Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
Подождите немного. Документ загружается.
88
V.
Grigoras and
C.
Grigoras
3. Case Studies
3.1. The Lorenz system
In this particular case, the emitter subsystem is a third order, analogue,
Lorenz type system:
j
X'
=
a·
(y -
x)
y' =
p·x-
y-x·
Z
z'=x-f3·z+x·y
For a large enough parameter range, this system exhibits chaotic behavior,
useful in ensuring the secrecy the modulating signal. To achieve
synchronization, the
y state variable is transmitted to the receiver subsystem,
designed by using the system partitioning method:
{
i'
=
a·(y-i)
z'=i-f3·z+i·
y
The second equation, for
y'
, will be introduced later on in the state description
of
the receiver for further use in the modulation/demodulation process.
Taking the autonomous case,
y =
0,
its state equations become linear, with a
state transition matrix
of
the form:
A
=(-a
0)
1
-13
The resulting eigenvalues are real and negative, for positive a and
p,
ensuring global asymptotic stability
of
the receiver. The error dynamics can be
analyzed by subtracting the receiver equations from their emitter counterparts.
The state errors,
E
J
= x -
x,
E3
= Z -
Z,
are described by the dynamic
equations:
{
EJ
' =
-a
. E
J
E3
' = E
J
-
13
.
E3
+ E
J
•
Y
Due to the fact that all state variables are bounded, it is easy to demonstrate
iteratively that all errors decay exponentially to zero. Thus the proposed
receiver synchronizes with the chaotic emitter. In order
to
use this
synchronizing setup to transmit
an
information signal, met), by the direct
modulation technique,
we
add the modulating signal to the second equation
of
the emitter:
j
X'
=
a·(y-x)
y:=
p·x-
y-x·
z+m(t)
Z
=x-p·z+x·y
At the receiving end
of
the communication channel,
we
demodulate the
received signal,
y(t), by subtracting from it the locally recovered second state
variable,
y:
y'=
p·x-
y-x·
z
Linear Communication Channel Based
Oil
Chaos Synchronization 89
Figure 1.a
The
PSD
of
the modulating signal
Figure
l.h
The
PSD
of
the chaotic signal for
amplitude
10
The error dynamics for the second state variable:
Figure
l.c
'nlC
PSD
of
the
chaotic signal for
amplltude
20
£2 • = P .
£J
- £2 - (x· z - i .
z)
+ m
leads to the differential eqnation for thc demodulated signal:
!n'
==
-/iI-(z
- P
)'£1
-.x·e,
+ m
Aft
er
the synchronization transient died
out
the demodulator equation can
be
approximated as follows:
iii'
""
-J'iI+m
This highlights the linear memory character
of
the resulting communication
channel, which can be characterized
by
the transfer function:
H(s)=!Yn~)
",_1
__
M(s) s+1
The dynamic range
of
the amplitude for the modulating signal must ensurc
both linearity
of
the communication system and the lack
of
visibility
of
the
modulating signal in the transmitted chaotic one. In order to track the
visibility
of
the modulating signal
in
the transmitted chaotic one, we studied the power
spectral density
(PSD)
of
the transmitted signal, for different amplitudes
of
the
modulating signal, depicted in figure
1.
As seen from the simulation results, if the amplitude
of
the modulating
signal is
as high as 10, its PSD is not visible in the one
of
the transmitted
signal. Beginning with amplitudes
of
the order
of
magnitude 20, the modulator
line becomes visible in the PSD
of
the chaotic signal. This gives a dynamic
range
of
modulating sigmtl amplitude comparable to the amplitudes
of
the
state variables and transmitted signal, thus large enough for practical
applications.
Figure 2.a The modulating Figure 2.b The demodulateD
signal signal
without equalizer
Figme 2.e The demodulated
signal with equalizer
90
V.
Grigoras
and
C.
Grigoras
To increase the bandwidth
of
the transmitted signal, we use for our
equalizer a transfer function
of
the form:
HE
(s)=
s+1
s+l
l/mc
·s+1
O.OI·s+1
In figure
2,
we give example results showing the demodulated signal
without and with the feed forward equalizer.
3.2. The modified Chen system
The Chen system is described by the state equations:
!
x,=-a.x+a.
y
< =
(c-a
)·x+c·
y-x·
z
z
=-b·
z+x·
y
where the standard values for the coefficients are a = 35, b = 3 and c = 28.
To achieve synchronization, the second state variable must be transmitted
and the synchronizing receiver is built around the first and third state
equations. Due to the fact that the
y coefficient in the second state equation,
c,
is positive, the proposed demodulation method cannot function since it will
lead to demodulator instability. In order
to
use our approach on the Chen
system, some coefficient modifications need
to
be made:
!
x,=-a
.
x+a.
y
y.'=d.x-c.y-x.z
z
=-b·z+x·
y
As a consequence
of
the sign change
of
the c coefficient, the values for the
other ones have to be modified to achieve chaotic behavior. For instance,
a =
15,
b =
3,
c = 1 and d = 32
is
a set
of
values leading
to
a chaotic attractor,
as
seen from the simulation results given in figure 3, where the 3D system
trajectory suggests chaos.
The resulting receiver
is
described by:
{
x'=-~
.x
+a.y
z'
=
-b
. z +
X·
Y
In the autonomous case, the receiver state equations are linear and the state
transition matrix is:
A
=(
-a
0 J
lo
-b
Obviously, the synchronizing receive is globally asymptotically stable, with
negative eigenvalues
-a
and
-b.
The error dynamics, in the case
of
parameter match, is given by:
{
C]
'=-a·c]
c
3
' =
-b
. c
J
+
c]
. Y
Linear Communication Channel Based
on
Chaos Synchronization
91
, - , -
-,-.
" ?
]I
.••••
•
~
.•.•.. ; .•. i ...
/
~)
.•.••
.. j
~..
...
....
.
......
.
..
...
....
.
..
..
...
.
~
~~;o~~20
o
~
<
.~
___
~
O
y
-4
0 -
20
o 200 4
00
600
800
1
000
Sample no.
Figure 3 The modified Chen attractor
Figure 4 Time evolution
of
the state
synchronization errors
The state errors,
EI
= X - i ,
E3
= Z - Z , exponentially converge
to
zero, with
the time constants
-1
/ a and
-1
/
b.
The demodulator
is
based on the second state variable:
y'=d·i-c·y-i·z
Subtracting from second equation
of
the emitter, the demodulated signal
dynamics results in the form:
m'=-c·m-(
z
-d)·E
-i·E
+m
1 3
After the synchronization transient,
of
time length
'r
Tr
= Max(
-1
/
a,
-J
/ b),
the state errors become negligible, and the time evolution
of
the recovered
signal can
be
approximated
by:
m'''''-
c
'm+m
The corresponding transfer function results
in
the form:
H(s)=M(s)
",,_1_
M(s)
s+c
Following the general method, the feed forward equalizer has the transfer
function:
HE(s)=
s+c
s+l
I/wc
·s+l
O.Ol·s+l
Simulation results confirm the design calculations previously presented. For
the same parameter values
as
above, the synchronization errors decay
exponentially with the predicted time constants, as presented in figure 4.
The efficiency
of
the proposed equalizer is highlighted in figure 5, where
the modulating signal and the demodulated one are compared.
. 1
50~
-iOO-
~",,
~",
;oo
'--
--';'-
~ooo
Sample no.
Figure 5.a. The Modulating signal
'o!ll-
~-
","
"'--
-660-
~DO
Sample no.
Figure 5.b. The demodulated
signal, using the equalizer
92
V.
Grigoras and
C.
Grigoras
4. Conclusions
We proposed an analog wide-band communication channel based on the
chaos synchronization and direct modulation principles. Using the system
partitioning synchronization and error dynamics methods,
we
demonstrated, in
a general enough setup that a linear dynamic relation exists between the
modulating signal and its demodulated counterpart. A feed-forward channel
equalization technique ensures that the modulating signals can have large
enough bandwidth, leading to the possibility
of
high speed digital
communication. The presented case studies highlight the feasibility
of
the
general method.
Further research is needed
to
approach noise and parameter mismatch
problems, specific
to
analog implementation
of
synchronizing transmissions.
A possible approach can be to extend the use
of
the communication setup
to
digital modulation.
References
[1]
T.
L. Carrol, L.
M.
Pecorra. 'Synchronizing Chaotic Circuits', IEEE
Transactions on Circuits and Systems,
Volume 38,
No.4
, April 1991, Page(s):
453-456.
[2]
Sparrow,
C.
'An introduction to the Lorenz equations', IEEE Transactions on
Circuits and Systems,
Volume 30, Issue
8,
Aug 1983 Page(s):533 - 542
[3]
T. Veta and
G.
Chen. "Bifurcation analysis
of
Chen's equation," International
Journal
of
Bifurcation and Chaos, Vol.
10,
Aug. 2000, Page(s): 1917-1931.
[4]
Gonzales, O.A.; Han, G.; de Gyvez, J.P.; Sanchez-Sinencio, E.; 'Lorenz-based
chaotic cryptosystem: a monolithic implementation'
IEEE Transactions
on
Circuits and Systems
I:
Fundamental Theory and Applications, Volume 47, Issue
8,
Aug. 2000Page(s):1243
-1247
[5]
Cuomo, K.M.; Oppenheim, A.V.; Strogatz, S.H.; 'Synchronization
of
Lorenz-
based chaotic circuits with applications to communications'
IEEE Transactions
on Circuits and Systems
II:
Analog and Digital Signal Processing. Volume 40,
Issue
10,
Oct. 1993 Page(s):626 - 633
[6]
Corron, N.J.; Hahs, D.W.;
'A
new approach to communications using chaotic
signals',
IEEE Transactions on Circuits and Systems
I:
Fundamental Theory and
Applications,
Volume 44, Issue
5,
May 1997 Page(s):373 - 382
Maxwell-Bloch
Equations
as
Predator-Prey
System
A.
S.
Hacinliyan
1
,2,
O.
O.
Aybar
1
,2,
i.
Ku:;;beyzi
1
,2 ,
i.
Temizer
1
,
and
E. E.
Akkaya
1
1 Yeditepe University
26
Agustos
Yerle:;;kesi
Kay:;;dagl
, istanbul, Turkey
(e-mail:
ahacinliyan@yeditepe.edu.tr
)
2 Gebze Instute of Technology
Kocaeli, Turkey
93
Abstract:
The dynamics of laser models based on the Maxwell Bloch equation
is
studied. Instances of stability and chaotic behavior are investigated. Special
solutions of the system one of which reduces to
th
e Lotka Volterra system are
derived. Absence of oscillating solutions in the reduced systems
is
studied.
Keywords:
Maxwell Bloch, Lotka Volterra, bifurcation.
1
Introduction
The
pure
Bloch
model
is
linear
so
that
chaotic
behavior
is
not
ordinarily
ex-
pected,
although
for
strongly
coupled
AC
components,
trajectories
that
are
dense
in
the
phase
space
can
result
in
seemingly
chaotic
like regions.
When
the
Bloch
system
is
coupled
to
the
Maxwell
equations
in
the
pr
ese
nc
e
of
ma-
terial
media,
chaotic
behavior
as evidenced
by
positiv
e
Lyapunov
exp
onents
can
be
seen.
These
systems
are
freque
ntly
quoted
as laser models.
Transi-
tion
to
chaos is
based
on
considering
the
resonance
between
the
laser
cavity
frequency
and
atomic
cavity
TEM
mod
es.
The
Maxw
ell
Bloch
equations
(to
be
referred
to
as
ME
henceforth)
as
given
by
and
involve
the
coupling
of
the
fundamental
cavity
mode,
E
with
the
collective
variables
P
and
L1,
repre-
senting
th
e
atomic
polarization
and
the
population
inversion.
The
eq
uations
are:
E =
-kE+gP
P =
-'l.P
+ gEL1
Li
=
-'
I
I(L1-
L1o)
-
4gPE
For
k =
(1,
Il.
=
g2/k
= 1,
g2L1o
/ k = r,
III
=
b,
the
system
can
be
transformed
into
the
Lorenz
system
about
the
equilibrium
poin
t
L1
=
L10
by
setting
x =
E,
y =
gP/k,z
=
L10
-
L1.
The
m
ea
ning
of
the
parameters
94 A.
S.
Hacinliyan et al.
The basic operating point given by the MINUIT Analysis showing "Throw
and Catch" chaotic behaviour similiar to LORENZ System.
p
7
.5
5
3
- 5
-
7.5
k=11.7S, g=6.06, gper=2.66, gpar=2.7S, d10=28
Fig.
1.
The
Str
ang
e
Attractor
for
basic
operating
point
with
given
parameters
in
the
original
eq
uations
are
given by Arecchi, while
(T,
r,
b
are
the
Lorenz
parameters
.
A basic
operating
point
given in
Figure
(1) was found
by
minimizing
the
negative
of
the
maximal
Lyapunov
exponent
using
the
CERN
MINUIT
package.
The
dominant
chaotic behavior
exhibited
by
the
syst
em is
the
so-called
Throw
and
Catch
mechanism
, in which two linearized equilibrium
points
with
a
pair
of
complex
conjugate
eigenvalues
with
slightly positive real
parts
surround
an
unstable
third
equilibrium
point
with
eigenvalues
(+
- -
).
Th
e chaotic behavior is
predictably
similar
to
the
LORENZ
System
in
Figure
1.
Rang
es
of
parameters
about
the
operating
point
for which a positive
max-
imal
Lyapunov
exponent
has
been
observed is given in
the
following Table
1.
Parameter
Lower
Upper
Hopf
Point
k 6.75 11.5
g
3
2.86
')'
.1
0 60
')'
11
0 60 5
Table
1.
Parameters
about
the
operating
point
for
which
a
positive
maximal
Lya-
punov
exponent
has
been
observed
Results
of
a numerical
study
of
the
Hopf
bifurcation
point
is also given
in
the
same
Table
1.
Maxwell-Bloch Equations as Predator-Prey System
95
This
range
of
parameters
is
true
for Class C lasers (for infrared). Class
A lasers require
1'.1..
CC:'
I'
ll »
k.
Class B lasers require
1'.1..
»
I'
ll
CC:'
k.
2
Special
solutions
of
the
MB
equations
and
its
relation
to
the
LV
equations
It
is possible
to
seek special solutions
to
the
MB
equations
using
subs
idiary
conditions
that
replace
eithe
r of
the
two nonlinear
terms
P E
and
E.6. by
the
remaining variable which yields a two by two
system
and
an
expon
ent
ial so-
lution
for
the
third
variable. We investigate these solutions
and
their
possible
relation
to
the
Lotka
Volterra (Henceforth referred
to
as LV) equations.
One
of
these
has
been studied.
2.1
Trivial
subsidary
condition
decoupling
Ll
By
the
change
of
variables
PE
=
.:\.6.
The
MB
eq
uations
can
be
reduced
to
the
following
system
Li
= -
I'
ll
(.6.
-
.6.
0
)
-
4gP
E
EE
=
-kE2
+
gP
P =
-l'l..P
+
gPE
2
f.:\
The
change
of
variables I =
E2,
j =
2EE
gives
1 .
"21
=
-k1
+
gP
P =
-l'
l..
P + gP1f.:\
.6.
is
thus
decoupled
to
give:
This
gives
the
following solution for
.6.
:
.6.
=
1'1
1.:\
+
Aexp[-bl
l +
4g.:\)t]
I'll + 4g
The
remaining
system
consists
of
the
two equations given by
the
following
two by two
matrix
system:
96 A.
S.
Hacinliyan et al.
We seek solutions of
the
form:
Here
fJ,
are
the
eigenvalues
of
the
system
given by:
2 92'II
fJ,
- ('II +
k)
fJ,
+
k'il
- 4
.A
= 0
'I
I + 9
To have complex
conjugate
eigenvalues,
that
can
give oscillatory behavior,
the
discriminant
must
be
neg
ative
.
This
is impossible. So
the
approximate
solution for
the
MB
equations
has
been
deriv
ed,
and
shown
to
reach
steady
state
without
oscillation.
This
special solution is
related
to
the
Typ
e
II
Lasers.
2.2
Trivial
subsidiary
condition
decoupling
P
The
MB
equations
can
be
transformed
into
a
system
proposed
by
,
that
resem-
bles
the
Lotka
Volterra
problem
with
I
representing
the
intensity
of
the
laser
field
and
N
denoting
population
density,
the
dimensionless
rate
equations
for
I
and
N
are
obtained
by
assuming I = E2
and
the
subsidiary
condition
gP
=
kE11. Multiplying
the
first
MB
equation
by
E gives
EE
=
-kE2
+
gP
E,
P = -
'.l.P
+ gE11 = - '.l.P +
g2p/k
The
subsidiary
condition
enables
one
to
decouple
and
solve for
the
polarization
as
P =
Po
exp
[-
b.l.P
-
g2
/ k
)t]
and
the
third
equation
becomes
Li
=
-'II
(11
-
11
0
)
- 4kE211. Finally,
the
following
eq
uations
for
the
intensity
an~
population
density
are
obtained
by
defining
11
=
N,
2kt
=
T,
A =
110
and
1=
2I,.
I =
(-1
+
N)I
N=
(A-N
-NIh
If
A = 0
then
these
equations
can
be
reduced
into
the
LV
equ
at
ions. For
negative values
of
" we
get
a
pair
of
purely
imaginary
eigenvalues
nam
ely
±v(hl)/2
The
normal
form
of
these
equations
are
in resonance
and
give
the
limit cycle N I = constant
for,
=
-l.
For values
of
A
=I
0,
there
also is a way in which
the
eq
uations
can
be
shown
to
be
compatible
with
the
LV
system.
Augmenting
the
system
by
the
equation
A = 0,
we
hav
e a 3-dimensional
system
which
can
be
studied
with
the
method
of
invariant
manifolds.
We
construct
th
e
augmented
system
::i::
= x(-
l+a+y)
Maxwell-Bloch Equations as Predator-Prey System
97
iJ
=
,(
-a:r
- y -
xy)
Q,
= °
The
linearized eigenvalues
are
the
original
eigenvalues
and
the
additional
zero
eigenvalue for
this
system,
(-1,
-,,0).
If,
>
0,
this
gives a
stable
invariant
manifold
on
which
the
flow is LV.
If,
< 0,
the
manifold
becomes
unstable
and
we
need
to
study
the
general
case.
3
The
general
case
for
arbitrary
A
I=(-l+N)I
IV=(A-N-NI)[
These
equations
possess
two
equilibrium
points,
I = 0, N = A
or
I =
,(A
- 1), N =
1.
The
first
equilibrium
point
when
linearized
has
eigenvalues
A-I,
-,
so
that
no
resonance
or
oscillation
is
expected.
At
the
second
equilibrium
point:
:...-
I=(-l+A+N)I
IV
=
(-I
-
AN
- N
1)[
An
integer
ratio
between
A-I
and
,
can
indicate
resonances
in
the
normal
form
expansion.
The
latter
point
when
linearized
has
eigenvalues
-A-"y±y!4-4A+A2,
th
h 4
4A
A2
h·
. . f
2 so
at
w
en
- +
,c
anges
SIgn, a transItIOn 0
domain
indicating
a
Hopf
bifurcation
can
occur.
The
normal
form
expansion
about
this
equilibrium
point
is
not
expected
to
yield a
resonance
condition.
4
Possibility
of
oscillatory
solutions
It
has
been
shown
that
oscillatory
solutions
for
the
special
solutions
of
the
MB
equations
using
subsidary
conditions
require
unphysical
parameter
val-
ues.
The
possibility for
oscillatory
solutions
can
be
investigated,
assuming
dominance
of
the
nonlinear
terms,
and
dropping
the
exponential
decay
caus-
ing
term
in
the
E
equations.
In
effect
this
is a
small
k, g
dominated
regime.
jj; =
-kE
+
g2
EiJ.
Li
= '1liJ.
o
-
4gPE
jj; =
-kE
+ l EiJ.
. . 2
iJ.
= '1liJ.
o
-
4(EE
+
kE
)