Table Of ContentPatterns of Chaos Syn
hronization
1 2 2 1
Johannes Kestler, Evi Kopelowitz, Ido Kanter, and Wolfgang Kinzel
1
Institute for Theoreti
al Physi
s, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
2
Department of Physi
s, Bar-Ilan University, Ramat-Gan, 52900 Israel
(Dated: De
ember 2007℄)
Small networks of
haoti
units whi
h are
oupled by their time-delayed variables, are investi-
gated. In spite of the time delay, the units
an syn
hronize iso
hronally, i.e. without time shift.
Moreover, networks
an not only syn
hronize
ompletely, but
an also split into di(cid:27)erent syn
hro-
nized sublatti
es. These syn
hronization patterns are stable attra
tors of the network dynami
s.
Di(cid:27)erent networks with their asso
iated behaviors and syn
hronization patterns are presented. In
parti
ular, we investigatesublatti
e syn
hronization, symmetrybreaking, spreading
haoti
motifs,
8 syn
hronizationbyrestoringsymmetryand
ooperativepairwisesyn
hronizationofabipartitetree.
0
0
2 I. INTRODUCTION
n
a
Chaos syn
hronization is a
ounter-intuitive phe-
J
nomenon. On one hand, a
haoti
system is unpre- FIG. 1: Two mutually
oupledunits.
3
di
table. Two
haoti
systems,startingfromalmostiden-
] ti
al initial states, end in
ompletely di(cid:27)erent traje
to-
D ries. Ontheotherhand,twoidenti
al
haoti
unitswhi
h theunitsinea
hsublatti
e
ansyn
hronizetoa
ommon
C are
oupled to ea
h other
an syn
hronize to a
ommon
haoti
traje
toryalthoughtheyarenotdire
tly
oupled.
.
haoti
traje
tory. The system is still
haoti
, but af- The
oupling of one sublatti
e is relayed by the
haoti
n
ter a transient the two
haoti
traje
tories are lo
ked to traje
tory of a di(cid:27)erent sublatti
e. The traje
tories of
i
l ea
hother[1,2℄. Thisphenomenonhasattra
tedalotof di(cid:27)erent sublatti
es are only weakly
orrelated, but not
n
resear
ha
tivities, partly be
ause
haos syn
hronization syn
hronized [21℄.
[
has the potential to be applied for novel se
ure
ommu- In this paper we want to investigate patterns of
haos
1 ni
ation systems [3, 4℄. In fa
t, syn
hronization and bit syn
hronization for several latti
es with uni- and bi-
v
ex
hange with
haoti
semi
ondu
tor lasers has re
ently dire
tional
ouplings with time delay. There exists a
8
been demonstratedovera distan
eof120km in apubli
mathemati
altheoryto
lassifypossiblesolutionsofnon-
8
4 (cid:28)ber-opti
ommuni
ation network [5℄. In this
ase, the linear di(cid:27)erential or di(cid:27)eren
e equations for a given lat-
0
oupling between the
haoti
lasers was uni-dire
tional, ti
e [22℄. However, this theory does not determine the
. the sender was driving the re
eiver. For bi-dire
tional stabilityofthesesolutions. Butinordertodes
ribephys-
1
ouplings, when two
haoti
units are intera
ting, addi- i
al or biologi
al dynami
networks, we are interested in
0
8 tional interesting appli
ations have been suggested. In stable patterns of
haoti
networks. The patterns whi
h
0 this
ase, a se
ure
ommuni
ation over a publi
hannel aredis
ussed in this paperareattra
torsin phase spa
e,
: may be established. Although the algorithm as well as any perturbation of the system will relax to these pat-
v
alltheparametersarepubli
,itisdi(cid:30)
ultforanatta
ker terns whi
h move
haoti
ally on some high dimensional
i
X to de
ipher the se
ret message [6, 7, 8℄. syn
hronizationmanifold.
r Typi
ally, the
oupling between
haoti
units has a Ourresultsaredemonstratedforiteratedmaps,forthe
a
time delay due to the transmissionof the ex
hanged sig- sake of simpli
ity and sin
e we
an
al
ulate the stabil-
nal. Nevertheless,
haoti
units
an syn
hronize without ity of these networks analyti
ally. But we found these
timeshift,iso
hroni
ally,althoughthedelaytimemaybe patterns for other systems, as well, for example for the
extremelylong
omparedtothetimes
alesofthe
haoti
Lang-Kobayashirateequationsdes
ribingsemi
ondu
tor
units. This (cid:21) again
ounter-intuitive (cid:21) phenomenon has lasers. Hen
e we think that our results are generi
.
re
ently been demonstrated with
haoti
semi
ondu
tor
lasers[9, 10, 11, 12℄, and it is dis
ussed in the
ontextof
orresponding measurements on
orrelated neural a
tiv- II. TWO INTERACTING UNITS
ity [13, 14, 15, 16℄.
Several
haoti
unitsmaybe
oupledtoanetworkwith We start with the simplest network: Two units with
delayed intera
tions. Su
h a network
an syn
hronize delayed
ouplings and delayed self-feedba
k, as sket
hed
ompletely to a single
haoti
traje
tory, or it may end in Fig. 1.
inastateofseveral
lusters,dependingonthetopologyof For iterated maps, this network is des
ribed by the
thenetworkorthedistributionofdelaytimes[17,18,19, following equations:
20℄. Re
entlyanotherphenomenonhasbeenreportedfor at =(1−ε)f(at−1)+εκf(at−τ)+ε(1−κ)f(bt−τ)
haoti
networks: Sublatti
e syn
hronization. If a small
bt =(1−ε)f(bt−1)+εκf(bt−τ)+ε(1−κ)f(at−τ) (1)
network
an be de
omposed into two sublatti
es, then
2
1
further identi
al units re
eiving the same drive would
relax to the same traje
tory. Thus Eq. (5) de(cid:28)nes the
0.8
region where identi
al units whi
h get the same input
syn
hronize to a
ommon traje
tory.
I
Now let us rewrite Eq. (1):
0.6
κ at =(1−ε)f(at−1)+ε(2κ−1)f(at−τ)+
0.4 II +ε(1−κ)f(at−τ)+ε(1−κ)f(bt−τ),
bt =(1−ε)f(bt−1)+ε(2κ−1)f(bt−τ)+ (6)
0.2 +ε(1−κ)f(bt−τ)+ε(1−κ)f(at−τ).
III
Both systems are driven by the identi
al signal
0
0 0.2 0.4 ε 0.6 0.8 1
st−τ =ε(1−κ) f(at−τ)+f(bt−τ) .
(7)
(cid:2) (cid:3)
α=3/2
FIG. 2: Phase diagram for (analyti
al result).
Comparing (4) with (6), one (cid:28)nds that for
κ˜ =2κ−1
f(x) (8)
where is some
haoti
map, for example the
Bernoulli shift,
theintera
tingunits (6)aredes
ribed bythe drivenunit
f(x)=αx mod 1
(4). The phase boundary of the driven system, region II
(2)
+III[Eq.(5)℄,andthephaseboundaryoftheintera
ting
α > 1
with . In this
ase, the system is
haoti
for all system, region I + II [Eq. (3)℄, are
onne
ted with ea
h
0 < ε < 1 0 < κ < 1 ε
parameters and . measures the other: WiththemappingofEq.(8), onephaseboundary
κ
total strength of the delay terms and the strength of τ
a→nb∞eobtainedfrom theothτer. Thisisnotonlytrue for
the self-feedba
k relative to the delayed
oupling. but for any value of .
a =b
t t
Obviously, the syn
hronized
haoti
traje
tory Additionally, this mapping does not only hold for the
τ
is a solution of Eq. (1). Its stability is determined by Bernoullishift but forany
haoti
system,providedthat
onditional Lyapunov exponents whi
h des
ribe the sta- the signal does not
hange the Lyapunov exponents of
bility ofperturbations perpendi
ular to the syn
hroniza- the driven system. Simulations with other maps, e.g.
tion manifold. For the Bernoulli map, these Lyapunov with the Tent map or the Logisti
map,
on(cid:28)rm the ap-
exponentshavebeen
al
ulatedanalyti
ally[21, 23℄, and pli
ability of the mapping (8). Sin
e the slopes of these
τ → ∞
for in(cid:28)nitely long delay, , one obtains the phase maps are not
onstant, the Lyapunov exponents depend
diagram of Fig. 2. on the traje
tories;in orderto gettraje
torieswhi
h are
In region I and II, i.e. for
omparable to the traje
tories of two mutually
oupled
α−1 2αε+1−α units,thedrivingsignalinthedriver/re
eiversetupmust
<κ< , ct
2αε 2αε (3)
ome from an identi
al unit (whi
h has an in
reased
self-
oupling due to the la
k of external
ouplings), i.e.
tjeh
etotwryoautn=itsbat.reAslytnh
ohurgohnitzheedttwooanuniditesnatri
ea
lo
uhpaloetdi
wtritah- st−τ = ε(1−κ˜)f(ct−τ) with (9)
τ ct = (1−ε)f(ct−1)+εf(ct−τ).
a long delay , they are
ompletely syn
hronized with- (10)
τ →∞
out any time shκif=t. 1For , this region is symmetri
about the line 2. EvenfortheLang-Kobayashiequations[24,25℄,whi
h
Complete syn
hronization
an be understsood by
on- des
ribe
oupled semi
ondu
tor lasers, we found a good
t
sidering a single unit driven by some signal : agreement for the mapping (8), see se
tion VII.
at =(1−ε)f(at−1)+εκ˜f(at−τ)+st−τ. Now let us
ome ba
k to the iterated equations (1).
(4)
Let us assume that we re
ord the syn
hronized traje
-
a = b
t t
tory of two intera
ting
haoti
units, regions I
Ifthesystemisnot
haoti
,i.e.ifitsLyapunovexponents
a
t and II of Fig. 2. Now we insert the re
orded traje
tory
are negative, then the traje
tory srelaxes to a unique bt at
t into Eq. (1). How will respond to this drive? We
traje
tory determined by the drive . For the Bernoulli
(cid:28)nd that in region II the unit A will syn
hronize
om-
shift, this region is de(cid:28)ned by the inequality bt
pletely to the re
orded traje
tory , whereas in region I
1+αε−α 1+αε−α
− < κ˜ < theunitAdoesnotsyn
hronize. Althoughthetwointer-
(cid:18) αε (cid:19) αε (5)
a
tingunitsAandBdosyn
hronize,theunitAdoesnot
followthere
ordedtraje
toryinregionI.Thisshowsthat
and indi
ated by II + III in Fig. 2. Sin
e the traje
tory bi-dire
tionalintera
tionisdi(cid:27)erentfromuni-dire
tional
whi
h A relaxes to is uniquely determined by the drive, drive.
3
FIG. 3: Ring of 4 units. FIG.4: Sublatti
esyn
hronizationinatriangularlatti
ewith
periodi
boundaries. The double lines signify bi-dire
tional
ouplings. The self-feedba
k is not drawn to simplify the il-
lustration.
III. SUBLATTICE SYNCHRONIZATION
The response of a single
haoti
unit to an external
fromrandominitialstatesitrelaxestothestateofsublat-
drive, Fig. 2, determines also the phase diagram of a
ti
e syn
hronization(regionIII of Fig. 2forthe ringof 4
ring of four
haoti
units. Additionally, it shows a new
units). The
haoti
traje
toriesofsublatti
esyn
hroniza-
phenomenon: sublatti
e syn
hronization [21℄. Consider AB
tion may be depi
ted as BA. Note that this stru
ture
the ring of four identi
al units of Fig. 3. The dynami
s
does not breakthe symmetryof the ring: The statisti
al
of unit A is des
ribed by
properties of the
haoti
traje
tory of A is identi
al to
those of B.
at =(1−ε)f(at−1)+εκf(at−τ)+
However, there are other solutions of the dynami
1 1
+ε(1−κ) f(bt−τ)+ f(dt−τ) ; equations, as well. These solutions are
lassi(cid:28)ed a
ord-
(cid:20)2 2 (cid:21) (11) ing to the theory of Golubitsky et al [22℄. For example,
AA
for the ring, the state B B is a solution as well. Su
h
a state breaks the symmetry of the latti
e. But we (cid:28)nd
thedynami
sofB,Cand1Darede(cid:28)nedanalogously. Note
theadditionalweighting 2 fortheexternal
oupling(to2 thatthisstateisunstable. Anytinyperturbationrelaxes
A B AA AB
neighbors)whi
h
ausesthetotalstrengthoftheexternal to the states BA in regionIII, AA in regionII and DC
ε(1−κ)
outside of II and III. In fa
t, we have never found a sta-
oupling to remain .
blestatewhi
hbreaksthesymmetryofthelatti
e. Only
Obviously, the two units A and C re
eive identi
al in-
when the
ouplings in the two dire
tions of the ring are
put from the units B and D. Consequently, they will re-
AA AB
spond with an identi
al traje
tory in the regions II and di(cid:27)erent,thestates B B and AB arestableinsome(dif-
ferent) parts of the parameter spa
e, see Fig. 5. Hen
e
III of Fig. 2. The sameargumentholdsforthe twounits
wepostulatethatspontaneoussymmetrybreakingisnot
B and D, whi
h both get the same input from A and
possible for (cid:28)nite latti
es of
haoti
units.
C. That leads to sublatti
e syn
hronization in region III
of Fig. 2: A and C have an identi
al
haoti
traje
tory
and B and D have a di(cid:27)erent one. Although there is a
delay of arbitrary long time of the transmitted signal, IV. SPREADING CHAOTIC MOTIFS
syn
hronizationis
omplete,withoutanytimeshift. The
syn
hronization of A and C is mediated by the
haoti
The response of a
haoti
unit to an external drive,
traje
tory of B and D and vi
e versa. But the two tra- Fig. 2, points to another interesting phenomenon. Con-
je
tories have only weak
orrelations, they are not syn- sider a triangle of
haoti
units with bi-dire
tional
ou-
hronized. Numeri
alτsimulationsoftheBernoullisystem plings as sket
hed in Fig. 6(a). Choose the parameters
with small values of have shown that there is no gen- su
h that the triangle is
ompletely disordered, but ea
h
eralized syn
hronizationeither [21, 26, 27℄. unit has negative Lyapunov exponents when it is sepa-
Sublatti
e syn
hronization has been found for rings rated from the two others. (Both
onditions are ful(cid:28)lled
with an even number of units, for
hains and also for in region III of Fig. 6(b), whi
h shows analyti
al results
other latti
es whi
h
an be de
omposed into identi
al fortheBernoullisystem.) Whenwere
ordthethreetime
a b c
t t t
sublatti
es [21℄. Forexample,thelatti
eof Fig. 4
anbe series , and we (cid:28)nd three di(cid:27)erent weakly
orre-
de
omposed into three sublatti
es. For some parameters lated
haoti
traje
tories. Nowwefeed the twotraje
to-
b c
t t
oftheBernoullisystemwe(cid:28)ndsublatti
esyn
hronization ries and into an in(cid:28)nitely large latti
e of identi
al
with three
haoti
traje
tories. Again, the syn
hronized unitswithuni-dire
tional
ouplingsasshowninFig.6(a).
units are not dire
tly
oupled, but they are indire
tly Ea
hunitre
eivestwoinputsignalsfromtwootherunits.
onne
ted via the traje
tories of the other sublatti
es. But sin
e all Lyapunov exponents are negative, the sys-
a b
t t
The sublatti
e traje
tories, des
ribed before, are sta- tem responds with the three
haoti
traje
tories ,
c
t
ble, i.e. the
onditional Lyapunov exponents whi
h de- and . Although the units of the initial triangle are
s
ribe perturbations perpendi
ular to the syn
hroniza- not syn
hronized, their pattern of
haos is transmitted
tion manifoldarenegative. Even when thesystem starts to the in(cid:28)nite latti
e. All units of the same sublatti
e
4
(a) Twodi(cid:27)erent
oupling
κ1 κ2
parameters and .
1
0.8 (a) Syn
hronization pattern (Sublatti
e
AA − syn
hronization) ofregionIII.
BB 1
0.6
κ
2 − 0.8
0.4
AA
AA 0.6 I
AB AB κ
0.2
BA AB
− 0.4
0 II
0 0.2 0.4 κ 0.6 0.8 1
1 0.2
α(b=)3P/h2asediεag=ra1m1/f2o0rBernoullisystemwith III
and (analyti
alresult). The 0
regionsofnosyn
hronisat−ion are labeledwitha 0 0.2 0.4 ε 0.6 0.8 1
dash( ).
α(b=)3P/h2asediagramforBernoullisystem,
FIG. 5: Ring of 4 units with two di(cid:27)erent
oupling parame- . Analyti
alresult
ombiningthe
ters. analyti
alsyn
hronization regionforaringof3
units((cid:22))andtheregionwhere identi
alunits
whi
hare drivenbyanidenti
alsignal,
syn
hronize (--).
are
ompletelysyn
hronizedwithouttimeshift,although
τ
the
oupling has a long delay time . Hen
e the
haoti
FIG. 6: Triangle (three bi-dire
tionally
oupled units) with
motif, three weakly
orrelated
haoti
traje
tories,
an a uni-dire
tionally atta
hed in(cid:28)nitely large latti
e. Double
be imposed on an arbitrarilylargelatti
e. Note that the linessignifybi-dire
tional
ouplingswhereasarrowsshowuni-
time for spreading a motif on a large latti
e in
reases dire
tional
ouplings. The self-feedba
k is not drawn to sim-
only linearly with the number of units be
ause of the plify theillustration.
unidire
tional
ouplings.
κ ε
For some parameters and , namely in regions I
and II of Fig. 6(b), the three units of the triangle are unidire
tional
ouplings.
ompletely syn
hronized. In region I, only the three
units are syn
hronized while the other units remain un-
syn
hronized. In region II, the other units, too, get syn- V. SYNCHRONIZATION BY RESTORING
hronized to the triangle (be
ause all Lyapunov expo- SYMMETRY
nents are negative), so the whole latti
e is
ompletely
syn
hronized. In general we expe
t that the larger the network is,
The phenomenon of spreading motifs is not restri
ted the smaller the region in the parameter spa
e is where
N =6
to a triangle. For example, a ring of 6 mutually
ou- the network syn
hronizes. For example, a ring of
(ε,κ)
pled units has also a region in the
orresponding units has a smaller region of syn
hronization than the
N = 4
phase diagram where there is no syn
hronization, but region II and III of Fig. 2 for . With in
reasing
N α
where isolated units have negative Lyapunov exponents, (andtheslope beingheld
onstant)syn
hronization
omparableto region III in Fig. 6(b) [21℄. Consequently, (cid:28)nally disappears
ompletely [21℄. However, we found
the pattern of six
haoti
traje
tories
an lead to sub- a
ounterexample where adding a unit restores syn
hro-
latti
e syn
hronization with 6 di(cid:27)erent sublatti
es on a nization.
orresponding latti
e of
haoti
units with short-range Consider the
hain of 5 units shown in Fig. 7(a). The
5
(a) Chainof(cid:28)veunitswithtwodi(cid:27)erent timedelays
andwithout self-feedba
k.
(b) Thelastunithasbeenremoved.
N
(a) There are di(cid:27)erentdelaytimeswhi
h
are pairwiseidenti
alforoneunitofsideA
FIG. 7: Symmetri
and asymmetri
hain without self-
andoneunitofsideB.
feedba
k.
oupling to the two outer units has a longer delay time
than the internal
ouplings. There is no self-feedba
k,
κ = 0
. Now remove unit E, Fig. 7(b), and res
ale the
oupling to unit D. (Unit D doubles the input from unit
C to
ompensate for the missing se
ond neighbor.) In N
this
ase, a syn
hronized solution does not exist. Nu- (b) There are di(cid:27)erentshiftsinthe
Bernoullimapwhi
hare pairwise
meri
alsimulationsoftheBernoullisystemandthelaser
identi
alforoneunitofsideAandone
equations show∆hig=hτ
or−reτlationsbetween units A and C unitofsideB.
1 2
with time shift , and between B and D with
zero time shift, but the
orrelation
oe(cid:30)
ient does not FIG. 8: Ea
h unit on one side is
oupled to all units of the
a
hieve the value one. On the other side, if we add unit otherside.
E we restore the symmetry of the
hain. In this
ase we
(cid:28)nd sublatti
e syn
hronization with time shift between
k
the outer units and the
entral one: and vi
e versa. Note tha1t/tNhe unit A re
eives only a
k
weak signal of the order from its
ounterpart B .
at =et =ct−∆; bt =dt .
(12) Nevertheless, we (cid:28)nd that the network syn
hronizes to
a =
k,t
τ τ a state of pairwise identi
al
haoti
traje
tories,
1 2 b ;k =1,...,N
If is greater than , the
entral unit is earlier than k,t
. FortheBernoullisystem,theregionof
the
haoti
traje
toryoftheouterones,itleads,whereas
pairwise syn
hronizationis similar to region II of Fig. 2.
for the opposite
ase it lags behind.
Thereisnosyn
hronizationamongunitsofthesameside.
Ea
hunitre
eivesthesumofall
haoti
traje
tories,but
it responds only to the tiny part whi
h belongs to its
VI. COOPERATIVE PAIRWISE
ounterpart. The syn
hronizationisa
ooperativee(cid:27)e
t.
SYNCHRONIZATION
As soon as a single unit is detuned, the whole network
loses syn
hronization.
Is it possible to syn
hronize two sets of
haoti
units
The se
ond setup, Fig. 8(b), is similar to the (cid:28)rst
withasingle
oupling
hannel? Infa
t,wefoundsu
hex-
one but allows analyti
al
al
ulations. In this se
ond
ampleswheretwosetsof
haoti
unitsarebi-dire
tionally
setup, all delay times are identi
al, while ea
h unit of
β k = 1,...,N
onne
tedbythesumoftheirunits,asindi
atedinFig.8. k
2N one side has a di(cid:27)erent shift , , in its
Thereare units, i.e.the numberofunitsonea
hside
N shifted Bernoulli map, see Eq. (14); the shifts are pair-
is . Ea
h side is the mirror image of the other. Only k k
N wise identi
alforunit A and the
orrespondingunit B
onesinglebi-dire
tionalsignal
omposedof all signals
so one side is again the mirror image of the other side.
from ea
hsideisdrivingthe otherside,and this leadsto
The shifted Bernoulli map is de(cid:28)ned by
ooperative pairwise syn
hronization.
f (x)=[α(x+β)] mod 1.
In the (cid:28)rst setup, Fig. 8(a), all units areidenti
al, but β (14)
thedelaytimesoftheir
ouplingsaredi(cid:27)erent. Theunits
k k
have pairwise identi
al delay times, i.e. A and B have Ea
h A unit re
eives the signal
2τ + τ
k
a
oupling delay time τsw=hi2
hτki+s eτnfor
ed by a 1 N
self-feedbaN
k=w1ith delay time . Hen
e, for st =ε(1−κ)N fβk(bk,t−τ) (15)
one pair, , we obtain the phase diagram of Fig. 2, Xk=1
where the two units are
ompletely syn
hronized in re-
N
gions I and II. For a large number of units, ea
h A
and vi
e versa. Analyti
al
al
ulations are possible for
unit re
eives the identi
al signal
this se
ond setup be
ause the delay times are equal (cid:21) in
N
1
ontrast to the (cid:28)rst setup (cid:21) and the di(cid:27)erent shifts in
st =ε(1−κ)N f(bk,t−(2τk+τ)) (13) themapdonothamperthe
al
ulations. IftheBernoulli
kX=1 maps were not shifted, the units would all be identi
al.
6
N
Then in regions II and III ofaFig=. 2a, all=u.n.it.s=ofathe of lasers
oupled semi
ondu
tor lasers, where the net-
1,t 2,t N,t
same side would syn
hronize, ; workstru
tureand
ouplingstrengthsaredeterminedby
b = b = ... = b w
1,t 2,t N,t j,k
be
ause they re
eive the same the weightings , whi
h are normalized so that the
input. Then one would e(cid:27)e
tively get two
oupled units strNengthwof th=e1total input for jea
h laser is the same,
rAegaionndIBIa(lwl2hNi
hunsyitns
whroounlidzebeinsyrneg
hiornosniIzeadn.dFoIIr),shsiofteind PNsko=ll,a1sGersN,jτ,k,αlef,γf,oΓr,epa,
ωh0laasnedrCsp[2a8r℄e.
Thohseenpaar
a
moredtienrgs
Bernoullimaps,theunitsofthesamesidearenotallowed to Ref. [25℄.
to syn
hronize by de(cid:28)nition. Nevertheless, the stability In a very simpli(cid:28)ed form, the equations above read
analysisregardingthelinearizedequationsisnota(cid:27)e
ted
by the di(cid:27)erent shifts, meaning that the same perturba- d
x=[ ]+λ[ ]
tions still are damped in the same regions; the only dif- dt internal dynami
s self-feedba
k
feren
e is that due to the shifts, most of the traje
tories +σ[ ].
external
oupling (19)
arenotallowedto
ome
losetogether. Onlythepairsof
a = b
k,t k,t
orresponding units
an syn
hronize, , and so
they do in region II. Comparing equations (1) and (19) yields a relation be-
tween the parameter spa
e of the Lang-Kobayashiequa-
{λ,σ}
tions, , and the parameter spa
e of the maps,
{ε,κ}
VII. ANALOGY OF LASER EQUATIONS TO , if the two following
onditions are
onsidered:
ITERATED MAPS
(i) The ratio of the self-feedba
k to the external
ou-
pling should be the same in both
ases.
Besides iterated maps, we
onsidered the Lang-
Kobayashi equations, whi
h des
ribe the dynami
s of
(ii) The ratioof the time-delayed terms to the internal
semi
ondu
tor lasers opti
ally
oupled to their own
or/and to the light of other semi
ondu
tor lasers. We dynami
s should be the same in both
ases.
used them for simulations in the following form, a
ord-
ing to Ref. [25℄: These two
onditionsyield the followingtransformation:
λ λ+σ
d 1 C γ[N +nj(t)] κ= , ε= .
E0,j(t)= GNnj(t)E0,j(t)+ sp Sol λ+σ v (20)
dt 2 2E (t)
0,j
+λE (t−τ)cos[ω τ +φ (t)−φ (t−τ)]
0,j 0 j j
These
ond
ondition(ii)isnotproperlyde(cid:28)ned be
ause
N
lasers the Lang-Kobayashi equations (19) (cid:21) in
ontrast to the
+σ w E (t−τ)cos[ω τ +φ (t)−φ (t−τ)],
j,k 0,k 0 j k
iteratedequations(1)(cid:21)aredi(cid:27)erentialequationsandthe
Xk=1
internal dynami
s is not only the (cid:28)rst termon the right-
(16) handside ofEq. (19)but isalso
ontainedin the
urrent
x v
state . Thereforethedenominator in(20v)i=sn1o8t0giv−en1
and has to be
hosen reasonably. We took ns
d 1 ε 0 1 λ + σ
φ (t)= α G n (t) so is between and [be
ause the sum (the
j N j
dt E2 (ltef−τ) strengthvo=f t1h8e0re-i−n1je
ted light) should not ex
eed the
−λ 0,j sin[ω τ +φ (t)−φ (t−τ)] value of ns in our
ase℄.
0 j j
E (t)
0,j In order to measure syn
hronization, we averaged the
E 1
0,j
N amplitudes of the ele
tri
(cid:28)elds, , in ns intervalls
lasers E0,k(t−τ) ∆ = 0
−σ wj,k sin[ω0τ +φj(t)−φk(t−τ)], and
al
ulated the un-shifted, ,
ross
orrelation
E (t)
Xk=1 0,j fun
tion de(cid:28)ned by
(17) C∆ = hxt·yt−∆i−hxihyi .
xy
hx2i−hxi2 · hy2i−hyi2 (21)
d p p
dtnj(t)=(p−1)γNsol−γnj(t)−[Γ+GNnj(t)]E02,j(t), A
ross
orrelationover0.99
anberegardedassyn
hro-
(18) nization in the
ase of lasers.
Figure 9 shows both the parameter transformation,
E (t) φ (t) {λ,σ} {ε,κ}
0,j j
where and are the amplitude and the Eq.(20), from to andthemapping,Eq.(8),
E (t) =
j
slowly varying phase of the ele
tri
(cid:28)eld from two mutually
oupled units to driven ones. From
E (t)exp{[ω τ +φ (t)]} n (t)
0,j 0 j j
i and is the
arrier num- the
omparison of this (cid:28)gure with Fig. 2, the analogy of
j =1,...,N
berabovethevalueforasolitarylaser, laserλs. laserequationstoiteratedmaps
anbeseen. Besidesthe
The strength of the self-feedba
k is determined by , analogy of features des
ribed in se
tion II, one
an also
while the strength of the external
oupling is de(cid:28)ned (cid:28)nd this analogy for the features des
ribed in se
tions
σ
by . The equations above
over the general
ase III, V and VI.
7
140 1
120
0.8
100
0.6
80
σ κ
/ n s −1
60
0.4
40
0.2
20
00 20 40 60 λ /8n 0 s −1 100 120 140 00 0.2 0.4 ε 0.6 0.8 1
⇆ ⇆
(a) A Bbeforetransformation(20) (b) A B
1 1
0.8 0.8
0.6 0.6
∼
κ κ
0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 ε 0.6 0.8 1 0 0.2 0.4 ε 0.6 0.8 1
⇆ κ→κ˜=2κ−1 ′← →
(
) A Baftermapping (d) A S A
F◦IG. 9: Phase diagrams forCth>e L0.a9n9g-Kobayashi laser equations. Every
ir
le/point represents one simulation. Open
ir
les
( ) show a
ross
orrelation , whi
h
an be regarded as syn
hronization in the
ase of the laser equations. Figure (a)
shows results for two mutually
oupled lasers before applying transformation (20), wheras (b) shows the same diagram after
theparametertransformation. Thesyn
hronizationregion oftwomutually
oupledunits(b)
anbemapped[usingEq.(8)℄to
a region (
)whi
h issimilar to thesyn
hronization region ofa driver/re
eiver system(d). Aftertheparametertransformation
(20), the syn
hronization regions for the Lang-Kobayashi equations, (cid:28)gures (b), (
) and (d), look very similar to the ones for
theBernoulli maps,Fig. 2.
VIII. SUMMARY
an be de
omposed into a few sublatti
es. Ea
h sublat-
ti
e is
ompletely syn
hronized, but di(cid:27)erent sublatti
es
are only weakly
orrelated. Syn
hronization is relayed
Small networks of
haoti
units with time-delayed
by di(cid:27)erent
haoti
traje
tories. The traje
toriesofea
h
ouplings show interesting patterns of
haos syn
hrony.
sublatti
e have identi
al statisti
al properties. Thus the
These patterns are stable attra
tors of the network dy-
symmetry of the latti
e is not broken. There are solu-
nami
s.
tions of the dynami
equations whi
h break the symme-
The phase diagram of two and four units with mu- try of the latti
e. However, we always found that these
tual
ouplings has been related to the properties of a solutions are unstable. Hen
e we postulate that stable
single driven
haoti
unit. Two intera
ting units with patternsof
haossyn
hronypossessthesymmetryofthe
self-feedba
k
an syn
hronize
ompletely, without time
orresponding latti
e.
shift,evenifthedelaytimeisextremelylarge. Whenthe Syn
hronization depends on the symmetry of the net-
haoti
traje
toryoftwointera
tingunitsisre
ordedand work. When the symmetry of an asymmetri
hain is
usedtodriveasingleidenti
alunit, itturnsoutthatthe restoredbyaddingunits,sublatti
eor
ompletesyn
hro-
driven unit syn
honizesonlyin asmall part ofthe phase nization is restored, too.
diagram. Hen
e intera
tion is di(cid:27)erent from drive.
Finally, a bi-partite network, where the two parts are
Sublatti
e syn
hronization is found for latti
es whi
h
oupled by a single mutual signal, shows pairwise
om-
8
plete syn
hronization, whereas the units of ea
h part do ingasingleunitdestroysthesyn
hronizationofthewhole
not syn
hronize. Ea
h unit responds to the weak
on- network.
tribution of its partner in the other part of the network. The work of Ido Kanter is partially supported by the
Pairwise syn
hronization is a
ooperative e(cid:27)e
t: Detun- Israel S
ien
e Foundation.
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