Breaking chirality in nonequilibrium systems on the lattice

0.29 MB·English

by  Diego Pazó

epl draft
Breaking chirality in nonequilibrium systems on the lattice
Diego Pazo´1 and Ernesto M. Nicola2
1 Instituto de F´ısica de Cantabria, IFCA (CSIC-UC) - Avda. Los Castros, 39005 Santander, Spain
2 Max-Planck-Institut fu¨r Physik komplexer Systeme - No¨thnitzer Straße 38, 01187 Dresden, Germany
8
0 PACS 05.45.-a–Nonlinear dynamics and chaos
0
2 PACS 47.54.-r–Pattern selection; pattern formation
PACS 02.30.Oz–Bifurcation theory
n
a Abstract. - We study the dynamics of fronts in parametrically forced oscillating lattices. Using
J
asaprototypicalexamplethediscreteGinzburg-Landauequation,weshowthatmuchinformation

7 about front bifurcations can be extracted by projecting onto a cylindrical phase space. Starting
1 from a normal form that describes the nonequilibrium Ising-Bloch bifurcation in the continuum

andusingsymmetryarguments,wederiveasimpledynamicalsystemthatcapturesthedynamics
]
S of fronts in the lattice. We can expect our approach to be extended to other pattern-forming
P problems on lattices.
.
n
i
l
n
[

Extended systems on lattices have played a major role The bifurcations linking different dynamics of the front
1
v in the development of nonlinear science. One may recall (including bistable regimes) are observed both in a pro-
9 celebratedmodels asthe discretesine-Gordonequationor jection of the system’s variables onto a cylindrical phase
8 the Fermi-Pasta-Ulam experiment. Usually the develop- spaceandinthenormalform. Ourresultsarerelevantfor
6
ment of the field has been associated with conservative experiments where discretisation is given as in arrays of
2
systems, but dissipative lattices have attracted growing coupledpendula[8,9],electroniccircuits[10,11],orchem-
.
1
attention in the last two decades (see [1] for a review). ical systems [12]; and also in systems usually modeled as
0
Two prominent examples of such lattices are provided by continuous, but that are intrinsically discrete (or behave
8
0 the discrete versionof the Nagumo andGinzburg-Landau likea lattice due to a spatiallyperiodic modulation ofthe
: partial differential equations. The former has been pro- medium).
v
posedasamodelofmyelinationofneuronalfibres[2];and Foralattice,theFCGLequation[13,14]takestheform:
i
X thelatterdescribes,amongothers,dissipativesolitons[3],
ar the dynamics of lines of vortices[4] and coupled wakes[5] A˙j = (1+iν)Aj −(1+iβ)|Aj|2Aj +γA∗j
in hydrodynamics. +κ(1+iα)(A +A 2A ), (1)
j+1 j−1 j
−
The complex Ginzburg-Landauequation [6] universally
where Aj ρjeiψj is a complex variable. The parameter
describes the dynamics of an extended medium in the ≡
γ measures the forcing strength, and κ controls the cou-
neighbourhood of an oscillatory instability. Under homo-
pling between neighboring units. Parameters ν, β, and
geneous resonant n : 1 forcing, different regions of space
α account for the detuning, the nonisochronicity, and the
maylocktothedrivingwithdifferentphaserelations;and
dispersion, respectively.
domain walls separate these regions. The prototypical
2 : 1 resonant case leads to the so called parametrically The continuum limit. – First of all we recall the
forcedcomplex Ginzburg-Landau(FCGL) equation. This results for the continuous version of (1):
equationhasbeensubjectofextensivestudysincethesem-
∂ A=(1+iν)A (1+iβ)A2A+γA∗+(1+iα)∆A. (2)
inal work by Coullet and coworkers in ref. [7]. There it t
− | |
wasfound a frontbifurcation which is the nonequilibrium
Vanishing values of ν, β, and α allow to cast (2) into a
analogue of the Ising-Bloch transition in ferromagnets.
variational form: ∂A/∂t= δ /δA∗. In this case, stable
− F
In this Letter, we show that the dynamics of fronts in front solutions minimise the energy functional , and de-
F
the FCGL equation on the lattice is captured by a nor- pending on the forcing γ they can be chiral (γ < γ =
IB
malformconsistingoftwoordinarydifferentialequations. 1/3) or achiral (γ γ ). If A vanishes at the centre of
IB
≥
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Europhys. Lett., 81 (2008) 10009
γ (forcing strength)(a)00..24 TIB Arnol’d Tongue (strength)c)00.1.25 Sκ=B0.25 I SB+TB I+TB (eU)I:: RIRImmee[[[[AAAA]]]] xx IImm[[AA]] RRee[[AA]]
(b)-00.5ν (frequenc0y detuning)0.5 rcing 0.1 TB 0.16 I I+TB Re[A] Im[A]
o x
sing) RIme[[AA]] x Im[A] γ (f γ SB+TB SB: Im[A] Re[A]
I (I Re[A] 0.05 0.14 SB TB Re[A]
(d) Im[A]
B (Bloch) IRme[[AA]] x Im[A] Re[A] 00 ν 0 .(0f5reque0n.0c2y0 .1det0u.04ninνg)0.01.506 TB: Im[A] x Re[A]
T
Fig. 1: Regions in the parameter space ν-γ where Ising (I) and travelling or stationary Bloch fronts (denoted by TB and SB,
respectively) appear: (a) in the continuum,and (c) on thelattice, κ=0.25 in eq. (1). Sketchesof all possible fronts (b) in the
continuum,and (e) on thelattice. Whereas thefronts I,TB and SBin (e) are stablefor some parameter values, thefront U is
always unstable.
the front the so-called Ising front is found, otherwise the fig. 1(e) and figs. 1(c,d) show their regions of stabil-
front is chiral and A does not vanish anywhere: two such ity. Stationary Bloch fronts (SB) exist on a finite region
Bloch solutions with opposite chirality exist. of parameter space around an interval of the line ν = 0.
The nonequilibrium Ising-Bloch (NIB) transition is ob- Additionally, two regions of bistability are found: In one
servedfornonzerovaluesofν,β,orα [7,15]. Thetermsν, of them Ising and travelling Bloch fronts coexist (I+TB).
β andαareperturbationstoagradientsystem,andcause The other bistable region (SB+TB) is shown in fig. 1(d),
theBlochfrontstomove,ineitherdirectiondependingon and it is a small triangle with stable stationary and trav-
their chirality (see [14] and references therein). elling Bloch fronts.
Front dynamics. – We restrict ourselves hereafter Cylindrical coordinates. – A projection of the 2N
to the case β = α = 0, so that ν remains as the pa- degrees of freedom onto a two-dimensional phase space
rameter that breaks the variational character of the sys- greatly simplifies the analysis of the fronts. One way of
tem;wenoteneverthelessthatthesamequalitativeresults performing this projection is
are obtained perturbing variationality through ν, β, and
N
α. Inside the region γ > ν (the Arnold tongue), see ρ
| | Φ = Re[X jei(ψj−ψS)], (3)
fig. 1(a), the local dynamics is bistable with two stable ρ
S
j=1
fixedpointsAS± = ρSeiψS withρS =[1+pγ2 ν2]1/2,
± − N
caonsd(2ψψS)∈=(−(ρπ2/2,π1/)2/)γ.theTshoelufitxioend opfoisnint(2aψtSt)he=orνi/gγin, C = Im[Xρρjei(ψj−ψS)]. (4)
A =S(0,0) is Seit−her completely unstable (γ2 < 1+ν2), j=1 S
O
or saddle (γ2 > 1 + ν2). We are interested in the dy- This corresponds to a rotation and compression of the
namics of fronts connecting the two stable fixed points: complexplaneAsuchthatthestablefixedpointsarenow
Aj→∓∞ =AS±. located on the real axis at 1. This projection permits
In the continuum, the locus of the NIB transition can us to discern if a front is sy±mmetric with respect to the
be calculated analytically [16]: γNIB(ν) =[√1+9ν2]/3, origin(insuchacaseC =0). Note thatΦ isacyclicvari-
seefig.1(a)[andfig.1(b)forasketchofbothfronttypes]. able that takes the same value when the front advances
As mentioned above, for ν = 0 we recover the variational or recedes one lattice unit (provided the front is far from
case,γNIB(ν =0)=γIB,andBlochfrontsarestationary. the boundaries) and thus can be defined modulo 2. Vari-
OurextensivenumericalsimulationsoftheFCGLequa- able C measures the deviation from stationarity and is
tion on the lattice1 have revealed several front types not intrinsically bounded.
present in the continuum. These fronts are sketched in Typical examples of the dynamics in the reduced coor-
dinates (3)-(4) are shown in figs. 2(a,b). Two stationary
1The numerical calculations were performed with zero flow fronts, both with C = 0, exist for all parameter values.
boundary conditions A0 = A1, AN+1 = AN, and a lattice size
One,the Ising front(I), is locatedat (Φ,C)=(0,0)if the
N large enough to neglect boundary effects on the front dynamics
(withN typicallybeing128). number of units N is even (or at (1,0) if N is odd) and
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Europhys. Lett., 81 (2008) 10009
2 2 TB media (e.g. arrays of FitzHugh-Nagumo units).
(a) TB (b)
Inthecontinuum,asthevariationallimitisapproached
SB (unstab.) thevelocityoftheBlochfrontdecreases(andbecomeszero
SB (unstab.) SB atvariationality). Consequently,weintroduce asmallpa-
rameter χ that accounts for the deviation from the vari-
C 0 I U I C 0 I U I ational case. For the situation considered here, χ is pro-
0 1 2 0 1 2
portional to2 ν. We have then at leading order:
SB
SB (unstab.)
φ˙ = χc ǫsinφ, (7)
SB (unstab.) −
c˙ = (µ+δcosφ c2)c. (8)
TB −
-2 Φ -2 TB Φ
The term ηsinφ present in (6) is absent in (8) due to the
invarianceofthediscreteFCGLequationunderthetrans-
Fig.2: Numericallyobtainedflowsprojectingthefrontdynam-
icsontothecylindricalphasespaceΦ-C definedbyeqs.(3)and formation (ν,β,α,A) ( ν, β, α,A∗). This requires
→ − − −
(4). In (a) an example of coexistence I+TB is shown (ν =0.1 thesymmetryunder(χ,φ,c) ( χ,φ, c)tobesatisfied.
→ − −
and γ = 0.183). In (b) SB+TB (ν = 0.045 and γ = 0.148). This considerably simplifies the normal form.
Symbols•,×,◦denotestable,saddleandunstablefixedpoints, Figure 3(a) shows the regions of the parameter space
respectively. χ-µ, and fig. 3(c) is a sketch of the corresponding phase
spaces. The normal form (7)-(8) qualitatively reproduces
frontdynamics,whichcanindeedbe projectedonacylin-
is a continuation for κ > 0 of the trivial solution at zero
der, e.g. the phase space Φ-C introduced above. There
coupling: [...,A ,A ,A ,A ,A ,A ,...]. The
S+ S+ S+ S− S− S− are two pitchfork bifurcations at µ = δ and the pa-
secondstationarysolution,denotedU,is at(1,0)(respec- ±
rameter space is organised by two codimension-2 points.
tively,at(0,0)foroddN),andisacontinuationoftheun-
We may see in fig. 3(b) that at a degenerate pitchfork
stable front solution: [...,A ,A ,A ,A ,A ,...].
S+ S+ O S− S− point located at χˆ2 = 2ǫ2/δ the pitchfork bifurcation
For some parameter values, there exist extra fixed points −
line switches between subcritical and supercritical. The
locatedoffthe Φ axis. We label these chiralsolutionssta-
second codimension-2 point is a saddle-node separatrix-
tionary Bloch (SB) fronts. Travelling Bloch (TB) fronts
loop point [22] where the saddle-node bifurcation on the
correspond to periodic orbits around the cylinder. Due
invariantcircle(SNIC)splitsintoanoff-cyclesaddle-node
to symmetry they always appear in pairs circulating in
(SN) and a homoclinic (Hom) bifurcations.
opposite directions.
In contrast to the continuous case, travelling Bloch
Normal form. – Next,wepresentasimpleordinary- fronts appear with nonzero chirality, and the (average)
differential-equation model that generates dynamics like velocity of the front v asymptotically follows: (i) the
the front dynamics on the FCGL lattice. We will obtain familiar square-root law when crossing the SNIC line;
this normal form via symmetry arguments. (ii) a logarithmic law below the homoclinic line: v−1 =
In continuous systems the NIB transition is a parity- a (1/λ )ln(γ γ ), where λ is the positive eigen-
u Hom u
− −
breaking (pitchfork) bifurcation coupled to a translation- value corresponding to the saddle Bloch front.
invariant coordinate. It is an example of the so-called Let us finally discuss the parameters modelling the dis-
drift-pitchforkbifurcationfoundinanumberofsituations cretisationstrength: ǫ and δ. In the continuum limit, our
(e.g. [17]), usually as a secondary instability. Its normal numericalsimulationsindicatethatdiscretisationentersin
form is [18,19] φ˙ =c, c˙ =(µ c2)c, where µ is the bifur- thenormalformthroughδatorderO(κ−1),whereasǫvan-
−
cation parameter, e.g., µ (γ γ). Coordinates φ, ishes much faster (possibly exponentially) as κ . In
NIB
∝ − → ∞
c representthe position and the velocity of the concerned addition,assuming opposite signsfor ǫ andδ correctlyin-
structure (the front in our case). heritsthebifurcationsintheanticontinuumlimit(κ 0).
→
Knobloch et al. [20] considered the breakdown of the Accordingly, our choice ǫ = 0.1 and δ = 0.2 in fig. 3(a)
−
continuous translational invariance to study the parity satisfies these constraints (i.e. ǫ < δ , ǫδ < 0). The or-
| | | |
breaking of a periodic pattern in the presence of an in- ganisation of the parameter space is qualitatively robust
homogeneity. This leads to the inclusionof smallperiodic to changes of both parameters in a wide range.
terms (sinusoidals in the simplest case) that preserve the
Phase equation. – Normal form (7)-(8) is obtained
invariance under inversion (φ,c) ( φ, c):
→ − − in a perturbative way including small terms that model
φ˙ = c ǫsinφ, (5) breakdown of translational invariance, and the departure
− from (ν,β,α;γ) = (0;γ ), the equilibrium Ising-Bloch
c˙ = (µ+δcosφ c2)c+ηsinφ, (6) IB
− bifurcation point. This means that we cannot predict the
behaviour of the bifurcation lines far from this point. In
where φ is now an angular variable. In [21] it was sug-
gested that this normal form could also be used to anal- 2AccordingtoEq.(4)in[7],inthevariationallimitonehasχ∝
yse parity-breakingbifurcations found in discrete bistable ν−β−(α−β)γ.
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Europhys. Lett., 81 (2008) 10009
(a) theFKmodel(seee.g.[23,24])statethatthe propagation
threshold (SNIC line) should approach the γ axis expo-
0 I nentially fast3 as γ 0.
→
Hom
PF I+TB Conclusions. – The nonequilibrium Ising-Bloch bi-
PFsub furcation in the parametrically driven complex Ginzburg-
µ I Hom I+TB Landau equation is one of the most studied pattern in-
SB 0.200 PF PF
PFsub stabilities. In the continuum, breaking of chirality causes
0.5 SB
µ the front to move. However,as shown here, on the lattice
m SB+TB
o a more complex scenario appears: specifically, two types
H
C N
NI S of bistability and a region with chiral (Bloch) stationary
S 0.201 TB (b) fronts. We have demonstrated that a normal form with
1 TB SNIC0.3 χ 0.32 tbwifourvcaartiiaobnlsesbectawpetuenredsitffheerednytnraemgiiocnsso.fItthies ftroonbteaenmdphthae-
0 0.2 0.4 χ 0.6 0.8 1 sised that the normal form (7)-(8) is based on symmetry
arguments that provide general qualitative results inde-
pendent of details as, for instance, the parameter of non-
variationalityor the discretisation-orderof the Laplacian.
(c)
The dynamics of patterns on lattices is typically much
hardertosolveanalyticallythanintheircontinuouscoun-
terparts. The continuum, usually serves as a zeroth order
approximation that is not necessarily accurate. Discrete-
nesstypicallyintroducesnewdynamicalregimesasshown
in the current letter for one spatial dimension4. Through
a modification of the normal form for the continuum, we
have been able to reproduce the dynamics of fronts on
a discrete medium and the structure of parameter space.
This approachshould also work in other problems on lat-
tices.
∗∗∗
We thank Manuel A. Mat´ıas, Juan M. Lo´pez, Ernest
Fig. 3: (a,b) Parameter space of the normal form (7)-(8) with Montbri´o and Luis G. Morelli for useful discussions and
ǫ=0.1 and δ =−0.2; the loci of local and global bifurcations critical comments. D.P. acknowledges support by Minis-
are depicted with solid lines. The black dots in the inset (b) terio de Educacio´n y Cultura(Spain)throughtheJuande
indicate the location of two codimension-2 bifurcation points. la Cierva Programme. This work was supported by MEC
(c) Schematic representation of the phase space φ-c for the (Spain) under Grant No. FIS2006-12253-C06-04.
five regions in (a,b). Gray-dashed lines indicate the link be-
tween different states via codimension-1 bifurcations (PF and
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