012)

Silicon-based three-dimensional photonic crystal nanocavity laser with InAs quantum-dot gain Appl. Phys. Lett. 101, 191107 (2012)

High single-spatial-mode pulsed power from 980nm emitting diode lasers

Appl. Phys. Lett. 101, 191105 (2012)

Blue 6-ps short-pulse generation in gain-switched InGaN vertical-cavity surface-emitting lasers via impulsiveoptical pumping

Appl. Phys. Lett. 101, 191108 (2012)

Additional information on Chaos

Journal Homepage: http://chaos.aip.org/

Journal Information: http://chaos.aip.org/about/about_the_journal

Top downloads: http://chaos.aip.org/features/most_downloaded

Information for Authors: http://chaos.aip.org/authors

Downloaded 16 Nov 2012 to 193.204.188.124. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Santo%20Banerjee&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Papri%20Saha&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=A.%20Roy%20Chowdhury&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://chaos.aip.org/?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.1755179?ver=pdfcovhttp://chaos.aip.org/resource/1/CHAOEH/v14/i2?ver=pdfcovhttp://www.aip.org/?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4764930?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4767128?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766288?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766267?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766290?ver=pdfcovhttp://chaos.aip.org/?ver=pdfcovhttp://chaos.aip.org/about/about_the_journal?ver=pdfcovhttp://chaos.aip.org/features/most_downloaded?ver=pdfcovhttp://chaos.aip.org/authors?ver=pdfcovhttp://chaos.aip.org/authors?ver=pdfcovhttp://chaos.aip.org/features/most_downloaded?ver=pdfcovhttp://chaos.aip.org/about/about_the_journal?ver=pdfcovhttp://chaos.aip.org/?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766290?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766267?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4766288?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4767128?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4764930?ver=pdfcovhttp://www.aip.org/?ver=pdfcovhttp://chaos.aip.org/resource/1/CHAOEH/v14/i2?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.1755179?ver=pdfcovhttp://chaos.aip.org/?ver=pdfcovhttp://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=A.%20Roy%20Chowdhury&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Papri%20Saha&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Santo%20Banerjee&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://aipadvances.aip.org/?ver=pdfcovhttp://chaos.aip.org/?ver=pdfcov


2/12

Optically injected laser system: Characterization of chaos, bifurcation,and controla…

Santo Banerjee,b) Papri Saha,c) and A. Roy Chowdhuryd)

High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata—700032, India

Received 17 December 2003; accepted 5 April 2004; published online 21 May 2004

A single mode semiconductor laser subjected to optical injection, described by a set of three coupled

nonlinear ordinary differential equations, exhibiting chaos is considered. By means of a recurrenceanalysis, quantification of the strange attractor is made. Analytical studies of the system using

asymptotic averaging technique, derive certain conditions describing the prediction of 1→2

bifurcation, which have subsequently been verified on numerical simulation. Furthermore, the locus

of points on the parameter phase space representing Hopf bifurcation has been derived. The problem

of control of chaos by a new procedure based on adaptive stabilization is also addressed. The results

of such control are shown explicitly. Though this analysis deals with a very specific set of equations,

the overall features that come out of the study remains valid for almost all laser systems. © 2004

American Institute of Physics. DOI: 10.1063/1.1755179

Different kinds of chaotic phenomena can be observed in

an optically injected semiconductor laser system. We

present here a different aspect of such laser model whose

parameter space is analyzed through bifurcation dia-

grams exhibiting period doubling route to chaos. Chao-

ticity of the system is quantified through recurrence

analysis which can identify hidden periodicity and distin-

guish between a random signal and deterministic state.

Transition points in bifurcation, obtained through nu-

merical simulations, is verified with the analytical calcu-

lations. A new type of control mechanism is constructed

and applied, and the results are discussed in detail. The

results may provide some new features in the study of

semiconductor lasers which has wide application in the

field of signal processing and data transmission.

I. INTRODUCTION

Optically injected semiconductor lasers exhibit a variety

of phenomena.1 Experimental and numerical research has re-

vealed different kinds of chaotic properties.2– 8 Studies on

single mode semiconductor laser show that they adopt a

number of different routes to chaos,9 such as period dou-

bling, torus break up and subsequently torus doubling route

to chaos. Analysis of a laser with injected signal consists of

Adler’s phase equations10 which in bifurcation terms exhibits

a saddle node infinite periodic bifurcation.

11

Numericalanalysis of the full three-dimensional equations produce

Hopf bifurcations,12,13 period doubling cascades,3,14,15 and

quasiperiodicity.12,16 Among the main features of laser in-

jected with signal, accounted for by the averaged equations,

there are saddle node bifurcations and relaxation oscillations

which originate in a Hopf bifurcation. Hopf-saddle-node bi-

furcation was discussed by Solari and Oppo.17

Another recent analysis of the laser system18 supports

the results based on the averaged equations of Ref. 17 ad-

vanced in Ref. 19, namely, that the special reinjection present

in the laser with injected signal together with the Hopf

saddle node type are responsible for the organization of bi-

furcations. Zimmermann et al.20 presented a consistent

perturbation-approximation framework to the laser equations

in order to incorporate larger degrees of complexity in the

model. It was also established that the homoclinic bifurca-

tions in such a model instead of involving the locking solu-

tions, involves the undamped relaxation oscillations associ-

ated with the Hopf bifurcations of the Hopf-saddle-nodebifurcation. The simplest invariant solutions of the equations

were derived by the averaging procedure, as was the locus of

the Hopf-saddle-node bifurcation and the normal form com-

putations were performed.

In this paper we present a totally different aspect of the

model analyzed by Krauskopf et al.9 The chaoticity of the

system is quantified through visual recurrence analysis21

which was first described by Eckmann, Kamphorst, and

Ruelle.22 This is a new graphical device for the qualitative

assessment of time series and one can graphically detect hid-

den patterns and structural changes in the data or see simi-

larities in patterns across the time series under study. Then

the identification of Hopf 23,24 and period-doubling25 bifurca-tion in the system of optically injected semiconductor laser

through analytical calculations is taken up. The motivation

was to verify that such transition points found through nu-

merical simulations, indeed matched the analytical results

obtained. We see that under certain parameter conditions,

Hopf bifurcation exists in the given system. A curve is de-

rived from analytical conditions such that each point on the

curve represents a Hopf point. On the other hand, in a sepa-

rate study the bifurcation of a period-1 orbit has been dem-

onstrated, based upon the asymptotic averaging techniques.

aThis work is dedicated to the memory of our beloved friend Mousumi

Saha.bElectronic mail: [email protected] mail: [email protected] to whom correspondence should be addressed. Electronic mail:

[email protected]

CHAOS VOLUME 14, NUMBER 2 JUNE 2004

3471054-1500/2004/14(2)/347/11/$22.00 © 2004 American Institute of Physics


http://dx.doi.org/10.1063/1.1755179http://dx.doi.org/10.1063/1.1755179http://dx.doi.org/10.1063/1.1755179


3/12

We have verified analytically that the points of 1→2 bifur-

cation on the bifurcation diagram actually satisfy the re-

quired conditions. Along with these, the method of adaptive

stabilization is implemented, which is different from the con-

ventional control strategies such as the OGY method,26 ex-

ternal harmonic forcing,27 adaptive and threshold control,28

etc. This method has shown good results for the control of

chaos in Lorenz system.29

The above analysis has been carried out on opticallyinjected laser system but such chaotic properties are also

observable in other laser types.30–32 The results are almost

valid for those systems and so the observations of these type

of analysis is equally applicable to them.

II. FORMULATION

The set of equations governing a single mode semicon-

ductor laser subjected to optical injection is given by

E ˙ 12 1i i E , 2.1

˙2 12 E 21 , 2.2

where E is the complex electric field, is the population

inversion, is the injected field strength, the detuning of

the electric field from the solitary laser frequency, is the

linewidth enhancement factor, is the cavity lifetime, and

is the damping rate. Substituting E x 1ix 2 and x 3 and

performing simple algebraic steps, Eqs. 2.1–2.2 are trans-

formed to three ordinary differential equations as

ẋ 1 x1 x 3

2

2 x 2 x 3 x2 , 2.3

ẋ 2 x 1

2 x 1 x 3

x2 x 3

2 , 2.4

ẋ 32 x3 12 x3 x12 x 2

21 . 2.5

Equations 2.3–2.5 show that 0,0,0 is not a fixed point.

Considering , the analytical fixed point is obtained as

x 12 p

p2 2 p 2 ,

x 22 p2

p 2 2 p 2 , 2.6

x3

p ,

where p is a real number satisfying the condition

p 2 21 4 p 2 2 4 2 2 0.

Assuming, 2 2 / ( 4 ) another analytical fixed

point arises as

x 1

x 20, 2.7

x 32

.

The system of equations under study are thus analyzed for its

dynamical behavior around the existing fixed points under

the effect of the parameters for its chaotic and periodic prop-

erties. Exploring through the parameter regime, Hopf bifur-

cation, intermittent transition to chaos and periodic doubling

routes were observed. Two bifurcation diagrams have been

depicted, which are part of the total bifurcation spectrum of

that particular parameter regime. Figures 1 and 2 demon-

strate a gradual period doubling and these are specially im-

portant to visualize the point of 1→2 bifurcation. These

points are later used in Sec. IV for analytical verification of

the 1→2 bifurcation. The points are indicated in the figures

by an arrow. In the figures, windows in between the chaotic

regions is also seen. On comparing, it is also seen that the

chaoticity is denser in the first than the second. This also

points out to the separation of the neighboring points in

phase space. For Fig. 1, the parameter is varied and thevalues of the other parameters are

2.0, 0.015, 0.035, 0.62, 2.8

whereas for Fig. 2, is varied and the other values are

2.5, 0.015, 0.05, 0.2. 2.9

Other than these, torus bifurcations and intermittency were

also found with the variation of the parameters and

FIG. 1. Bifurcation diagram showing the variation of the variable x with .

FIG. 2. Bifurcation diagram showing the variation of the variable y with .

348 Chaos, Vol. 14, No. 2, 2004 Banerjee, Saha, and Chowdhury



4/12

which is not depicted. Along with the analytic study of pe-

riod doubling, the case of analytic detection of Hopf bifur-

cation is also considered. Before proceeding to that, the

quantification of the chaotic region is studied, which gives a

visual understanding to the properties of the attractor.

III. QUANTIFICATION OF THE ATTRACTOR THROUGHRECURRENCE ANALYSIS

In the method of recurrence analysis, an attractor is re-

constructed from a given time series of the nonlinear dy-

namical system of equations. To expand a one-dimensionalsignal or time series into an m-dimensional phase space one

substitutes each observation in the original signal X (t ) with

vectors,

y i x i , x id , x i2d , . . . , x i m1 d ,

where i is the time index, m is the embedding dimension,41

and d is the time delay. As a result we have a series of

vectors,

Y y 1 , y 2 , y 3 , . . . , y N m1 d .

From this new reconstructed dynamical system, a recurrence

plot can be used to show which vectors in the reconstructed

space are close and far from each other. We calculate thedistance between all pairs of vectors and plot them. In the

diagram ‘‘dark’’ points show large distances between vectors

while ‘‘light’’ denote small distances. Since the distance of

the ith vector to the jth vector is the same as that of jth to ith

vector, recurrence plot is symmetric and the diagonal of the

plot is a white line. The recurrence plot carried out with

10 000 data points is shown in Fig. 3. The x-axis and the

y-axis represent the data points and so do not have any unit

and each point on the plot represent their mutual distance.

The time delay is estimated by the mutual information func-

tion. The point where the average mutual information

reaches its first minimum is considered as the optimal time

delay. The embedding dimension is obtained from the false

nearst neighbor method which was suggested by Kennel

et al.42 The dimension for which the proportion of false near-

est neighbors is minimum is chosen. The embedding dimen-

sion and time delay obtained in our case are 3 and 7, respec-

tively. On observing the plot we can see that there is no

definite repeating pattern. Such irregular patterns represent

chaoticity and cannot be seen in any periodic signals.

Contrary to the above one can also make a nonparamet-ric time series prediction without any assumption about the

functional form of the process that generated the observable

time series. The methodology behind this is that to predict a

point x(n1), we determine the last known state of the

system as represented by vector

X x n , x nd , x n2d , . . . , x n m1 d ,

where d and m have their usual meanings. Then the time

series is searched to find k similar states that have occurred

in the past, where ‘‘similarity’’ is determined by evaluating

the distance between vector X and its neighbor vector X in

the d -dimensional state space. If the observable signal wasgenerated by some deterministic map

M x n , x nd , x n2d , . . . , x n m1 d

x n1 ,

then the map can be recovered from the data by looking at its

behavior in the neighborhood of X . We find the approxima-

tion of M by fitting a polynomial which maps k nearest

neighbors of similar states of X onto their next immediate

values. Now using this map x(n1) is predicted. So with

the assumption that M is fairly smooth around X if a state

X x

n , x

nd , x

n2d , . . . , x

n m1 d

in the neighborhood of X resulted in the observation x(n

1) in the past, then the point x(n1) which we want to

predict must be somewhere near x(n1).

Normally the data set is divided into two parts. The first

part is used for model fitting and the second part for model

validation. In our case we have a total of 7530 data points

out of which the first 3367 are utilized for model fitting and

the rest are for model validation.

The embedding dimension has to be chosen properly be-

cause a low embedding dimension would not reveal the un-

derlying system properties uniquely and a high embedding

dimension would lead to increased computation time, ampli-fication of noise, and deteriotion in the quality of predictions.

The time delay determines the time separation of the com-

ponents in the reconstructed vector of the system state is to

be chosen properly. If time delay is set to a low value, then

the corresponding vector components may be too redundant

and a high time delay can led to corresponding vector com-

ponents too independent. Embedding dimension and time de-

lay was estimated as 3 and 7, respectively.

Other than these, a kernel have also been chosen which

is essentially a weighting function used to assign the contri-

bution of each neighbor to the prediction of the kernel re-

gression and locally weighted linear predictors.

FIG. 3. Recurrence plot for the chaotic regime with 10 000 data points.

349Chaos, Vol. 14, No. 2, 2004 Chaos, uncertainty, control



5/12

To weigh the contribution of each of the k nearest neigh-

bor to the resulting prediction we have, k (Epanechnikov)

0.75*(1distance/ h)2. h is the bandwidth of the neigh-

borhood formed by k neighbors which is simply the distancefrom the reference state. The predictor was the nearest neigh-

bor basis and ‘‘one-step-ahead’’ forecast was used. The dis-

tance was measured by correlation, the root mean square

error0.0083 and the normalized error0.0006. The varia-

tion of the predicted and the actual model with time is shown

in Fig. 4, where we can see that actual values and the pre-

dicted values almost overlap each other and is only differen-

tiable at the end of the curve. The time represents the inte-

gration step of the Runge–Kutta–Fehlberg method which

was considered to be 0.01.

IV. PREDICTION OF 1\2 BIFURCATION THROUGHASYMPTOTIC AVERAGING TECHNIQUES

In Sec. II we have explored the various parameter re-

gions where the system shows period doubling before attain-

ing chaos. In this process the system undergoes subsequent

bifurcations and those are depicted in the bifurcation dia-

grams. It is possible to predict those points of period dou-

bling through partly analytical and partly numerical methods.

In the other sense it is possible to verify that those points

where bifurcation occur actually satisfy the required analyti-

cal conditions. This can be done by the asymptotic averaging

technique which is briefly discussed. The differential equa-

tion modelled by the formdX

dt F X , 4.1

where F ( X ) F 1( X ) F 2( X ) F 3( X )T and X X 1 X 2 X 3

T

can be partitioned according to

X X 1 X 2 , F X F X 1 G X 1 , X 2 ,

4.2dX 1

dt F X 1 ,

dX 2

dt G X 1 t , X 2 . 4.3

Equation 4.2 is for the period-1 orbit and Eq. 4.3 is for

the period-2 orbit, since X (1 )( t ) is the period one solution

of period T 2 / , in the region X (2 )0. In the parameter

space where X (2 ) is born X (1 ) loses its stability. Basically,

this technique of asymptotic averaging requires transforma-

tion of Eq. 4.1 in a form that supports the averaging pro-

cedure. The system coordinates ( X 1 , X 2 , X 3) are transformed

to the dynamical coordinate ( x , y , z) such that the system of

Eq. 4.1 can be expressed as a single third order differentialequation according to

X i X i x, y , z i1,2,3, 4.4

dx

dt y ,

dy

dt z , and

dz

dt W x , y , z . 4.5

These transformations are treated as they are extensively

used to construct embeddings of scalar experimental data

into multidimensional phase spaces. If the system of Eqs.

4.1 were to be linearized around a fixed point, choosing

X i x for any i, one obtains W g1 xg 2 yg 3 z , where g 1 1 2 3 , g 2 ( 1 2 2 3 3 1), and g 3 1 2

3 with 1 , 2 , and 3 as eigenvalues of the secularequations. In the environment of period one orbit, W appears

as the sum of linear and a quadratic form. The fundamental

procedure described in Ref. 33 consists of writing each of the

dynamical variables x, y, z in the form x x 1 x2 , y y 1 y 2 , and so on, where x 1 , y 1 ,... stands for the period 1 and

x2 , y 2 those for period 2. For our system we set X

( X ,Y , Z ),

X ˙ XZ

2

2 Y Z Y ,

Y ˙ X

2

X Z Y Z

2

, 4.6

Z ˙2 Z 12 Z X 2Y 21 .

The transformation of the system to a single third order dif-

ferential equation is done by choosing x Z which leads to

the following exact but complicated result:

dx

dt y ,

dy

dt z ,

dz

dt W , 4.7

where

W x , y , z 2 z y2 2 12 x y 24

xy

xz

1

2 x 4 yz

4 xz4 y 2 4 2 y 2 y2 x 12 x 2

4 y x 12 x x 1

2 x 2 12 x , 4.8

where and are given by

1

2 12 x 2 x 2 xy2 x 2 x z2 y

12 x 2 y y2 x 12 x 2,

1 y2 x 12 x 1 X 21/2.

FIG. 4. Variation of the actual bold line AV and predicted thin line

PV values of the model with time.




6/12

We now assume that the parameter is very small, so that

we approximate Eq. 4.8 by

W x , y , z g 1 xg2 yg3 zhq11 x2q 12 x yq 13 xz

q22 y2q 23 y zq 33 z

2, 4.9

where

h2 22 , g1 4 4 24 ,

g 2 1, g 32 ,

q 118 2 2

4 2 4

0.5,

q 122 4 44

2 2

,

q 134 1

2 0.5, q 234

4 ,

q 33

4 .

The above transformation yields the following equations:

dz 1

dt g1 x 1g2 y 1g3 z 1hQ x1 , y 1 , z1

equation for period one orbit 4.10

so that x 2 evolves according to

dz 2

dt F 1 t x 2F 2 t y 2F 3 t z2Q x2 , y 2 , z 2

4.11

and

F 1 t g 12q11 x1 t q 12 y 1 t q 13 z 1 t ,

F 2 t g 22q22 y 1 t q 12 x 1 t q 23 z 1 t

equation for period two orbit,

F 3 t g 32q33 z1 t q 13 x1 t q 23 y 1 t .

To proceed further we assume a simplest form of period one

orbit without any higher harmonics which when used in Eq.

4.10 yields the following conditions:

q 12q13 D 02

2q11g 2q 13g3q 12 D 0 g 1g2g 304.12

and

D 12

2 g1 D 0q 11 D 02h

q11 2 q 22q13

4.13

with

2 g 2q12 D 0, where x1 t D 0 D 1 cos t .4.14

A similar analysis is also performed for the period 2 case and

with the following substitutions:

x2u cos t 2 v sin t

2 , 4.15

y 2

2 u sin t

2

2 v cos t

2 , 4.16

z2 2u

4

2

d v

dt cos t

2

2v

4

2

du

dt sin t

2 .

4.17u and v are determined by the inverse transformations.34 An

elaborate computation leads to the following two differential

equations for u and v:

d 2u

dt 2 A 11 t

du

dt A 12 t

d v

dt A 13 t u A 14 t v

U u,v , . . . , t , 4.18

d 2v

dt 2 A21 t

du

dt A 22 t

d v

dt A 23 t u A24 t v

V u ,v, . . . , t . 4.19

The strategic point of the asymptotic averaging technique is

to approximate the solutions of these two equations by the

corresponding ones of the averaged differential equations

written as

d 2 x

dt 2

1

T

0

T

F dxdt

, x,t dt . 4.20Here x(u,v) with T 4 / . Hence the next stage is to

introduce the following second transformation from (u,v) to

new coordinates ( y , z) defined by generating functions 1 ,

2

as

u y 1 y , z , d ydt ,dz

dt , t ,

4.21

v z 2 y , z , dydt ,dz

dt , t .

The quadratic terms with nonzero averaging values are now

critical for the onset of period 1→2 bifurcation. The details

calculations leads to a secular equation which controls the

stability of the period 2 equation as

A4 4 A 3

3 A2

2 A 1 A 00, 4.22

where A4 1 f 11 1 f 22 f 12 f 21 ,

A3 d 11d 22 d 11 f 22d 22 f 11d 12 f 21d 21 f 12,

A2 d 11d 22d 12d 21 e11 e 22 e 11 f 22e 22 f 11

e21 f 12e 12 f 21,

A1 d 11e 22 d 22e 11 d 12e 21 d 21e12 ,

A0e 11 e22 e12 e21 .

This equation determines the conditions for stability of the

period two orbit evolving from period one orbit, parallel to




7/12

the secular equation which determines the conditions for

evolution to a dynamical orbit evolving from instability of a

fixed point. The first analytic condition for the onset of pe-

riod one to period two bifurcation is A 00, where one of the

roots of the eigenvalue Eq. 4.22 passes through a zero. For

the development of the period two orbit there must exist one

real negative root and a pair of conjugate complex roots with

a negative real part. As the period two develops, it is stable

as long as all roots are negative or have negative real parts.When any real part becomes positive, the period two orbit

itself becomes unstable. The calculation of that parameter

point at which the period two orbit becomes unstable re-

quires explicit inclusion of the nonlinear terms which is suf-

ficient to establish the necessary analytic condition A 00 for

the onset of period two. The control parameter or at the

1→2 bifurcation point is denoted by * or * so that A k *

Ak ( *) or A k * A k ( *) and

A 4* 3 A 3*

2 A 2* A1*0.

The necessary and sufficient condition that all eigenvalues

have negative real parts and therefore the bifurcation is

stable when A 2* A3* A 1* A 4* . 35 The quantities 0( y 0 , z0)determines the amplitude and phase of the period two orbit

where 0 is the fixed point of the averaged equation. The two

conditions for the onset of 1→2 bifurcation are

A 0 0 0 and

4.23 0 A 2 0 A 3 0 A 1 0 A 4 0 0

in the limit 0→0.

As 0→0 grows there are now four roots to Eq. 4.22.

The extent of the period two orbit is determined by that value

0 such that one or more of the roots acquires a positive real

part, at which point the period two orbit becomes unstable.25

The coefficients and terms used in this section are explicitlygiven in the Appendix.

Figures 1 and 2 are the bifurcation diagrams in the two

parameter regimes of and , respectively. In the first one it

is observed that the period 1→2 bifurcation occurs at the

point where 0.4325, whereas in the latter it occurs at a

point where 0.4202. These points along with their cor-

responding other parameter values were used to calculate

numerically the coefficients A1( 0), A 2( 0), A 3( 0), and

A 4( 0). Substituting all these values in Eq. 4.23, it is found

that A 00 and ( 0)0 for both the cases thus satisfying

our numerical results.

V. ALGEBRAIC DETECTION OF HOPF BIFURCATION

The n-dimensional system of ordinary differential equa-

tion defined by

ẋ f x, ,

where x x(t )Rn and f : U →Rn is a C 2-smooth function

defined on a subset U Rn which depends upon a vector of

parameters, Rk . The equilibrium point of the system is

xRn such that f ( x , )0. As the parameters are varied,

equilibrium points can undergo bifurcation. There are two

types of elementary bifurcations that occur in one parameter

families of systems—saddle nodes and Hopf bifurcations.

The necessary condition is that the Jacobian evaluated at an

equilibrium point possesses a simple zero eigenvalue in case

of former and a pair of pure imaginary eigenvalues in case of

latter. For the case of saddle-node bifurcation one can obtain

the bifurcation locus by augmenting the equation f ( x, )

0 with a procedure to calculate equilibrium points where

the Jacobian has numerical rank ( n1). An inflated system

for the detection of saddle nodes has been previously done

using det( D x f ) as the augmenting equation.36

Here we exam-ine a procedure for locating Hopf bifurcations. For this we

seek explicit criteria that specify whether a ( nn) matrix

with coefficients that may depend upon parameters has a pair

of pure imaginary eigenvalues. For this purpose the desired

necessary conditions are to be developed.

Let J be the Jacobian matrix for f and its characteristic

polynomial be given by

p C 0C 1¯C n1n1 n.

p has a nonzero root pair , if and only if is a com-

mon root of the two equations p() p() and p()

p(). Making the substitution z2 and rearranging,

we construct two new polynomials. If n is even, then

r e z C 0C 2 zC 4 z2¯C n2 z

n2 /2 z n /2,

5.1r 0 z C 1C 3 zC 5 z

2¯C n1 z

n2 /2,

while for odd n

r e z C 0C 2 zC 4 z2¯C n3 z

n3 /2 z n1 /2,

5.2r 0 z C 1C 3 zC 5 z

2¯C n2 z

n3 /2 z n1 /2.

Then p has a nonzero root pair , if there exists a z that

satisfies

r e z

r 0 z 0.The two polynomials have a common root if they have a

common factor. Since the degree of r 0 is less than or equal to

r e , we use the Euclidean algorithm.

We construct a sequence of polynomials,

P0r e , P 1r 0 , P 2 , P 3 , . . . , P k ,

such that P i1 is the remainder of the division of P i1 by

P i . Thus the degrees of P i are strictly decreasing and there

are polynomials Q i , i1,2,...,k such that

P i1Q i P iP i1 .

The Sylvester matrix can be constructed from the coeffi-

cients of the polynomials. The determinant RS det(S ) is a

polynomial in the coefficients of p referred to as the

Sylvester resultant of r e and r 0 share a common root.

For i0, 1 let S i denote the matrix obtained from S by

deleting the rows 1 and n /2 and the columns 1 and i2 for

even n. The relationship between the characteristic polyno-

mial and its corresponding matrices S , S 0 , S i leads to the

results.

Theorem 1: Let S be the Sylvester matrix for the poly-

nomials r e and r 0 . Then the jacobian matrix J has precisely

one pair of pure imaginary eigenvalues if




8/12

RS 0 and det S 0•det S 1 0.

If RS 0 or det(S 0)•det(S 1)0, then p() has no purely

imaginary roots.

Theorem 2: Let 1 , 2 , . . . , n be the roots of the poly-

nomials p() and let S (s i j) be the (n1)(n1)

Sylvester matrix of the associated pair ( r e , r 0). If

m n1 /2

2 n

2 ,then

RS 1 m

1i jn i j .

Theorem 3: For (r e) and ( r 0) defined as above, let

r 0(C r e)C if n is even and r e(C r 0)C if n is odd. If RC is

used to denote det(C ), then RC RS .

To develop a third criterion, we consider the Bezout re-

sultant. For n even the two polynomials specified by Eq.

5.1 we define the brackets

i, j det C 2i C 2 jC 2i1 C 2 j1 , 5.3where 0i , jn /2 and we take C n1, C n10. The Be-

zout matrix B corresponding to the polynomial pair ( r e , r 0)

is an n /2-dimensional square symmetric matrix with entries

constructed as sum of bracket products in the coefficients C ias follows:

For 1i jn /2 set

k minmax 0,i j n21 ,5.4

k maxi1.

Then,

B i j k k min

k max

i jk 1,k B ji . 5.5

The only modification required is the definition for the case

of n-odd is that C n has the value prescribed by the charac-

teristic polynomial p and C n1 is taken to be unity. The

Sylvester and the Bezout matrix are connected by a relation 37

which leads to the following theorem:

Theorem 4: For the Bezout matrix corresponding to the

polynomial pair (r e ,r 0) let R B denote det( B). Then we have

R B1 nRS . 5.6

The coefficients of the linear remainder term from the Eu-

clidean algorithm can be expressed as determinants of certain

Bezout submatrices, in analogy to the submatrices defined

for the Sylvester form. We consider the two submatrices, B ifor i0 and 1, obtained by deleting the first column and the

ith row of B. We now state another theorem:

Theorem 5: Let B be the Bezout matrix for the polyno-

mials r e and r 0 . Then J has precisely one pair of pure imagi-

nary eigenvalues if

R B0 and det B 0•det B 1 0.

If R B0 or det( B0)•det( B1)0, then p() has no purely

imaginary roots.

For our system 2.3–2.5 if we choose the fixed point

2.7, the corresponding eigenvalue equation becomes

3 2 2 2 2

2

2 2 2

2

2

2

2 2

2

0.Therefore Eq. 5.2 can be written as

p C 0C 1C 223,

r e zC 0C 2 z ,

r 0 zC 1 z,

where z 2 and

C 0 2 2 2

, C 1 2

2 ,

C 2 2 2 2

2

2

.The Sylvester matrix S and the two relevant submatrices, S 0and S 1 , are given by

S C 0 C 2C 1 1

,

Now by Theorem 1, p has a pair of purely imaginary roots if

Rsdet(S )C 0C 1C 2 vanishes and the product S 0S 1C 10.

The Bezout matrix associated with (r e , r 0) is just a

single element given by

FIG. 5. Locus of the points satisfying the Hopf bifurcation condition in the

, phase space.




9/12

B 1,0det C 2 C 01 C 1

C 2C 1C 0 1 3 C 0C 1C 2 R s .

The Bezout subresultant condition provided by the expres-

sion, det( B0)det( B1)C 1 is necessarily positive when p has a

pair of imaginary conjugate roots. In our case C 1( 2 2 / 2)( 2 / 2)) is always positive. Therefore C 0C 1C 20 is the only condition for a pair of purely imagi-

nary conjugate roots. Figures 5 and 6 represents the curve of

Hopf bifurcation for versus and versus , respec-

tively. Each point on the two curves represents a Hopf bifur-

cation point. Tables I and II provide the eigenvalues for some

points of Figs. 5 and 6, respectively.

Thus we have a method of computing the singularity

condition necessary to establish that the Jacobian of the dy-

namical system 2.3–2.5 has a pair of purely imaginary

eigenvalues, which is the condition for the Hopf bifurcation.

VI. CONTROL OF CHAOS BY ADAPTIVESTABILIZATION

In this section the task of controlling chaos is consid-

ered. The aspect of controlling chaos in a semiconductor

laser system is important due to its possible applications in

synchronization, communication, and data transfer. Different

control techniques have been developed so far,38,39 most of

which require the system parameters to be known exactly.

This requirement makes the practical application of such

control procedures difficult. So we have discussed a new

control strategy using invariant manifold theory which can

deal with systems where the parameter values are unknown.

This makes the procedure feasible to experimental setup. The

essential requirement is that the steady or equilibrium state

of the system is to be known and so this procedure is imple-

mentable to fixed points and not to periodic solutions. The

amplitude of perturbation given to the dynamical system in

the form of three control inputs is determined by the method

itself.

To stabilize the system we introduce three control inputs

in the system 2.3–2.5 as follows:

dx 1

dt x1 x 3

2 x 2 x 3 x2u 1 , 6.1

dx 2

dt

x 1

2 x1 x3

1

2 x 2 x3

u 2 , 6.2

dx 3

dt 2 x3 12 x3 x1

2 x 2

21 u 3 . 6.3

Now we consider the fixed point 2.6 of the system. To

avoid the numerical complications we choose 0.7. There-

fore the fixed point 2.6 becomes

a a 1 ,a 2 ,a 3T 6.4

a 1

1.4 p

p 2 2 p 2

a 21.4 p2

p 2 2 p 2

a 3 p . 6.5

The three manifolds given below are selected as invariant

manifolds:

M 1 xR3 x 1a 10, 6.6

M 2 xR3 x 2a 20, 6.7

M 3 xR3 x 3a 30. 6.8

Therefore the system is also stable on the intersection of

these manifolds, M c M 1 M 2 M 3

xR3 x 1a 10, x2a 20, x 3a30. 6.9

Our motivation is to design a control which drives the sys-

tem trajectories along these three manifolds to the set M c .

Then by Lasalle’s invariant set principle40 we can con-

clude that x will converge to the equilibrium a asymptoti-

cally, where a is the only stable equilibrium of the set M c .

To design such a stabilizing controller we construct a

Lyapunov function as follows:

V x 12

T x T x , 6.10

FIG. 6. Locus of the points satisfying the Hopf bifurcation condition in the

, phase space.

TABLE I. 1.0.

Eigenvalues obtained

0.2 0.19183 0.10870 0.2, 5.81090.9798i

0.25 0.23386 0.14286 0.25, 5.961080.96824i

0.31 0.27862 0.19188 0.31, 2.3031080.95073i

0.44 0.34444 0.35901 0.44, 1.078081080.89799i

0.56 0.341921 0.75107 0.56, 5.5321090.82840i

TABLE II. 1.5.

Eigenvalues obtained

3.0 0.35355 0.5 0.5, 2.42881070.86602i

4.0 0.31792 0.26087 0.375, 2.1231060.92702i

5.2 0.26337 0.17303 0.28846, 2.3031080.95073i

6.3 0.22419 0.13427 0.23810, 2.81060.97124i

7.9 0.18290 0.10231 0.18987, 1.14241070.98181i




10/12

where T ( x) 1( x) 2( x) 3( x) x 1a 1 x 2a 2 x 3a3 , T R

33 is a positively definite matrix. Now we

want the following differential equations holds so that the

controller can be designed

T ˙ x x 0. 6.11

Therefore the time derivative of V ( x) becomes

V ˙ x T x x . 6.12

This implies that V ˙ ( x)0 for all x M c , meaning the sys-

tem trajectory can be driven to M c asymptotically. To find

such a controller, the system Eqs. 2.3–2.5 into Eq. 6.11

gives

u u 1u2u3 0.7 x1 x 3

2 x 2 x3 x 2

x 1

2 x1 x 3

1

2 x 2 x3

2 x 3 12 x 3 x 12 x 2

21

T 1 x 1a 1 x 2a 2

x 3a 3 . 6.13

If the parameters , , are unknown, the controller 6.13

should be able to realize the stabilization. However the con-

troller 6.13 is not implementable because parameters , ,

are assumed to be unknown. This problem cannot be

solved via the conventional adaptive control theory, because

the basic assumption of the adaptive control is that the equi-

librium is fixed and unknown. Here we shall develop a tai-

lored adaptive control for our system in the following. Be-

cause the parameters , , are unknown, we use themodified three invariant manifolds M 1 , M 2 , M 3 which are

defined, respectively, as ˆ 1( x) x 1â 1 , ˆ

2( x) x2â 2 ,

ˆ 3( x) x3â 3 , where â 1 , â 2 , â 3 are given by

â 11.4 p̂

p̂ 2 2 ˆ ˆ p̂ 2 , 6.14

â 21.4 ˆ p̂ 2

p̂ 2 2 ˆ ˆ p̂ 2 , 6.15

â 3 p̂ , 6.16

where p is real satisfies the relation

p̂ 2 ˆ 21 4 p̂ ˆ ˆ 0.98̂4 ˆ 20.490. 6.17

For a adaptive control design we construct the following

parameter-estimate-dependence Lyapunov function:

V x, ˆ , ˆ , ̂12

ˆ T x T ˆ x eT P1e, 6.18

where e ˆ with T and ˆ ˆ ˆ ̂ T , ˆ T ( x)

ˆ 1 ˆ

2 ˆ

3 and PR33 is a positive definite matrix. Let

us denote

f x 0.7 x1 x 3

2

x2 x 3

2

x12 x 2

21

,

F x x 2 x 3

2 x2 0

x 1 x3

2 x 1 0

0 0 2 x 3 x12 x 2

21 ,

and

g x 1 0 0

0 1 0

0 0 1 .

Then the time derivative of the Lyapunov function is given

by

V ˙ x, ˆ , ˆ , ̂ ˆ T T ˆ

x f x F x g x u

ˆ T T ˆ

ˆ ˆ ˙e T P1 ˆ ̇

ˆ T T ˆ

x f x F x ˆ g x u

ˆ T T ˆ

x F x e ˆ T T

ˆ

ˆ ˆ ˙e T P1 ˆ ̇.

Let u consists of two parts, i.e., uueu c . Also let

ˆ

x f x F x ˆ g x u eT

1 ˆ , 6.19

ˆ T T ˆ

x g x uc

ˆ T T ˆ

ˆ ˆ ˙ , 6.20

ˆ T T ˆ

x F x eeT P1 ˆ ̇. 6.21

Then

V ˙ ˆ T ˆ 0

for all x such that 0, so the stability of the system state

with respect to the invariant manifolds ˆ 10, ˆ

20, and

ˆ 30 is realized. From 6.19–6.21 we obtain




11/12

u e u e1ue2ue3

0.7 x1 x3 ˆ

2 x 2 x3 ˆ x 2

ˆ x 1 ˆ

2 x 1 x3

1

2 x 2 x3

2̂ x 3 12̂ x 3 x 12 x 2

21

T 1

x 1â 1 x

2

â 2

x 3â 3 6.22

and

u c u c1uc2uc3

ˆ 1

ˆ

ˆ 1

ˆ

ˆ 1

̂

ˆ 2

ˆ

ˆ 2

ˆ

ˆ 2

̂

ˆ 3

ˆ

ˆ 3

ˆ

ˆ 3

̂

P

x 2 x32

x 2 0

x1 x3

2 x1 0

0 0 2 x 3 x 12 x2

2

T x 1â

1

x2â 2 x3â 3

6.23and the update law for parameter estimation

ˆ ̇ ˆ ̇

ˆ ˙

̂̇ P ˆ

x F x

T

T ˆ

P x 2 x3

2 x 2 0

x1 x 3

2 x1 0

0 0 2 x 3 x 12 x2

2

T

x 1â 1 x2â 2

x3â 3

. 6.24

For our system we consider the parameter values as 2.0,

0.69, 0.005. Also constant matrices were set to be

T 1 0 0

0 1.43 0

0 0 0.5 , P

1 0 0

0 1.05 0

0 0 0.25 .

The initial conditions were x(0)(0.5,0.5,1.0)T and

( ˆ 0 , ˆ 0 ,̂0)(2.5,0.7,0.01). Using Eqs. 5.22–5.23 and

then integrating the system 6.1–6.3 along with 6.24 we

get a controlled scenario from a chaotic state. The time evo-

lution of the variables are shown in Fig. 7 which depicts their

convergence to a steady state.

VII. CONCLUSION

Properties of single mode optically injected semiconduc-

tor lasers are very much important from various experimen-

tal points of view. In this paper, we have made a detailed

simulation study of its chaotic aspects, the various mecha-

nism of onset of chaos—period doubling bifurcation and

Hopf bifurcation and have given a complete characterization

of the attractor with an eye to the availability of a single time

series from experimental data. Our results are checked in a

dual way—both analytically and numerically. Finally a new

method has been applied to ascertain the controlling property

of the chaotic scenario. The construction of a control mecha-nism is nowadays a well researched and formulated topic.

Already there exist a few authentic monographs on the sub-

ject and all of them actually start from the ansatz of a suit-

able Lyapunov function. The net effect and the conditions

therein, for the influence on the phase space, are classified in

stages. We have not attempted to deal with this ourselves, but

refer to the literature.43

ACKNOWLEDGMENTS

One of the authors P.S. is grateful to the CSIR Gov-

ernment of India, for the senior research fellowship, and the

work of A.R.C. is partially supported by a research project

FIG. 7. Time variation of the system variables on the application of adaptive

stabilization which shows convergence to a steady value.




12/12

sponsored by the CSIR. The authors are also thankful to the

anonymous reviewers for their valuable suggestions.

APPENDIX: COEFFICIENTS f ij , d ij , e ij

f 11C 0

2

2 , f 12

C 03

2 ,

f 21

C 03

2 , f 22

C 02

2 ,

d 11 k 23 k 11 C 0

2

k 11m 1

k 21m 2

c 11 ,

d 12 k 24 k 12 C 0

2

k 12m 1

k 22m 2

c 12 ,

d 21 k 13 k 21 C 0

2

k 11m 3

k 22m 4

c 21 ,

d 22 k 14 k 22 C 0

2

k 12m 3

k 22m 4

c 22 ,

e 11k 13C 0

2

k 13

m 1

k 23

m 2c 13 ,

e 12k 14C 0

2

k 14

m 1

k 24

m 2c 14 ,

e 21k 23C 0

2

k 13

m 3

k 23

m 4c 23 ,

e 22k 24C 0

2

k 14

m 3

k 24

m 3c 24 .

1 S. Wieczorek, B. Krauskopf, and D. Lenstra, Opt. Commun. 172, 279

1999.2 G. H. M. van Tartwijk and D. Lenstra, Quantum Semiclassic. Opt. 7 , 87

1995.3 F. T. Arecchi, J. Gea, and F. Romanelli, Opt. Commun. 36, 144 1981.4 L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt.

Commun. 46, 64 1983.5 T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, Phys. Rev. A 53,

4372 1996.6 V. Anoovazzi-Lodi, S. Donati, and M. Manna, IEEE J. Quantum Electron.

3D, 1537 1994.7 T. B. Simpson, J. M. Liu, K. F. Huang, and K. Toi, Quantum Semiclassic.

Opt. 9, 765 1997.8 A. Gavrielidis, V. Kovanis, P. M. Varangis, and T. Erneux, Quantum Semi-

classic. Opt. 9, 785 1997.

9 B. Krauskopf, S. Wieczorek, and D. Lenstra, Proc. SPIE 3944, 612 2000.10 R. Adler, Proc. IRE 34, 352 1946; reprinted in Proc. IEEE 61, 1380

1973.11 Y. Kuznetsov, Elements of Applied Bifurcation Theory, No. 112, in Ap-

plied Mathematical Sciences Springer, New York, 1997.12 G. L. Oppo, A. Politi, G. L. Lippi, and F. T. Arecchi, Phys. Rev. A 34,

4000 1986.13 H. Zeglache and V. Zehble, Phys. Rev. A 46, 6015 1992.14 J. R. Tridicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, J. Opt. Soc.

Am. B 2, 173 1985.15 A. Politi, G. Oppo, and R. Badii, Phys. Rev. A 33, 4055 1986.16 P. A. Braza and T. Erneux, Phys. Rev. A 40, 2539 1989.17 H. G. Solari and G. L. Oppo, Opt. Commun. 111, 173 1994.18 M. K. S. Yeung and S. Strogatz, Phys. Rev. E 58, 4421 1998.19 M. G. Zimmermann, M. A. Natiello, and H. G. Solan, Physica D 109, 293

1997.20 M. G. Zimmermann, M. A. Natiello, and H. G. Solan preprint.21 M. C. Casdagli, Physica D 108, 12 1997.22 J. P. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 4 , 973

1987.23 J. Guckenheimer, M. Mayers, and B. Strumfels, ‘‘Computing Hopf bifur-

cations I’’ preprint.24 J. Guckenheimer and M. Mayers, ‘‘Computing Hopf bifurcations II’’ pre-

print.25 P. E. Phillipson and P. Schuster, Int. J. Bifurcation Chaos Appl. Sci. Eng.

8, 471 1998.26 E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 1990.27 A. RoyChowdhury, P. Saha, and S. Banerjee, Chaos, Solitons Fractals 12,

2421 2001.28 S. Banerjee, P. Saha, and A. RoyChowdhury, Phys. Lett. A 291, 103

2001.29 Y. Tian and X. Yu, Complexity International http://life.csu.edu.au/

complex/ci/vol6/tian.30 G. L. Lippi, J. R. Tredicce, N. B. Abraham, and F. T. Arecchi, Opt. Com-

mun. 53, 29 1985.31 Instabilities and Chaos in Quantum Optics, edited by F. T. Arecchi and R.

G. Harrison Springer-Verlag, Berlin, 1987.32 P. Glorieux, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9 , 1749 1998.33 P. E. Phillipson and P. Schuster, Int. J. Bifurcation Chaos Appl. Sci. Eng.

10, 1787 2000.34 C. Holmes and P. Holmes, J. Sound Vib. 78, 161 1981.

35 E. A. Jackson, Perspectives in Nonlinear Dynamics Cambridge Univer-sity Press, New York, 1991.

36 J. P. Abbott, J. Comput. Appl. Math. 4, 19 1978.37 S. Barnett, Polynomials and Linear Control Theory Marcel Dekker, New

York, 1983.38 F. T. Arecchi, S. Boccaletti, M. Ciofini, C. Grebogi, and R. Meucci, Int. J.

Bifurcation Chaos Appl. Sci. Eng. 8, 1643 1998.39 S. Boccaletti and F. T. Arecchi, Physica D 96, 9 1996.40 J. P. Lassalle, J. Diff. Eqns. 4 , 57 1968.41 P. Grassberger and I. Procaccia, Physica D 7 , 473 1983.42 M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys. Rev. A 45, 3403

1992.43 M. Krsti, I. Kanellakopoulos, and P. V. Kokotovi, Nonlinear and Adaptive

Control Design Wiley, New York, 1995.


Documents

Optically injected laser system: Characterization of chaos, bifurcation, and control