Ordered and self-disordered dynamics of holes and defects in the one-dimensional complex Ginzburg-Landau equation

4 Pages·0.15 MB·English

by  Martin van Hecke

Ordered and self–disordered dynamics of holes and defects in the one–dimensional
complex Ginzburg–Landau equation
Martin van Hecke1 and Martin Howard2
1Center for Chaos and Turbulence Studies, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
2Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
(February 7, 2008)
We study the dynamics of holes and defects in the 1D complex Ginzburg–Landau equation in
ordered and chaotic cases. Ordered hole–defect dynamics occurs when an unstable hole invades a
planewavestateandperiodically nucleatesdefectsfromwhichnewholesareborn. Theresultsofa
detailed numericalstudyoftheseperiodicstatesareincorporated intoasimpleanalyticdescription
1 of isolated “edge” holes. Extending this description, we obtain a minimal model for general hole–
0
defectdynamics. Weshowthatinteractionsbetweentheholesandaself–disorderedbackgroundare
0
essential for theoccurrence of spatiotemporal chaos in hole–defect states.
2

n PACS numbers: 05.45.Jn, 05.45.-a, 47.54.+r
a
J

5 The formation of local structures and the occurrence 150 100 100
1 of spatiotemporal chaos are the most striking features
of pattern forming systems. The complex Ginzburg– 100 90 90
] t t t
D Landau equation (CGLE)
50 80 80
C
At =A+(1+ic1)∇2A−(1−ic3)|A|2A (1) 0(a) 70(b) 70(c)
.
n -150 0 x 150 70 80 x 90 70 80 x 90
i providesaparticularlyrichexampleofthesephenomena.
l FIG. 1. (a) A space–time grey–scale plot of |A| (dark:
n The CGLE describes pattern formation near a Hopf bi-
A≈0), showing the propagation of incoherent holes into a
[ furcation and has become a paradigmatic model for the
plane wavestate. The dark dots correspond to defects. Note
2 study of spatiotemporalchaos [1–7]. Defects occur when theroughly constant velocities at which theholes propagate.
v A goes through zero and the complex phase ψ:=arg(A) Parameter values are c1=0.6,c3=1.4 , with an initial condi-
1 is no longer defined. In two and higher dimensions, such tion given by Eq. (2), with γ=1,qex=−0.03. This non–zero
3 defects can only disappear via collisions with other de- qex breaks the left–right symmetry and results in the differ-
0
fects,andactaslong–livingseedsforlocalstructureslike ing periods of the left and right moving edge holes. (b–c)
2
spirals [3] and scroll waves [4] whose instabilities lead to Close–up of |A| (b) and the complex phase ψ (c).
0
0 various chaotic states [3,4]. For the 1D CGLE, however,
0 defects occur only at isolated points in space–time (see grow out to defects occur (Fig. 1 and 2b).
n/ Fig. 1) and intricate dynamics of defects and local hole In this Letter we study the hole→defect and defect
i structuresoccurs,especiallyintheso–calledintermittent →holes dynamical processes of the 1D CGLE [9]. We
l
n and bi–chaotic regimes [5]. The holes are characterized present a minimal model for hole–defect dynamics that
: by a local concentration of phase–gradient q:= ∂ ψ and describesthefull“interior”spatiotemporalchaoticstates
v x
i adepressionof|A|(hencethename“hole”),anddynam- of Fig. 1a, where holes propagate into a self–disordered
X
ically connectthe defects (Fig. 1). We divide these holes background. Similar “self–replicating” patterns are ob-
r into two categories: coherent and incoherent structures. served in many other situations, e.g., reaction–diffusion
a
Coherent structures-Bythiswemeanuniformlyprop- models [10], film–drag [11], eutectic growth [12], forced
agating structures of the form A(x,t) = e−iωtA¯(x−vt) CGLE [13] and space–time intermittency models [14].
[8]. Recently,holesolutionsofthisformcalledhomoclinic Hole→defect - Let us consider the short–time evolu-
holeswereobtained[6]. Asymptotically,homoclinicholes tion of an isolated hole propagating into a plane wave
connect identical plane waves where A ∼ ei(qexx−ωt). state. Holes can be seeded from initial conditions like:
With c ,c and q fixed, unique left moving andunique
1 3 ex
rightmovingcoherentholes arefound. Left (right)mov- A=exp(i[qexx+(π/2)tanh(γx)]) . (2)
ing holes with q = Q (q = −Q) are related by the
ex ex
left–rightq↔−q symmetryoftheCGLE.Coherentholes The precise form of the initial condition is not impor-
have one unstable core mode [6]. tant here as long as we have a one–parameter family of
Incoherent structures - In full dynamic states of the localized phase–gradient peaks. This is because the left
CGLE, one does not observe the unstable coherent ho- moving and right moving coherent holes for fixed c ,c
1 3
moclinicholes,unlessonefine–tunestheinitialconditions andq areeachuniqueandhaveoneunstablemodeonly.
ex
(see Fig. 2d). Instead evolving incoherent holes that can As γ is varied three possibilities can arise for the time
1
(a) Γ 1.5 (b) 150 (a) 5 (b) 0.9 (c)
(d(b))Wns WWDuenfsects 0q.5 10τ0 100/qm 0.qo4
(c) Wu c1 c3
50 0.61.4
Decay -0.5 00..8611..42
-10 25 x 60 0 1.01.2 0 -0.1
1.5 (c) 1.5 (d) 10-7 10-5q e x - q co h 10-1-0.6 -0.3 ∆ t 0 0 1 q 2
FIG.3. (a)Log–linearplotoftheperiodτ asafunctionof
q q
qex−qcoh. (b)100/qm asafunctionofthetime∆tbeforethe
0.5 0.5
formation of a defect; (c) q˙m as a function of qm.
constant initial conditions for their daughter edge holes,
-0.5 -0.5 similar to fixing γ in Eq. (2). The period τ will depend
-10 25 x 60 -10 25 x 60 on the location of the defect profile with respect to the
FIG. 2. (a) Schematic representation of the phase space
stablemanifoldofthe coherenthole. Whenq isvaried,
ex
of the CGLE around the homoclinic hole solution, showing:
both this manifold and the defect profile may change,
the 1D unstable manifold Wu; the high dimensional neu-
tral/stable manifold Wns that separates decaying from de- and for a certain value of qex which we call qcoh, the de-
fect generates an initial condition precisely on the stable
fectforming states;themanifold Γrepresentingthefamilyof
manifold of the coherent hole. The lifetime of the result-
peaked initial conditions of the form (2). (b–d) Four snap-
shots (∆t=10) of the q-profile of a right moving hole where ing daughter hole then diverges (see Fig. 2d).
qex=0, c1=0.6, and c3=1.4. The peaked initial condition is To substantiate this intuitive picture, we have per-
givenbyEq.(2): (b)Aholeevolvingtoadefect(γ =0.568), formednumericsonthe dynamicsof“edge–holes”invad-
(c)adecayinghole(γ =0.5),and(d)aholeevolvingcloseto ing a plane wave state where A ∼ ei(qexx−ωt). We have
a coherent structure(γ =0.5545). performed runs for many different parameters, but will
evolution of the initial peak: evolution towards a defect only discuss a representative subset here. Our results
(as in Fig. 1a), decay, or evolution arbitrary close to a indicate that the τ divergence is of the form
coherent homoclinic hole (see Fig. 2).
The hole propagation velocities are much larger than τ ∼−sln(qex−qcoh)+τ0 . (3)
the typical groupvelocities in the plane wavestates: the
holes are thus only sensitive to the leading wave. Their This equation, and in particular the value of s can be
internal, slow dynamics determines their trailing wave. understoodbyconsideringtheflownearthesaddlepoint
A (nearly)coherenthole will, due to phase conservation, shown in Fig 2a. Just after the hole has been formed,
haveatrailingwave(nearly)identicaltotheleadingwave it first evolves rapidly along the stable manifold. Sec-
(Fig. 2); hence the relevance of the homoclinic holes. ondly it evolves slowly along the unstable manifold be-
Defect→holes - What dynamics occurs after a defect fore being shot away towards the next defect. For val-
has been formed? A study of the spatial defect profiles ues of qex close to qcoh, the holes approach the coherent
revealsthattheyconsistofanegativeandpositivephase– structure fixed point very closely, and τ will be domi-
gradient peak in close proximity (the early stage of the nated by a regime of exponential growth close to this
formation of these two peaks can be seen in Fig. 2b; see fixed point. Small changes in qex will have a negligible
also Fig. 4d of [6]). The negative (positive) phase gradi- effect on the durationof the first phase (τ0), but the du-
ent peak generates a left (right) moving hole. The life- rationofthesecondphasewilldivergelogarithmicallyas
times oftheseholesdepend ontheir parentdefectprofile −(1/λ)ln(qex−qcoh). Here λ, which depends on c1 and
(analogous to what we described in Fig. 2) and also on c3,denotestheunstableeigenvalueofthecoherentstruc-
c1,c3 and qex. Hence the defects act as seeds for the tures at qex=qcoh. In Table 1 we list some numerically
generation of daughter holes (see also Fig. 1). determined values for qcoh, 1/λ, and s. We obtained s
andq from a fit ofτ to Eq.(3), whereasλ is obtained
Periodic hole-defect states - When an incoherent hole coh
from a shooting algorithm, see Ref. [6]. The agreement
invades a plane wave state and generates defects, stable
between s and 1/λ is quite satisfactory.
periodic hole→defect→hole behavior can set in at the
edges of the resulting pattern [15] (Fig 1a). The asymp- We will now construct a phenomenological model for
toticperiodτ ofthisprocessdepends onc ,c ,theprop- isolated incoherent holes. (i) We will ignore their early
1 3
agation direction and the wavenumber q of the initial time attraction to the unstable manifold, and think of
ex
conditiononly;wefocushereonrightmovingholes. The their location on WU as an internal degree of freedom,
period τ diverges at a well–defined value of qex = qcoh parameterized by the phase–gradient extremum qm. (ii)
(Fig. 3a). This can be understood in the phase space Clearlythemodelshouldhaveanunstablefixedpointfor
picture presented in Fig. 2. Suppose we fix c1 and c3. values of qm corresponding to coherent holes. We have
The edge defects that are generated periodically yield found that, in good approximation, coherent holes have
2
c1 c3 qcoh 1/λ s Minimal model - To illustrate our picture of self–
0.6 1.4 -0.0362 8.42 8.4 disordered dynamics, we will now combine the various
0.8 1.4 -0.0727 9.91 9.5 hole–defect properties with the left–right symmetry and
0.6 1.2 0.0538 12.72 12.7 local phase conservation of the CGLE to form a mini-
1.0 1.2 -0.0200 17.71 18.7 mal model of hole–defect dynamics. From our previous
analysis,we see thatthe followinghole–defectproperties
Table 1. Comparison of 1/λ with s (see text for details). should be incorporated: (i) Incoherent holes propagate
either left or right with essentially constant velocity (see
q =q +gq where q denotes the value of q for a Fig. 1a). (ii) For fixed c ,c , their lifetime depends on
m n ex n m 1 3
coherent hole in a q =0 state, and g is a negative phe- the profile of their parent defect, the direction of prop-
ex
nomenological constant. (iii) When approaching a de- agation, and on the wavenumber of the state into which
fect, q diverges as (∆t)−1 [16]; we have confirmed this they propagate. (iii) Eq. (4) captures essentially all as-
m
byaccuratenumerics(Fig.3b). Anappropriateequation pectsoftheevolutionoftheirinternaldegreeoffreedom.
incorporating these three features is When q diverges, a defect occurs.
m
In our model we will assume that all the defects have
q˙m =λ(qm−(qn+gqex))+µ(qm−(qn+gqex))2 , (4) the same profile and so act as unique initial conditions
for their daughter incoherent holes. While in principle a
whereg andµarephenomenologicalconstants. Thefirst defect profile could depend on the entire history of the
term on the RHS of (4) results from the linearization hole which preceded it, for simplicity we have chosen to
near the coherent fixed point. Nonlinear terms of higher neglect this. We have observed that for some regions of
thanquadraticorderontheRHSofEq.(4)areruledout thec ,c parameterspace,thedefectprofilesfromthein-
1 3
by the (∆t)−1 divergence of qm. Our numerical data for terior spatiotemporal chaotic patterns show a surprising
q˙m versus qm indeed shows quadratic behavior for large lackofscatter[9]. Thereforewebelievethattreatingthe
enough values of qm (Fig. 3c). For smaller values of qm, defect profiles as constant, and only including the effect
the curves are quite intricate; this corresponds to the of the backgroundin the hole dynamics incorporatesthe
rapid early time evolution along the stable manifold not essence of the coupling to a disordered background.
includedinmodel(4). FromEq.(4),itisstraightforward We discretize both space and time by coarse-graining,
to show that the hole lifetime τ (the time taken for qm andtakea“staggered”typeofupdaterulewhichiscom-
to diverge) displays the required logarithmic divergence pletely specified by the dynamics of a 2 × 2 cell (see
as qex is tuned towards a critical value qcoh. Fig. 4a). We put a single variable φi on each site,
Disordered dynamics - If the patches away from corresponding to the phase difference across a cell di-
the holes/defects were simply plane waves with fixed vided by 2π. Local phase conservation is implemented
wavenumber,then one would expect, following the argu- by φ′+φ′ =φ +φ , where the primed (unprimed) vari-
l r l r
mentsgivenabove,quiteregulardynamics. Thecoupling ables refer to values after (before) an update.
betweenholesandthebackgroundinducedbyphasecon- Holes are represented by active sites where |φ| > φ ;
t
servation becomes the key ingredient to understand dis- here φ plays the role of the internal degree of freedom.
order in hole–defect dynamics such as shown in Fig 1a. Inactive sites are those with |φ| < φ , and they repre-
t
Letusintroduceavariableφ:=R dxq thatmeasuresthe sent the background. The value of the cutoff φ is not
t
phasedifference across a certain interval. veryimportantaslongasitis muchsmallerthantypical
Consideragainanedgeholeevolvingtowardsadefect. values of φ for coherent holes. Here φ is fixed at 0.15.
t
While the peak of the q-profile grows, the hole creates Without loss of generality we force holes with positive
′ ′
a dip in its wake (see Fig. 2b) in order to locally con- (negative) φ to propagate only from φ (φ ) to φ (φ ).
l r r l
serve φ. Clearly the trailing edge of this incoherent hole Depending on the two incoming states, we have the
is not a perfect plane wave. In the interior of states such following three possibilities:
as shown in Fig. 1a, unstable holes move back and forth Onesite active: Withoutlossofgeneralityweassume
throughabackgroundofdisorderedq andamplify this that we have a right moving hole. We implement evolu-
ex
disorder. Nevertheless,aswepointedoutearlier,thedis- tionsimilarto Eq.(4), but neglectthe quadratictermof
orderingdynamicsissufficiently slowsuchthatthe holes Eq. (4); even though q diverges, the local phasediffer-
m
remainapproximatelyhomoclinicformuchoftheir lives. ence φ does not diverge near a defect. Hence the finite
m
Although the typical range of values for the disordered timedivergenceofthe localphasegradientq thatsignals
q is small, the hole lifetimes depend on it sensitively. a defect can be replaced by a cutoff φ for φ. Therefore,
ex d
Hence the variation in q and φ is sufficient to explain when φ <φ , the internal hole coordinate φ is taken to
ex l d
′
the varying lifetimes found in the interior states such as evolve via φ = φ +λ(φ −φ −gφ ). Here λ sets the
r l l n r
thatshowninFig1a. Thustheessenceofthespatiotem- time scales and can be taken small (fixed at 0.1). This
poralchaoticstateshere liesinthe propagation of unsta- evolution equation, combined with the local phase con-
ble local structures in a self–disordered background. servation, means that an incoherent hole propagating
3
′ ′
φ/ φ 300 c3, |φl| and φr are typically larger than φn so that most
l r/ “daughterholes” will grow out to form defects and hole-
φ φ t defect chaos spreads (Fig. 4c,d). For sufficiently small
l r 150 values of c and c , on the other hand, φ is large and
1 3 n
both daughter holes will decay. For intermediate values
(b) ′
(a) 0 of c1 and c3 it may occur that |φl| is significantly larger
0 200 x 400 than φ′, leading to zigzag states [6] (Fig. 4d).
r
300 400
In conclusion, we have studied in detail the dynamics
of local structures in the 1D CGLE. We have obtained a
t t
150 250 quantitative understanding of the edge holes, unraveled
theinterplaybetweendefectsandholes,andputforward
(c) (d) a simple model for some of the spatiotemporal chaotic
0 100
states occurring in the CGLE.
0 200 x 400 0 200 x 400
M.v.H. acknowledgessupport from the EU under con-
FIG.4. (a) Grid model geometry showing the sites (dots) tractERBFMBICT972554. M.H.acknowledgessupport
and hole propagation direction (arrows). The update rule is from the Niels Bohr Institute, the NSF through the Di-
definedwithina2×2cell. (b–d)Dynamicalstatesinthegrid vision of Materials Research, and NSERC of Canada.
model, for φn=0.6 and φad=0.75. Initial condition: center
site has φ=0.7, everywhere else φ=0 (hence the symmetric [1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65,
patterns). (b) g=0 and φd=1. (c) Disordered dynamics for 851 (1993).
nonzerocoupling(g=−3,φd=1). (d)Zigzagstructuresoccur [2] B. I.Shraiman et al., Physica D 57, 241 (1992).
for g=−3,φd=0.98. [3] I. S. Aranson, L. Aranson, L. Kramer and A. Weber,
Phys. Rev. A 46, R2992 (1992); G. Huber, P. Alstrøm
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[4] I.S.Aranson,A.R.BishopandL.Kramer,Phys.Rev.E
in its wake. When φ >φ , a defect occurs and two new
l ′ d ′ 57, 5276 (1998); G. Rousseau, H. Chat´e and R. Kapral,
holes are generated: φ = φ , and φ = φ −1−φ .
r ad l d ad Phys. Rev.Lett 80, 5671 (1998); K. Nam, E. Ott, P. N.
The factor −1 reflects the change in winding number at
Guzdar and M. Gabbay,Phys. Rev.E 58, 2580 (1998).
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relevantdynamicsisphasediffusion. Thisisimplemented [7] L. Brusch, M. G. Zimmermann, M. van Hecke, M. B¨ar
via: φ′r=Dφl+(1−D)φr. The value of D is fixed at 0.05 and A.Torcini, Phys. Rev.Lett. 85, 86 (2000).
and is not very important. [8] W.vanSaarloosandP.C.Hohenberg,PhysicaD56,303
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annihilation of both holes (see Fig. 1a), which we imple- [10] W.N.Reynoldset al.,Phys.Rev.Lett.72,2797(1994);
′ ′ M. Zimmermann et al., Physica D 110, 92 (1997); A.
ment here via phase conservation: φ =φ =(φ +φ )/2.
r l l r Doelman et al., Nonlinearity 10, 523 (1997); Y. Hayase
The coupling of the holes to their background, g,
and T. Ohta, Phys. Rev. Lett. 81, 1726 (1998); Y.
should be taken negative (although its precise value is
Nishiura and D.Ueyama, Physica D 130, 73 (1999).
unimportant). For g=0 the lifetime τ becomes a con-
[11] D.P.Vallette,G.JacobsandJ.P.Gollub,Phys.Rev.E
stant, independent of the φ of the state into which the 55, 4274 (1997).
holes propagate,and moreover,the dynamicalstates are [12] S. Akamatsu and G. Faivre, Phys. Rev. E, 58, 3302
regularSierpinsky gaskets(Fig. 4b). Nevertheless,start- (1998).
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hole dynamics leads to a background state with a disor- 17 (1999).
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ton, (Kluwer) 273 (1991).
the crucialimportance ofthe coupling betweenthe holes
[15] Thefullstatethatdevelopsinthewakeoftheincoherent
and the self–disordered background.
holes is chaotic, i.e. one has exponential sensitivity. For
The essential parameters determining the qualitative
theedgeholes,however,localperturbationsoftheinitial
natureofthe overallstate areφ , φ andφ . Thesepa-
n d ad conditionleadonlytoaspace–timeshift,andtheasymp-
rameters determine the amount of phase winding in the
totic period τ is independentof such perturbations.
core of the qex =0 coherent holes (φn) and in the new [16] To see this note that A is a smooth function of x and t,
holes generated by defects (φad,φd − 1 − φad). When andexpressqasafunctionofthereal(u)andimaginary
varying the CGLE coefficients c1,c3, these parameters (v) parts of A: q=(u∂xv−v∂xu)/|A|2. As a function of
change too; for example, φ typically decreases when c time, u and v are both moving linearly through zero at
n 1
orc areincreased. Asaresult,forlargevaluesofc and the defect, and hence we finda 1/∆t divergenceof qm.
3 1
4