Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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98 A.
S.
Hacinliyan et
01.
E=gP
P = gE11
Li
=
-4
gP
E
£ = g2E11
Li
=
-4E
E
11
=
-2E2
+ C
E =
lE(-2E2
+
11
0
)
E
2
/2
= g2(E
4
/2
+
11
oE2/
2)
+
cat
E =
EaE
= 0
EE
=
)(g2)E2(110
- E2)
5
Periodic
Analysis
of
Maxwell
Bloch
Model
When
we
separate
expon
e
ntially
decaying
part,
we have
steady
stat
e so
lution
from
If
E = p
th
en
If
C2 = 0
then
E=gP
P = gE11
Li
= -
4gPE
P = g- l£
Li
=
-4gEg
-
1
E
Li
+4EE
= 0
11
+ 2E2 =
C1
.
2··
P=gE(Cl-2
E ) =
E/g
£ =
lE(
Cl
- 2E2)
p(dp/dE)
= l
EC1
-
2g2
E3
p2/2
=
g2
C1
E
2
/2
-
g2
E4 /2 +
C2
/
2
E = gE(Cl - E2)1
/2
E = 4
Fr
exp(
Frgt
+
Frc2)/(1
+ 4
exp(2Fr(gt
+
C2))
If
C2 = 0 t
hen
the
equation
is
elliptic
and
app
roximat
ely periodic.
Maxwell-Bloch Equations as Predator-Prey System 99
The
equilibrium
points
are
E = 0
and
E = )
cl/2.
For
E = 0 we
have
runaway
solution.
..
2
E = 9 C1
E
E = A exp(gylClt)
For
large E E = _2g2 E3 for E =
)cl/2
+
~
Near
the
other
equilibrium
point
after
qualitative
analysis, we see
that
the
system
has
periodic
solutions
with
2g)
cl/2
6
Impossibility
of
trivial
resonance
k =
-b.L
+
'II
+
1'1
+
21'2)
k = - ('.L +
'II)
is
the
requirement
for resonance.
The
eigenvalues
are
0:
±
pi,
TIf
-0: =
2,
then
characteristic
polynomial
is
(0:
+
pi
- ,)(0: +
pi
- ,)(
-2,
-
A)
Since
the
sum
of
the
roots
is 0, we
have
-k
- '.L -
'II
= 0 which implies
that
one
parameter
is
unphysical.
7
Conclusion
In
conclusion,
the
Maxwell-Bloch
system
that
is
usually
used
as a
model
for
the
Helium-Neon
L&<;er
has
been
studied.
This
system
has
a rich
structure
as a
function
of
its
parameters.
We
have
shown
that
this
system
exhibits
several
types
of
instability.
The
Lyapunov
spectrum
is
compatible
with
the
eigenvalues
of
the
linearized
system,
which
consist
of
a
negative
value
plus
a
pair
of
complex
conjugate
eigenvalues
with
slightly
negative
real
part.
These
parameters
correspond
to
far
infrared
lasers
where
Lorenz
like
chaos
has
been
observed.
The
bifurcation
mechanism
that
characterizes
the
possible
transition
to
chaos
from
this
operating
point
is
studied
by
the
MATCONT
package
and
Hopf
bifurcation
is identified
in
several
instances.
Emphasis
has
been
placed
on
one
operating
point
where
chaotic
nature
as
evidenced
by
a
positive
maximal
Lyapunov
exponent
is
shown
while
the
change
of
the
varying
parameters
are
plotted
for
each
item.
However,
this
behavior
is sensitive
to
variations
in
the
parameters
and
rapidly
changes
to
steady
state.
As
each
parameter
is varied,
transition
to/from
chaotic
behavior
can
be
seen. We
have identified several
instances
of
Hopf
bifurcation
at
this
operating
point,
the
operating
point
is
within
limits
of
the
parameter
range
for
which
chaotic
behavior
has
been
studied
in
the
He-N e laser.
100 A.
S.
Hacinliyan et al.
References
•
A.
Hacinliyan, N.
Z.
Perdah<;l, G.
~ahin
,
H. A.
Yddmm.
Transforming
to Chaos
by
Normal Forms. CMDS 10, Shoresh, Israel, 15-20, 2003.
•
F.
T. Arecchi. Chaos and Generalized Multistability
in
Quantum
Optics.
In A.F.
Round,
editor, Physica Scripta, 85-92, 1985.
• H.Haken. Light, Volume
2,
1985.
• A. Wolf, J. B. Swift,
H.
L. Swinney
and
J. A. Vastano. Routes to chaotic
emission in a
cw
He- Ne Laser. Physics Letters A, Vol. 28,2, 1983.
• P. Grassberger, editor. Non-linear time sequence Analysis. Int.
J.
Bifurcation
and
Chaos, pages 521, 1991.
•
J.
P.
Eckmann,
S.
O.
Kamphorst,
D.
Ru
elle. Ergodic
Theory
of
Chaos
and
Strange
Attractors.
Reviews
of
Modern Physics, Vol. 57, pages
617-656, 1985.
• H.
Kantz,
editor. A robust method to estimate the maximal Lyapunov
exponent
of
a time series. Physics
Lett
ers
A,
Vol. 185, pages 77-87,
1994.
•
J.
Chrostowski. Noisy Bifurcation in acousto-optic bistability. Physics
Letters
A,
26, 1982.
•
Thomas
Erneux,Kozyreff G, editor. Nearly Vertical Hopf Bifur-cation for
a Passively
Q-Switched Microchip Laser. Vol.
101
, 2000.
•
Erneux,
T.,
Kozyreff, G. Nearly Vertical
Hopf
Bifurcation for a Passively
Q-Switched Microchip Laser
Journal
of
Statistical Physics,
Vol.
101,
Nos.
1/2,
2000.
• Krise, S., Choudhury,
S.
R.
Bifurcations
and
chaos in a
predator-pr
ey
mod
el
with
delay
and
a laser-diode
system
with
self-sustained pulsations
Chaos, Sol'itons and Fractals 16 59-77, 2003.
• Clarici, G., Griese, E. Analytical Laser Model for
Optical
Signal In-
teg
rity
Analysis
Int
ernational
Students
and Young Scientists Workshop
"Photonics and Microsystems", 2005.
• Siegman,
A.
E. Lasers Univer-sity Science Books
ISBN
0-9357022-11-5,
1986.
•
A.
Wolf,
J.
B. Swift,
H.
L. Swinney
and
J.
A.
Vastano. Deter-mining
Lyapunov exponents from a time
ser-ies.
Physica D,
pag
es 285- 317, 1985.
Identifying
Chaotic
and
Quasiperiodic
Time-Series
Candidates
for
101
Efficient
Nonlinear
Projective
Noise
Reduction
Nada
Jevtic
1
,2
and
Jeffrey
S.
Schweitzer2
1
Physics
and
Engineering
Technology,
Bloomsburg
University
400
East
Second
St,
Bloomsburg,
PA
17815,
USA
(e-mail:
nj
evtic©bloomu.
edu)
2
Physi
cs
Department
U-3046, U
niver
s
it
y
of
Connecticut
Storrs,
CT
06269,
USA
(e-mail:
schwei
tz©phys.
uconn.
edu)
Abstract:
One
of
t.he
principal
problem
s
in
identifying
chaos
using
nonlinear
time-
series
analysis
of
real-world
data
is
additive
noise. A
straightforward
procedure
in
traditional
phase-space
is
introduced
that
can
be
used
to
identify
data
sets
amenable
to
nonlin
ear
projective
noise
reduction.
This
methodology
can
be
used
both
as
a
tool
for
identifying
candidates
for noise
reduction
and,
with
some
adapt.a
tion,
for
zeroth
order
noise
quantifi
cat
ion.
Results
for
simulation
data
are
compared
with
that
for a
quasiperiodic
measured
time
series
to
illustrate
when
series
can
benefit
most
from
nonlinear
projective
noise r
eduction.
Keywords:
nonlinear
time-series
analysis,
Poincare
section,
Poincare
sectio
n re-
turn
time,
lo
ca
l
projective
nonlinear
noi
se
reduction,
PG
1351+489.
1
Introd
uction
During
the
nineteen
eighties
the
study
of
chaotic
deterministic
systems
re-
s
ulted
in
the
first nonlin
ear
tools for
the
analysis of
time
series
generated
by
these
systems.
It
, however,
soon
became
clear
that
chaotic
systems
were
very
hard
to
find in
the
real world. Since
the
th
eoret
ical basis for
these
methods
was initially limited
to
dynamics
that
can
be
described by a sys-
te
m
of
autonomous
differential
equations
(Takens
[5
]),
ent
husiasm
flagged.
Since
then,
the
validity
of
the
embedding
theorems
has
been
extended
to
deterministically
driven
(Stark
[4])
and
stochastically
driven
systems
(Stark
et
al.[I]). Moreover, now
the
computing
power needed
to
do
th
ese
ana
lyses
is available
on
our
desktops.
One
of
the
principal
problems
in identifying
chaos using nonlinear time-series analysis
of
real-world
data
is
additive
noise.
Additive
noise
has
also limited
our
capab
ility
to
study
both
nonlin
ear
and
linear systems. We
introdu
ce a
straightforward
procedure
in
traditional
re-
constructed
phase-space
that
can
be
used
to
identify
data
amenable
to
non-
linear local
projective
noise reduction.
Simulated
solar
data
are
compared
with
a quasi-periodic
measured
time
series
of
a variable
star
light curve in
classical
reconstructed
time-delay
ph
ase
space
to
illu
strate
when
series
can
benefit
most
from
nonlinear
projective noise
reduc
tion
. We also discuss how
this
methodology
might
be used for zero
th
order
noise quantification.
102
IV levtic and
1.
S.
Schweitzer
2
Time-Delay
Reconstructed
Phase
Space
Of
the
tools developed for time-series
data
analysis in time-delay
reC011-
:,;tructed phase space of interest
to
this discussion
are
Poincare sections
and
locally projective nonlinear noise reduction.
2.1
Poincare
sections
Flow
data
such as in
the
example
of
variable white
dwarf
PG
1351
+489
which
will be used here
are
characterized by
the
fact
that
errors in
the
direction
tangent
to
the
t.rajectory
neither
shrink
nor
increase exponentially.
This
is a
so called marginally
stable
direction
and
thus
possesses one Lyapunov expo-
nent
which
is
zero. Any
perturbation
in
this
direction
can
be
compensated
by
a simple shift
of
the
time.
In
many
data. analysis ta.sks
this
direction is of
marginal interest, a.nd
it
can
be
eliminated.
The
theoretical concept
to
do so
is called
the
Poincare
section Thus, a Poincare section consists of points
in
phase space generated by
the
sequence
of
intersections of a
trajectory
with
a
generalized surface.
In
three
dimensions,
the
generalized surface
is
a plane.
Points
A
and
B in Fig. 1
at
which
the
trajectory
intersects
this
plane in
the
same
direction
constitute
the
Poincare Section.
This
is
the
'space' aspect
of
11
Poincare section.
It
is importa.nt
to
note
here
time
is
an
irnplicit variable.
Fig.
1.
A
phase-space
trajectory
intersects
the
Poincare
plane
at
points
A
and
B.
In
order
to
recoup time information,
time
intervals beb,veen successive
crossings
through
the
Poincare surface (between
points
A
and
B)
may
be
used.
These
times meaningful observables, too.
They
will be referred
to
as
"Poincare
section
return
times" or
PSRT.
The
inverse of a Poincare sec-
tion
return
time (PSHT) will be introduced as
the
Poincare section
return
frequency
or
PSRF.
This
is
the
'time'
aspect
of
a Poincare section.
2.2
Locally
projective
nonlinear
nois~~
reduction
It
is assumed
that
the
measured
data
are
the
output
of a low-dimensional
dynamical
sy:,;tem
and
random
high-dimensional noise spreads
the
data
off
Identifying Chaotic and Quasiperiodic Time-Series Candidates
103
Fig.
2.
In
locally
projective
nonlinear
noise
reduc-
tion,
if
the
pha.5e-space
trajectory
lies
on
a surface,
points
that
have
been
forced off
the
surface
by
noise
are
orthogonally
projected
back
onto
the
surface.
the
manifold
that
the
phase-space
trajectory
lies on. FbI' small
amplitude
noise,
the
data
points are
distributed
closely
around
the
manifold (Fig. 2).
The
goal in local projective nonlinear noise reduction is
to
identify
the
manifold
and
orthogonally
project
the
data
points
onto
it.
In
this procedure
we assume
that
if
the
dynamical
system
with
noise forms a q-dimensional
manifold containing
the
tmjectory,
and
the
delay vectors lie inside
another
lower m-dilnensional manifold
with
q > m. As a result overembedding
is
used in
an
iterative process
t.o
obtain
the
highest degree of noise reduction.
3
Analysis
of
Measured
Data
0.15
~,1(l
+-c---r-.,......,.~.,.~.--~
.
.....,.......-,~
!J
5001000
1500 2000 2500
;>Q!lC
3WQ
400U
.tSQO
soon
t(sj
Fig.
3.
PG
1351+489
relative
light
in-
tensity
curve
of
10 s
binned
data.
Fig.
4.
Power
spectrum
o[PG
1351
+489
of
30 s
integratpd
data.
The
measured
data
is
the
light curve of DB variable white
dwarf
PG
1351
+489
that
was observed
by
the
vVhole
Earth
Telescope. A section of
the
light. intensity curve
is
shown in Fig. 3.
The
light curve
data
are available
a.t
two
integration
t.imes:
10
sand
:
:\0
s.
The
lOs
binned light curve is over
40
periods long.
The
PG
1351
+489
power
spectrum
consists
of
the
fundament.al frequency
fo
= 2.041 mHz, (period of 489 s), its harmonics, subharmonies
and
other
nonlinearly generated lines (Fig. 4)
]04
N.
Jevtic and J.
S.
Sch1,>'eitzer
3.1
Phase-space
analysis
of
PG
1351+489
light
curve
The
:3-D
time-delay phase space
portrait
of
the
10 H
integrated
data
is shown
in
Fig
. 5.
Fig.
5.
;j-D time-delay
phase
space port.rait
of
the
PG
1:551+4S9 lOs
integrated
data.
As
part
of
a comprehensive nonlinear analysis
of
PG
1351+489
data
Poincare
sections were
obtained
at
the
average
of
the
data
for two direct.ions.
The
Poinca.re section 'space'
aspects
in
the
two directions
are
shown in Fig. 6.
The
effect
of
noise is evident
in
the
form
of
'upward'
point.s
appearing
in
the
'downward'
bundle
of
trajectories
and
vice versa.
-1:!lD·
«I)
c·
IJpw.ind
...
d
('<W
fil,!'l,l
"ff
d
,
HUl
Fig.
6. 'Space'
Poincare
section
of
the
PC
1:351+489 lOs
binned
data
at
average
of
data.
The
Poincare
Section
Return
Times (PSRT)
are
shown
in
Fig. 7 (left)
311dFig. 7 (right) for
the
30
sand
10 s
binned
data,
respectively.
nIH
period
points
at
48D
s
and
some noise
at
short
P
SRT
are
present
in
Fig. 7 (left).
In
Fig. 7 (right) however, in a.ddition to
the
full
period
and,
at
short
PSRT,
noise,
there
is
an
excess
of
points
at
about
half
the
fundamental
time.
In
theF'ourier
power
spectrum,
the
fundamental
and
the
first
harmonic
are
in
a,
10:1 ratio. Here
the
ratio
is
about
1:1.
ldenti/ving Chaotic and Quasiperiodic Time-Series Candidates J
05
b.
~-'-'-'T-"'--""'---''''''''''''-I
II
:;rl
.0(;'
CI
;n
=
Cifl)"urt/>t$rlb<:t
Fig
. 7.
Poincare
Section
Return
Times
(PSHT)of
the
PG
1:351+489
at
average of
the
data:
left -
:~o
s
binned
data
and
right
- 10 s
binned
data.
4
Analysis
of
Model
Data:
Flux
Transport
Solar
Dynamo
with
Meridional
Circulation
At
the
time
·work was progressing
on
the
analysis
of
model
data
f()l'
the
mag-
netic
field energy
of
a flux
transport
solar
dynamo
with
meridional circulation
generated
by
Iv1.
Dikpati
at
the
HAO,
NCAR
where stochastiC'fiuctuatiolls
were
added
to
the
meridional velocity.
Fig.
8.
Flux
transport
dynamo
model
with
stochastic
fluctuations
added
to
the
merid-
ional
circulation
velocity:
Phase
space
por-
trait
(left)
and
PSRT
(right).
The
phase
-space
portrait
for
the
model
data
for
the
magnetic
fieJd
energy
of
a.
flux-transport
solar
dynamo
with
stochastic
fluctuations
added
to
the
meridional circulation velocity
is
shown in Fig. 8 (left).
The
Poincare
section
return
times
are
shown in Fig. 8 (right).
These
exhibit
the
sa.me kind
of
increase in
the
number
of
PSRT
at
the
times corresponding
to
about
the
first
harmonic as
the
data
for
PC
13.51+489
with
the
advantage
that
it
is clear in
Fig. 8 (right)
that
the
increase
of
the
first ha.rmonic
times
is
at
the
expense
of
the
fundamental
period
points.
The
sum
of
t.wo
adjacent
ha.lf-times were
shown
to
be
equal
to
the
fundamental
PSRT.
5
Comparison
of
Model
and
Measured
Data
Insights gained for
the
dynamo
model
w(;)re
then
applied
to
the
PG
1351
+489
data.
There,
as well, it was shown
that
adjacent
fractional
PSRT
times
or
the
'first
harmonic
times'
add
to
the
fundamental
period. Fig. 8.h.
In
an
attempt
to
better
understand
the
observations, a series
of
Poincare
pla.nes were positioned along
the
phase-space portra.it
in
one specific direction
] 06
N.
levtic
and
1.
S.
Schweitzer
a)
b)
i
ll
_
-·
1·
§ & •
••
..
..
..
y"
"·
..
f&.···
_if
, ' ,
, , ,
~-
. :
~
~
j
" "
i~
j
:'"
i i
~
·1
,;
:;
-,--.
-···
r
--···
··. --
....
--,-·-·
r-···-'
O,2()401I080
1
00
Inters
ection
number
Intersection
number
Fig.
9.
Comparison
of
the
PSRT
for
the
sol
ar
dynamo
with
resetting
(left)
and
PG
1351+489 (10 s
binned
data)(right).
and
histograms were (Fi?;. 10)
obtained
for
the
PSRT
at
each position.
Fig.
10
.
Histograms
of
the
PSRT
of
PG
1351+489
PSRT
as
the
Poincare
plane
sweeps
the
portrait.
At
the
average of
the
data
the
number
of full period
PSRTs
was smaller
(Fig. 10)
than
the
data
requires
and
there
were
too
many
almo
st
equa
l-
time
PSRTs
of 200-300 s. As
the
plane swept
the
representation
towards
the
apex, two
return-time
groups differ
entiate
whose
PSRT
time
s
add
to
the
fundamental period.
These
diverge, one
group
of
PSHT
times becomes
shorter
and
the
other
becomes longer.
They
revert
to
two groups of times,
one corresponding
to
the
fundamental period a
nd
the
other
to
noise
at
the
very
apex
of
the
figure.
6
The
Effect
of
"Resetting"
on
the
PSRT
The
observed
PSHT
diagrams
and
histograms are
best
explained in
the
fol-
lowing
mann
er.
If
a
pha
se-space
point
in Fig.
11
(left)
at
C spikes off
the
traject
or
y
due
to
noise
and
then
falls back
through
the
plane towards D,
it
returns
to
the
trajectory
at
A.
Two Poincare
return
times result or
time
resetting occurs
and
as a result
the
process has been
dubbed
resetting. For
positions of
the
Poincare plane
near
the
average of
the
data,
thes
e
return
times
are
about
half
the
fundam
ental
period.
Identifi;ing Chaotic and Quasiperiodic Time-Series Candidates
107
Fig.
11.
The
Effect,
of
"R.esetting"
on
the
PSHT.
Poincare
plane
at.
the
average
of
the
data
(left.)
and
Poincare
plane
towards
the
apex
(right).
As
the
Poincare
plane
is
taken
toward
the
apex
of
the
phase
space
portrait
(Fig.
10
(right)),
the
difference
in
the
two times grows:
one
time
grows longer
(B
to
C) while
the
other
grows
shorter
(A
to
B).
The
single
branch
centered
around
the
first
harmonic
at
the
average of
the
data,
as
the
plane
is moved,
spreads
out
to
become
two
branches.
These
diverge
towards
lower
and
higher frequencies.
7
The
Three-Dimensional
PSR
-
Frequel1cy
Spectrum
The
previous discussion justifies
the
introduction
of
the
Three-Dimensional
PSR-F'reqllency
Spectrum
shown
in Fig. 12.
This
spectrum
can
be
obtained
using rela.tively simple softwa.re
and
integrates
the
above
procedure. A 3-D
spect.rulll
obtained
in
the
following
manner:
Fig.
12
.
;3-D
Poincare
Section
Return
Frequency Spw;trum.
a. A series of
Poincare
planes
is
taken
along a selected
direction
and
PSRT
obtained
on
a fine
grid
of
positions.
b,
In
order
to
enhance
the
visibility
of
the
noise-induced branches,
a.
number
of
these
histograms
are
added
(binned).
c.
R.eturn times
are
converted
into 'frequencies'
at
each
position.