Dynamic Phase Transition from Localized to Spatiotemporal Chaos in Coupled Circle Map with Feedback

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by  Abhijeet R. Sonawane

Dynamic Phase Transition from Localized to Spatiotemporal Chaos in
Coupled Circle Map with Feedback.
Abhijeet R. Sonawane1 and Prashant M. Gade2
1)Center for Modeling and Simulation, University of Pune, Pune-411 007, India
2)PG Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Campus, Nagpur-440 033,
India
We investigatecoupledcircle maps in presenceof feedback andexplorevarious dynamicalphases observedin
this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to
spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence
acts as an excellent quantifier to describe this transition. Taking the locationof the fixed point of circle map
(which does not change with feedback) as a reference point, we compute number of sites which have been
1 greater than (less than) the fixed point till time t. Though local dynamics is high-dimensional in this case,
1 this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In
0 most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional
2 scaling as seen in second order phase transitions. This indicates that persistence could work as good order

n parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in
a eigenvalue spectrum of the Jacobian of localized state.
J

5 PACS numbers: 05.45.Ra,05.70.Fh
2

]
D Coupled map lattices have been investigated ex- and have uncovered interesting modifications of equilib-
tensively in past few decades. The dynamics can rium, bifurcations and stability properties of collective
C
show very novel features when local dynamics is spatiotemporal states. These systems have been stud-
n. higher dimensional. Using feedback to make the ied extensively6,7. However, while we have arrived at a
i local dynamics higher dimensional makes it pos- goodunderstandingoflowdimensionalchaosinpasttwo
l
n sible to systematically increase dimensionality of decades,andunderstoodroutestochaosinseveralseem-
[ localmapwithoutchangingseveralotherfeatures ingly disparate systems in an unified manner, the same

such as location of fixed point. Investigation of is not true for spatiotemporal systems.
1
v phase diagram of such a system shows possibility As one would expect, there is practical relevance to
6 of existence of partially arrested state in which these models. Certain simplified models of spatiotem-
5 certain sites are near the fixed point while others
poral dynamics have been studied extensively in recent
7 arenot. Westudythenonequilibriumphasetran-
past. We would like to make a special mention of cou-
4
sitionfrom such a state to a state ofspatiotempo-
. pledmap lattices whichhavebeen studiedextensively in
1 ral chaos and propose that persistence works as
the hope that the understanding we have developed in
0 an excellent order parameter for such transitions.
low-dimensional systems could be useful in understand-
1
ing spatiotemporal dynamics. Majority of the studies
1
: are in coupled one-dimensional maps. At times, they
v
have been successful in modeling certain patterns ob-
i
X I. INTRODUCTION served in spatiotemporal systems. For example, the
phenomenaofspatiotemporalintermittencyinRayleigh-
r
a Benard convection has been modeled by coupled map
Pattern formation in spatially extended systems has
lattices8,9. They have been used to model spiral waves
been an object of extensive study in past few decades.
in B-Z reaction10 and even in crystal growth11.
The reasonsfor the interestinpatternformationarenot
Wefeelthatthesestudiescouldbehelpedconsiderably
fartoseek. Patternformationhappensinseveralnatural
by quantitatively identifying different phases in the sys-
systemsandscientists areinterestedinunderstandingit.
tem and attempting to understand transitions between
Examples could be flame propagation or occurrence of
those. A lot of attention has been devoted to transi-
spiralsinachemicalreactionsorpatternsonthe skinsof
tion to synchronization in these systems which is easy
animalsmodeled byreaction-diffusionprocesses. Several
systemslikebiologicalpatterns1,chargedensitywavesor to identify and analyze. However, other spatiotemporal
JosephsonJunction Arrays2–4, lasers5 have been studied patterns are far less analyzed and understood.
extensively from the viewpoint of dynamics and pattern It is worth noting that in several spatiotemporal sys-
formation. The practical importance of understanding tems the field is high dimensional. For example, for a
abovesystemscannotbeoveremphasized. Partialdiffer- chemical reaction, we would require information about
ential equations, coupled ordinary differential equations, concentrationsofallreactantsandproducts atallpoints
coupledoscillatorarraysandcoupledmapshaveallbeen tobeabletosimulatethesystem. Thereareseveralphe-
used to model different physical and biological systems nomenawhicharepossibleonlywhendynamicsishigher
2
10 chronization, there have been extensive studies to ascer-
tain whether or not this transition is in the universality
9
10 class of directed percolation25. Transition to synchro-
8 FP-P/C 8 nization, locking in of the dynamics of several systems
S.T.C
7 6 to a collective oscillation, is an important but minuscule
6 4 part of overall spatially extended systems. Several non-
y synchronization transitions have been observed such as
Dela 5 2 FP-P/C spiral state to spatiotemporal chaos22, traveling wave to
4 00.235 0.237 0.239 0.241 spatiotemporalchaos23. Thespatiotemporaldynamicsis
3 far richer and other transitions deserve attention.
SFP
2 FRP S.T.C As suggested by Pyragas, feedback in the dynamical
systems can play a central role in controlling chaos by
1 I II III IV
stabilizing UPO’s embedded in chaotic attractor26. The
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 scheme has been efficiently implemented in wide range
Coupling strength of experimental applications. Feedback leading to syn-
chronization is a well established fact supported by sev-
FIG.1. Weplotthephasediagramofcoupledcirclemapwith eral researchers27. Though feedback has been an impor-
delayintwoparameterspaceofcouplingstrengthǫanddelay
tantaspectofstudiesincontrolandsynchronization,not
time τ. The abbreviation SFP, FRP, FP-P/C and STC de-
much has been said about its effect on different dynam-
note Synchronousfixed point, Frozen random pattern, Fixed
ical phases. It is then important to find the effects of
point-Periodic/Chaotic cycles and Spatiotemporal chaos re-
spectively. In the inset, we magnify the phase boundary be- feedbackinterms ofalterationsinsystemdynamicsthat
tween FP-P/C regime and STC regime. We can have only may lead to origin of different dynamic phases.
integer values of feedback time and the lines in phase dia- In this work, we investigate coupled maps with feed-
gram are guide to the eye. The black solid circles indicate back. As mentioned before, coupled higher dimensional
pointsinphasespacewherespatiotemporalfixedpointphase
maps havenotbeen studied adequately. Using feedback,
is seen.
we can systematically vary dimensionality of local map
using feedback and study its effect on overall dynamics.
We note that the feedback as defined in our model does
dimensional. On the other hand most of the existing not change the location of fixed point of system with-
studies in coupled map lattices are about coupled one- out feedback. We have used it extensively as a reference
dimensional maps and very few involving high dimen- point. This allows us to study spatially extended higher
sional maps, for example coupled Chialvo’s map12, cou- dimensional systems with known properties. We investi-
pledHenonmap13,Arnold’scatmap14,modifiedBaker’s gate a model of coupled circle maps with feedback. The
map for Ising model15, etc. Here we try to study cou- strength of feedback and the delay time are the param-
pled higher dimensional maps. We try to systematically eters in the system. We will present a general phase
change the dimensionality of the map and study cou- diagramof the system. In particular, we observea novel
pled maps with feedback where the feedback time could transition from localized chaos to spatiotemporal chaos.
be varied. In coupled map lattice (CML)16 time evolu- Bylocalizedchaos,wemeanthatthelaminarandturbu-
tion is conflict between competing tendencies: the dif- lent phases coexist in the same space domain for weaker
fusive coupling between units tends to make the system couplings. The reason to call it localized chaos is that
homogeneous in space, while the chaotic map produces one can see there are well defined boundaries of turbu-
temporal inhomogeneity, due to sensitive dependence on lent sites which do not mix with laminar ones. A similar
initial condition. Such systems display a rich phase dia- patternhasbeennamedspatialintermittency(SI)inpre-
gram and origin and stability of various phases in these viousstudiesandsomeofitspropertiessuchaseigenvalue
systems are of theoretical and practical interest. The spectrum of Jacobian are found to be related to its dy-
transitionsbetweenvariousphasesinthesesystemscould namical properties20,21. Interestingly, we find that per-
be viewed as dynamic phase transitions. Though these sistenceactsineffectivewaytocharacterizeandquantify
are clearly non-equilibrium systems, different techniques certaindynamicalphasesandtransitionbetweenthemin
used in the study of equilibrium phase transitions have this system. Thus, persistence can be taken as excellent
beenappliedtoexplorethevarietyofphenomenologically order parameter for certain phase transitions.
distinct behaviors. Such non-equilibrium systems range The plan of paper is as follows. We introduce the
from growths of surfaces17 to traffic jams in vehicular modelinthesectionII.InSectionIII,wemakeadetailed
traffic and granular flows18,19. For such analysis, there analysisofoneofthe phases,namelylocalizedchaosand
is a need to define an order parameter that will charac- calculatequantitiessuchasturbulentlengthdistribution.
terize the transition. However,not many dynamic phase In section IV, we present our investigations in transition
transitions have been studied from this viewpoint20–24. betweenlocalizedchaosregimeandspatiotemporalchaos
Most of the studies are devoted to transition in models and we present evidence that persistence acts as an ex-
with so-called absorbing states. In the context of syn- cellentorderparametertocharacterizethetransition. In
3
section V, we demonstrate that some characteristic fea-
tures in eigenvalue distributions of Jacobian can distin-
guish between different dynamical phases.
II. MODEL
We define model of coupled circle maps with feedback
as follows. We assigna continuous variable x (t) at each
i
site i at time t where 1 ≤ i ≤ N. N is the number of
lattice points. The evolution of x (t) is defined by,
i
ǫ
x (t+1)=(1−ǫ−δ)f(x (t)) + [f(x (t))
i i i−1
2
+f(x (t))]+δf(x (t−τ)) (1)
i+1 i
Theparameterǫis thecouplingstrengthandthe local FIG. 2. Bifurcation diagram of Spatial pattern at an instant
dynamical update function f(x) is the sine-circle map, oftime(aftertransience),withrespecttocouplingstrengthǫ
for delay time τ =1.
k
f(x)=x+ω− sin(2πx) (2)
2π
phase diagram Fig. 1. The first region in phase dia-
Where, k is the strength of nonlinearity and ω is the
gram labeled as SFP (Synchronous fixed point) where,
windingnumber. Theparameterδisstrengthoffeedback
for smaller coupling strengths ǫ ≤ 0.06 the entire sys-
andτ isdelaytime. Ifweincreaseτ,we canincreasethe tem goes to local fixed point x∗. This is an absorbing
dimension of map. In our simulations, we confine values
state and system settles in this state very quickly. We
of parameters to δ = 0.1, k = 1 and ω = 0.068. The
have checked that this state exists at least for τ ≤ 10.
couplingtopologyconsideredhereisdiffusivei.enearest-
For higher coupling strengths, The system goes to spa-
neighborconnections. Forcoupledcirclemap,weconfine
tiotemporal chaos as shown in phase diagrammarked as
thedynamicsintheinterval[0,1]usingthefollowingrule.
STC.Thesedifferentbehaviorscanalsobevisualizedus-
If int[x (t)] = m, x (t) = x (t) − m if x (t) > 0 and
i i i i ing the bifurcation diagram Fig. 2 obtained for different
x (t) = x (t)−m+1 if x (t) < 0. We study the system
i i i coupling strengths for τ = 1. This bifurcation diagram
with random initial conditions. The system is updated
shows the state of all sites at a given instant of time.
synchronously with periodic boundary condition. The
Forveryhighcouplingstrengths,the systemshowssome
feedbackdoesnotchangethefixedpointsolutionoflocal
stability islands for some values of τ though they consti-
map f(x) without feedback. The fixed point solutionfor
tute a small part in phase diagram, we find them worth
the local map f(x) is given by, x∗ = 1 sin−1(2πω). The
2π k a mention.
parametervalues consideredhere yield the value offixed
In the intermediate coupling strengths, the system
point as x∗ = 0.07026. This point acts as an excellent
shows a rather interesting behavior. We can see that
reference point in visualizing the dynamical behavior of
some of the sites approach local fixed point and some of
systemandalsoinsymbolicdynamicsofthe system. We
the sites take on different values which do not change in
candefinethesitedynamicstobelaminarif|x (t)−x∗|<
i time. Thisbehaviorisshowninphasediagrammarkedas
η for small enough neighborhood η = 0.001 and to be FRP which stand for frozen random pattern28. This re-
turbulent otherwise.
gionischaracterizedbyaspatialsequenceofdifferentdo-
mainswhosesizeandperiodicitymaygreatlyvary. Upon
further increaseofcoupling strengths,one enters ina re-
III. SPATIOTEMPORAL DYNAMICS gionwhere we cansee coexistence of different dynamical
regimes. Wheresomeofthesitesindeedgotofixedpoint,
Weexaminethephenomenologyofdelaycoupledcircle some of the sites show periodic/chaotic behavior. This
mapwith feedback under variationof coupling strengths is an interesting state where one can see that different
and delay time. Various spatiotemporal patterns are dynamical phases do not mix with each other. The re-
clearly evident from phase diagram Fig. 1 and the bi- gionsmarkedasFRP andFP-P/Cin the phase diagram
furcation diagram shown in Fig. 2. Note that on y axis collectively represent the dynamics which we refer to as
ofFig. 1,the time isdiscrete. The delaytime τ canonly Localized chaos. Chaos is localized in certain dynamical
be some integer. The phase boundary of various regions regimes of turbulence and does not interfere with evolu-
can be seen. tion of laminar states. In localized chaosregime, chaotic
Increasingvalue ofτ is equivalentto increasingthe di- domainsandperiodicdomainsarefixedinspace. Thisis
mensionality of local dynamics. There are clearly four evident from spatialdistribution pattern. The perturba-
distinct behaviors shown by the system as seen from tions do notpropagatein space. Essentially,the dynam-
4
ics is non-spreading and non-infective in nature, i.e. the in persistence in coupled map lattices by Menon et al.
spatially intermittent solutions have zero velocity com- and by Gade et al.. Menon et al. found that it was
ponents in spatial direction, and modes traveling along a good indicator for a transition from fully synchronous
the lattice are absent. This can also be considered as a state to spatiotemporal chaos while Gade et al. found
case of spatial intermittency20. The important result is that its dynamical behavior is able to characterize even
that spatial intermittency is seen over a large parameter traveling wave state which can be seen as a partially ar-
regime in presence of feedback. For each feedback time, rested state in moving frame of reference23,29. It was
wehaveawelldefinedregionwherewegetthetransition. observed by us in preliminary version of these studies30
These spatiotemporal patterns are clearly evident from that persistence acts as an good quantifier. Recent re-
space-time density plots below. Each figure represent a sults by Mahajan and Gade show that local persistence
particular region in phase diagram. can also characterize the transition from clustered state
Fig. 3 shows space-time density plots of x for six to spatiotemporalchaosin small worldnetworks24. This
i,t
choices of parameter values. The site index is plotted definitionofpersistenceconcentratesonasinglevariable
along the x-axis and the time evolves along the y-axis. and one would wonders if it is adequate for the high di-
As mentioned previously, laminar regions are blue (dark mensional local dynamics we are studying in this case.
grey) in this representation. The colorbar shows differ- However, we find that persistence acts as an excellent
ent value of x(i) according to given scale. Fig. 3(a) orderparameterfordistinguishingbetweenpartiallyand
gives the space-time density plot for parameter values fully arrested states in this system. This demonstrates
wherethesystemgoestospatiotemporalfixedpoint. Fig. a generic utility of persistence as order parameter even
3(b) shows the plot for non infective and non spread- in transitions from partially arrested phase. Thus, it is
ing behavior called as frozen random pattern. Fig. 3(c) possible that persistence is a good characteristic for sys-
and(d)showscoexistenceofdifferentdynamicalregimes tems in which dynamics is partially or fully localized.
for which the dynamics essentially remains ‘arrested’. Conventionally, persistence in the context of stochastic
Fig.3(e) shows density plot for ǫ=0.24097which is crit- processeshasbeen definedasthe probabilityP(t)that a
ical point for the transition, we can see the spreading stochastically fluctuating variable with a zero mean has
behavior just emerging in the system. Fig.3(f) depicts not crossed a threshold value up to time t31. For a spin
fully grown spatiotemporal chaos and here we can see system with discrete states, such as Ising or Potts spins,
that the state variable encompasses much larger part of it is defined in terms of the probability that a givenspin
available phase space. has not flipped out of its initial state up to time t. The
The number of turbulent sites increase as coupling power-law tail of P(t), i.e.
strength increases. Note here that, all those sites which
are not laminar are considered turbulent. We calculate P(t)∼1/tθ, (4)
distributionofturbulentphaselength. Itisobservedthat
defines the persistence exponent θ. The exponent
turbulentlengthdistributionisgovernedbytheexponen-
θ is dimension-dependent, non-trivial exponent. For
tial law,
spatially-extended systems, the time evolution of the
T(l)=Aexp(−βl) (3) stochastic field at one lattice site is coupled to its neigh-
bors. This makes the effective single-site evolution es-
where, exponent β is the rate of decay and l is the the sentially non-markovianand renders the nontriviality to
cluster size of turbulent sites. Fig. 4(a) shows Cumula- the exponent θ. This nature of local persistence prob-
tivedistributionofturbulentlengthsforcouplingparam- ability relates it to infinite point correlation function in
eter values ǫ=0.1,0.16,0.23,0.245. We can see that the time direction32.
distributionforsmallercouplingstrengthsfallsofquickly A natural generalization of the idea to CML as sug-
as opposed to higher value of coupling strength where gested by Menon et al.29, defines local persistence in
the exponential decay is slower indicating large number terms of the probability that a local state variable x
(i,t)
of turbulent phases. Fig. 4(b) shows how the exponent does not cross fixed point value x∗ up to time t. With
β varies with respect to ǫ. There is a sharp increase in this definition, we study persistence in the coupled map
value of beta near ǫ∼0.24 . latticedefinedthrougheq(1),viasimulationsonsystems
with various lattice sizes. The persistence defined above
isreferredaslocalpersistenceinliterature. Inthis work,
IV. PERSISTENCEAS AN ORDER PARAMETER we study only local persistence and do not study any
other definition of persistence. In this paper, by persis-
As it can be seen the system displays a dynamical tencewemeanlocalpersistenceonly. Westartwithlocal
phase transition i.e. from localized chaos regime to spa- x distributed randomly over interval [0,1]. We find the
i
tiotemporal chaos. It is important to find a good quan- normalizedfractionofpersistentsites,definedassitesfor
tifier to quantify this transition. This is a partially ar- whichx hasnotcrossedx∗ uptotimet. Thus,sitesfor
i,t
rested state and in partially or fully arrested dynamical which the sign of (x −x∗) has not changed unto time
i,t
states, we believe that persistence acts as a good order t are persistent at time t. This persistence probability
parameter. There have been couple of previous studied P(t) is averaged over an ensemble of over 2000 random
5
FIG. 3. (Color online)Space time density plot of coupled circle map lattice with feedback defined through eqs. (1)-(2), in a
system of size L = 200, for parameters k = 1, δ = 0.1, ω = 0.068 for τ = 1 and (a)ǫ = 0.05, (b) ǫ = 0.1, (c)ǫ = 0.16, (d)
ǫ=0.23,(e) ǫ=0.24097, and (f) ǫ=0.245
initial conditions. demonstratethevalidityofthisscalingformnumerically.
Fig. 5(a) shows plots of P(t) at three different values
of coupling parameter ǫ. It can be easily noted that for
As stated by Fuchs et. al.34, a finite 1+1 dimensional
values below critical point P(t) saturates. For the value
contact process at criticality reaches the absorbing state
above critical point i.e. in chaotic domain, P(t) decays
within finite time so that there is a finite probability for
exponentially. Exactly at the critical point of transi-
persistentsites to surviveforever. Therefore,persistence
tion between localized chaos to spatiotemporal chaotic
probability saturates at a constant value. One can eas-
regimes, P(t) decays as a power law, which defines the
ilycarrythisanalogytononspreadingregimeincoupled
persistence exponent θ = 1.31. The power law decay
evolutionofdelayedcirclemapsinlocalizedchaosregime.
can also be seen for other values of τ33 . Fig. 5(b)
When the dynamics is such that no site is laminar, the
shows persistence distribution for τ = 4,7 at the crit-
persistence decays to zero. The scaling function at crit-
ical points ǫ = 0.2386 and ǫ = 0.23847 respectively.
c c icality ∆ = 0 for various system sizes L, i.e. plotting
The observedpower-lawbehaviorin 1+1dimensionsug-
P(t)Lθz against t/Lz, shows a excellent data collapse at
gests the further investigation of parameters involved in
longer times for z =1.62 and θ =1.31 in Fig. 6.
phenomenological scaling laws. The distance from criti-
cality ∆=|ǫ−ǫ | and lateral system size L can be used For a continuous phase transition below (above) the
c
as parameters to study scaling properties of the system. critical point, we expect the persistence probability to
One would expect a asymptotic scaling law of the form saturate(decayexponentially). Accordingto abovescal-
ing form, the curves should collapse in both cases when
P(t)≃t−θF tL−z,t∆ν|| (5) P(t)∆−θν|| is plotted against t∆ν||. This is indeed the
case as shown in Fig. 7. The simulations were carried
(cid:16) (cid:17)
where, F is a scaling function and z = ν /ν = 1.62 out for lattice size L = 5000. The size is large enough
|| ⊥
is the dynamical exponent. We consider two cases to so that finite size corrections could be neglected. Also
6

0
10
-1
10

1
10
-2
10

-3
) 0 (t) 10
l( 10 P
T -4
10
-5
-1 10
10
-6
10
-2 -7
10 10
100 200 300 400 500 600 700 800 900 100 101 102 103 104 105
(a) l (a) t

0
10
0.14
-1
0.12 10
0.10
-2
10
0.08 P(t) =4, = 0.23860
c
-3
0.06 10
=7, = 0.23847
c
0.04
-4
10
0.02
-5
0.00 10
0 1 2 3 4 5
0.09 0.12 0.15 0.18 0.21 0.24 10 10 10 10 10 10
(b) coupling strength (b) time t
FIG. 4. (a) The distribution of turbulent phase lengths for FIG. 5. (a) Persistence probability P(t) is plotted as a func-
coupling values ǫ = 0.14,0.16,0.23,0.245 for τ = 1. The tionoftimetonalogarithmic scale fork=1,ω=0.068 and
fit to this distribution is exponential with exponent β. The τ = 1. The lattice size is 5000. The curves shown are for
ordinate axis is presented in logarithmic scale. (b) This plot ǫ = 0.24040 (upper), ǫ = 0.24097 (middle), and ǫ = 0.24300
shows exponentβ vscoupling strength ǫ for τ =1. (lower). The line with slop 1.31 is plotted for reference. (b)
(Color online) Log P(t) vsLog t for τ =4, τ =7.
ensemble averaginghas been performed over5000 differ-
ent initial conditions to ensure convergence. Persistence
acts as an excellent order parameter for transition from
spreading to nonspreading regimes. This point can fur- state to spatiotemporal chaos (shown to be in DP uni-
therbesubstantiatedbysimilarpowerlawdecayseenfor versality class) is found to 1.529 while in transition from
δ =0caseofspatialintermittency,asseenbyJabeenand laminar state to traveling wave state in negatively cou-
Gupte21. Figure8showspersistenceprobabilitybehavior pledcirclemaps,theexponentis123. Thustheexponent
for parametervalues of coupled circle map without feed- is clearly different from the exponents observed in pre-
backwherespatialintermittencyisseen. Thepersistence viouslystudied transitions. Howeveras notedpreviously
exponent observed is 1.2. We would like to note that in literature, persistence is known to be least universal
persistenceexponentobtainedintransitionfromlaminar exponent among various dynamical exponents.
7

0 0
10 10
10-1 = 0.24000
-1 = 0.24015
10 -2
10
-3
10
P(t)10-2 P(t) -4 = 0.24050
L = 20 10
L = 40 -5 = 0.24600 = 0.24300
10-3 L = 60 10
L = 80 = 0.24400
-6
L = 100 10
-4 -7
10 -1 0 1 2 3 4 5 10 0 1 2 3 4 5
10 10 10 10 10 10 10 10 10 10 10 10 10
(a) (a) t
t

4
10 7
10
103 105
z
3
P(t)L102 L = 20 P(t)10 = 0.00097
L = 40 101 = 0.00082
= 0.00047
101 LL == 6800 10-1 == --00..0000230033
L = 100 = -0.00503
0 -3
1010-4 10-3 10-2 10-1 100 101 102 103 104 1010-6 10-5 10-4 10-3 10-2 10-1 100 101 102
||
(b) t/Lz (b) t
FIG. 6. (a) The persistence probability P(t,L) is plotted FIG. 7. (a) The persistence probability P(t,L) is plotted
against time t for different lattice sizes L. (b)The scaling against time t for different coupling strengths ǫ. (b) The
functionf(x) = P(t,L)Lzθ is plotted against dimensionless scaling functionf(x) = P(t)∆−θν|| is plotted against dimen-
scaling variable x = tL−z. The data for different L values sionless scaling variable x = t∆ν||. A good data collapse is
were found to collapse for longer times for z=1.62. obtained for ν =1.51.
||

V. STABILITY ANALYSIS 100
-1
10
Computing Jacobian matrix of the system is useful -2
10
from several viewpoints. Lyapunov exponents are es- )
(t
sentially related to eigenvalues of product of Jacobians P -3
10
overinfinite period. For a synchronizedor periodic solu-
tions,one cangetconsiderablesimplificationsby analyz- -4
10
ing the Jacobian matrix for those states. The localized
statehasbeenpreviouslyobservedincoupledcirclemaps -5
by Jabeen and Gupte21 and has been named as spatial 10100 101 102 103 104 105
intermittency and they observed certain differences be- t
tween eigenvalue distributions of one-step Jacobian ma-
trix. It has been observed that eigenvalue spectrum of FIG. 8. (a) Persistence probability P(t) is plotted as a func-
the Jacobian of the system at any time T gives a good tion of time t on a logarithmic scale for k = 1, ω = 0.062
indicatorofthis phase. We showthatthe eigenvaluesre- and ǫ = 0.26634. The lattice size is 5000. The persistence
main real for our systems as well. This spectrum shows exponentθ=1.2.
distinct features in the phases concerned.
8
For τ =1, our equations can be put in the form. The Jacobian J at time t is given by,
t
A B
J = t N
t I 0
N N
(cid:18) (cid:19)
ǫ
xi(t+1)=(1−ǫ−δ)f(xi(t)) + [f(xi−1(t)) where 0N is an N ×N Null matrix with all entries 0.
2
I is N-dimensional identity matrix. The matrix B is
N N
+f(xi+1(t))]+δf(yi(t)) defined as B(i,i) = δf′(yi(t)) and B(i,j) = 0, if i 6= j.
y (t+1)=x (t) The matrix A is given by,
i i t
ǫ f′(x (t)) ǫ f′(x (t)) 0 ... ǫ f′(x (t))
s 1 n 2 n N
ǫ f′(x (t)) ǫ f′(x (t)) ǫ f′(x (t)) ... 0
n 1 s 2 n 3
 0 ǫ f′(x (t)) ǫ f′(x (t)) ǫ f′(x (t)) ... 
A = n 2 s 3 n 4
t
. . . . .
 .. .. .. .. .. 
 
ǫ f′(x (t)) 0 ... ǫ f′(x (t)) ǫ f′(x (t)) 
 n 1 n N−1 s N 
 
where, ǫ = 1 − ǫ − δ, ǫ = ǫ and f′(x (t) = 1 − for τ = 1. Gupte and Jabeen already noticed the gap
s n 2 i
Kcos(2πx (t)). where, x (t) is the state variable. The for case without feedback. We would also like to note
i i
diagonalizationofJ givesthe 2N eigenvaluesofthe sta- that these are stretching transformations and eigenvec-
t
bility matrix. The eigenvalues of the stability matrix tors which are localizedin one frame of reference remain
were calculated for values of state variable x (t) for all i localized in other frame of reference as well. This obser-
i
atonetimestepondiscardingsufficientlylongtransients. vationwillbeusefulinlatersection. Thusitispossibleto
Fig. 9(a) shows that the eigenvalue distribution for lookforgapintheeigenvaluespectrumeveninthiscase.
ǫ = 0.2, where spatial intermittency or localized chaos Unfortunately the argument breaks down for τ >135.
is seen. This distribution has gap or zero probability Now let us try to understand the reason for gap in
regionsin the eigenvalue spectrumas reportedearlierby eigenvaluespectrum inA(t) inJabeenandGupte’s case.
Jabeen et al.21. On the other hand, Fig. 9(b) shows We choose a diagonal part of matrix as an unperturbed
that there is no such gap for coupling values ǫ = 0.3 matrixandalloff-diagonalentriesasaperturbation. Our
wherefullygrownspatiotemporalchaosisseen. Consider choiceforlth eigenvectorforthediagonalpartisavector
matrix J′ given by whose lth component is unity and all other components
t
are zero. We define A′ = Ad +Ap, where diagonal en-
J′t = BA′′t B0N′ tries of Adt are given bty Adt(ti,i) =t At(i,i) and rest of
(cid:18) N N (cid:19) its entries are zero. Let us consider an idealized case in
which kth and Nth sites are turbulent and all other sites
where B′(i,i) = δf′(y (t)) and B′(i,j) = 0, if i 6= j.
i arelaminar. Similaranalysiscanbe carriedoutformore
The elements of the matrix A′ are A′(i,i) = A(i,i) and
p that two turbulent sites as long as they are not adjacent
A′(i,j)= A(i,j)A(j,i)whichcanbeobtainedfromthe
sites.
transformationA′(t)=C′AC′−1 wherematrixC′(i,i)=
p
ǫnf′(xi(t)). a b 0 0 0 0 0 ... 0 bN
One can verify that B′2 = B. Thus, it is easy to b a b 0 0 0 0 ... 0 0
spee that matrix J′(t) and matrix J(t) have same char-  ... ... ... ... ... ... ... ... 0 0 
acteristic polynomial and hence they have same eigen-  0 0 ... a b 0 0 ... 0 0 
hrveaallsautereess.aJl(cid:18)teiagCn0ed′nvJCat′l.′0uBNe′so.(cid:19)wTitshhuetshm,etahstiermixmilJaar′ti(rttiy)xitsJras(tyn)msfaomlrsemotrsiwhcohauinclhdd A′t = 00... 00... ......... b0...k ab...kkk ba...k 0b... ......... 00... 00... 
have real eigenvalues. The assumption has been that  0 0 0 0 0 0 0 ... a bN 
f′(xi) ≥ 0,f′(yi) ≥ 0 which is justified for k ≤ 1. The  bN 0 0 0 0 0 0 ... bN aN 
feedback strength δ is already assumed to be positive in  
our work. Thus A(t) and A′(t) as well as Jt and Jt′ are where a = (1−ǫ)f′(x∗), b = ǫnf′(x∗) and ai = (1−
related by similarity transformation and since A′(t) and ǫ)f′(xi), bi =ǫn f′(xi)f′(x∗), where i = k and N sites
J′ are symmetric, A(t) and J should have real eigenval- respectively. ThematrixAdhaseigenvalueawith(N−2)
t t p t
ues. Thus we can look for gap in eigenvalue spectrum folddegeneracyandtwonondegenerateeigenvaluesgiven
for these matrices and we observe the gap in our case by a and a . The perturbation is Ap =A′ −Ad. If all
k N t t t
9
80

70 (a) 70 (b) 3
15
λ) 60 λ)60 2.5
P( P( 2
Density 4500 P(λ)10 ensity 4500 λP()1.5
ability 30 5 bility d30 0.15
b a
Pro 20 00 0.5 1 1.5 Prob20 0 0.2 0.4 0.6 0.8λ 1 1.2 1.4 1.6
λ
10 10
−00 .2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Eigenvalues λ Eigenvalues λ
FIG. 9. The eigenvalue distribution for (a) ǫ=0.2, (b) ǫ=0.3 is plotted for 50 different realizations for lattice size L=1000
with bin-width = 0.001. Gaps are seen in localized chaos eigenvalue distribution whereas the eigenvalue distribution for
Spatiotemporal chaos region shows no such gaps. The inset in figure(a),(b) show themagnified region of respective plots
.
sites are laminar except kth and Nth site (since we have it can be seen that
periodic boundary conditions, by appropriate relabeling
0 b 0 0 ... 0 0 0 0 0 0 ... 0
of sites, one can always bring the site in turbulent state
to Nth site), the matrix Ad = diag(a,a,a ...,a,a ) , b 0 b 0 ... 0 0 0 0 0 0 ... 0
t k N  
0 b 0 b ... 0 0 0 0 0 0 ... 0
the perturbation will be,
0 0 b 0 ... 0 0 0 0 0 0 ... 0
0 b 0 0 0 0 0 ... 0 bN  ... ... ... ... ... ... ... ... ... ... ... ... ... 
b 0 b 0 0 0 0 ... 0 0  
 ... ... ... ... ... ... ... ... 0 0  PTApP=00 00 00 00 ...... 0b 0b 00 00 00 00 ...... 00
 0 0 ... 0 bk 0 0 ... 0 0  t 0 0 0 0 ... 0 0 0 b 0 0 ... 0
Apt = 0 0 ... bk 0 bk 0 ... 0 0  0 0 0 0 ... 0 0 b 0 b 0 ... 0
b00N... 000... .00..... 000... b00...k 000... 00b... ............ b00N... b00N...  00... 00... 00... 00... ......... 00... 00... 00... 0b... 0b... .0b.. ......... 00b 
   
0 0 0 0 ... 0 0 0 0 0 0 b 0
 
 
We perform a simple degenerate state perturbation
theory to our perturbationmatrix Apt, we canconsider a torT,hais(Ngiv−es2u)s×fo(rNea−ch2)(Nm−at2ri)xfoolfdpdeergtuenrberaattioeneiAgepn.vWece-
matrix P of (N −2) eigenvectorsof degenerate eigenval- t
can see that the above matrix contains two blocks. The
ues as,
firstblockisof(k−1)×(k−1)sizeandthesecondblock
has size (N −k−1)×(N −k−1). The eigenvalues of
1 0 0 0 0 0 0 0 thismatrixwillbefirstordercorrections. Theperturbed
0 1 0 0 0 0 0 0 eigenvalues after first order perturbation for degenerate
 ... ... ... ... ... ... ... ...  states in this case will be,
 
0 0 0 1 0 0 0 0  λ =a+2bcos(θ ) (6)
0 0 0 0 0 0 0 0  l l
P= 
0 0 0 0 1 0 0 0  where36,
0 0 0 0 0 1 0 0 
 
 ... ... ... ... ... ... ... ... 
  lπ for l=1,..,k−1
00 00 00 00 00 00 00 10  θl =( (k−1)l+π1 for l=1,..,(N −k−1) (7)
  (N−k−1)+1
 
10
We have two non-degenerate eigenvectors, an unit vec-
tor in Nth direction v = [0,0,...,1]T and an unit TABLEI. TableshowingEigenvectorsshowingpeaksatcor-
N respondingtwoturbulentsitesinotherwisesteadystate,here
vector in kth direction v = [0,0,...1,...,0]T. Now
k 300 and 500. In second column larger eigenvalues are given
vTApv = vTApv = 0. Thus the eigenvalues a and
N t N k t k k which match with thevalues shown in last column.
a are unperturbed after first order perturbation. Now
N i Eigenvector Eigenvalue (1−ǫ)f′(x)
if these values are far from a±2b, we will observe a gap
in the spectrum. 500 1 1.0861 1.0817
Now let us generalize this argument to finite fraction 300 2 0.4177 0.4132
f of turbulent sites. The number of turbulent sites is
m=fN andtheyareinterspersedinthelattice. Wealso
assume that no two turbulent sites are adjacent. In that
case, we have a = (1−ǫ)f′(x ), b = ǫ (f′(x )), where
i i i n i
i=l ,l ,l ,...,l sitesrespectively,wherel ,l ,..,l are
1 2 3 m 1 2 m
indices of m turbulent sites andl site is Nth site which
m
can be obtained due to periodic boundary conditions.
AgainthematrixAd willhaveeigenvalueawith(N−m)
t
fold degeneracy and m nondegenerate eigenvalues given
by a ,a ,...,a . The matrix P becomes matrix of (N −
1 2 m
m)eigenvectorsofdegenerateeigenvalues. Thematrixof
perturbationAp changes accordinglyto (N−m)×(N−
t
m). We canseethatthematrixobtainedfromoperation
PTApP contains m blocks. The square blocks will be of
t
corresponding size, i. e. for l , the size will be (l −1),
1 1
for l it is (l − l − 1) and in the same way for mth
2 2 1
block correspondingto turbulent site l , the size will be
m
(l −l −1). The generalized case for m turbulent
m (m−1)
sites goes as,
FIG. 10. (Color online) Overlapping histograms of the cal-
lπ for l =1,..,l −1 culated eigenvalues. In red (gray), histogram of eigenvalues
l1 1 obtained by theory of pertubative analysis. In blue(black),
θl = lll234lll−−−πππlll123 fffooorrr lll ===111,,,......,,,lll234−−−lll123−−−111 (8) haicsagisnrgteoievagmelrsnaoe.mnbtTeobfhesneetwuevnmeaeleanurweitcashayelflooyfrrryootmbuatrnabtdihunelenedunbmtaeinegsdrietince[vssaac−gluaivne2esbnb,oeafbmc+yleaǫ2atnbrrlf]iy.x′(AsxTej(eh)tne))
lπ for l =1,..,l −l −1 for degenerate as well as non-degenerateeigenvalues.
Inanycaslemt−hle(me−i1g)envaluesarebemtweenm−[a1−2b,a+2b].
Nowletuslookatnondegeneratestates. Arguingsimilar
to the case of two site, after first orderperturbation, the
tries on diagonal, upper diagonal, lower diagonal and
neiogtenwviathluinesthareeraanl1g,eala2,±..2.ba,lwme. wIfillthoebsseereviegeangvaalpueins tahree elements A′t(1,N) and A′t(N,1). In this case, ratio of
width of band to system size tends to zero as N → ∞.
spectrum. Thus,wepredictthateigenspectrumofJ will
t The eigenvalue spectrum of such a matrix tends to stan-
begivenbyabandofeigenvaluesbetweena+2banda−2b dard Gaussian distribution37. It is worth noting here
superposedwithvaluesǫ (f′(x )),wherej isaturbulent
n j that, persistence acts as a good order parameter for this
site. Localizedeigenstates for A′(t) will also be localized
transition for all values of τ including τ = 0, whereas it
for A(t) since they are related by stretching transforma-
is not clear that dynamic characterizer of gaps in eigen-
tion. If we make the histogram of eigenvalues as shown
values spectrum might work for higher value of τ.
in figure 10, we find that this is indeed true. We can see
histogramsofeigenvaluescomputednumericallyandthe- SimilarcalculationscanbecarriedoutforourJacobian
oreticallyfromtheeqns. 6and8obtainedbypertubative J′. The localizedeigenvectorsexplainwhy localizedpat-
t
analysis for 20 turbulent sites in the lattice of 100 ele- terns are stable. The eigenvectors having largest eigen-
ments. The values for turbulent sites given by ǫ f′(x )) value are localized. Thus any perturbationfrom laminar
n j
canalso be seenawayfrom the band[a−2b,a+2b]. For state is likely to be growing in the direction of local-
generalcase,thereisabandofeigenvaluesfollowedbym izedeigenvectorswhichdoesnotdisturbvaluesatnearby
non-degenerate eigenstates which are far from the band. sites. Theaboveanalysisofeigenvaluedistributionforlo-
We checked that these eigenstates are localized and the calized chaos is restrictedto demonstrate the analogyto
corresponding eigenvectors are demonstrated in Fig. 11 spatial intermittency. The eigenvalue spectrum clearly
for case of two turbulent sites. differentiatesinfective tonon-infective phase. The above
On the other hand, for spatiotemporal chaos, we have analysisexplainswhytheeigenvaluespectrumisdifferent
a symmetric random matrix A′ which has nonzero en- for spatiotemporal chaos and spatial intermittency.38
t