Pattern Selection in the Complex Ginzburg-Landau Equation with Multi-Resonant Forcing
aa r X i v : . [ n li n . PS ] J un Pattern Selection in the Complex Ginzburg-Landau Equation with Multi-ResonantForcing
Jessica Conway and Hermann Riecke Engineering Sciences and Applied Mathematics,Northwestern University, Evanston, IL 60208, USA
We study the excitation of spatial patterns by resonant, multi-frequency forcing in systems un-dergoing a Hopf bifurcation to spatially homogeneous oscillations. Using weakly nonlinear analysiswe show that for small amplitudes only stripe or hexagon patterns are linearly stable, whereassquare patterns and patterns involving more than three modes are unstable. In the case of hexagonpatterns up- and down-hexagons can be simultaneously stable. The third-order, weakly nonlinearanalysis predicts stable square patterns and super-hexagons for larger amplitudes. Direct simula-tions show, however, that in this regime the third-order weakly nonlinear analysis is insufficient,and these patterns are, in fact unstable.
PACS numbers: 82.40.Ck, 47.54.-r, 52.35.Mw, 42.65.Yj
The variety of patterns and planforms that have beenobserved in surface waves on vertically vibrated fluid sur-faces (Faraday waves) is remarkable [1, 2, 3, 4]. As elu-cidated in various theoretical investigations [5, 6, 7] thisextreme variability is a result of the fact that the waveform of the vibration’s forcing function allows for a de-tailed tuning of various aspects of the interaction betweendifferent plane-wave modes, which can stabilize complexpatterns like super-lattice patterns and quasi-patterns.Motivated by this richness of patterns we are investigat-ing here the effect of time-periodic forcing with differentwave forms on systems undergoing a Hopf bifurcation tospatially homogeneous oscillations.In order to describe a forced Hopf bifurcation withina weakly nonlinear framework the forcing must be suffi-ciently weak. For it to have a qualitative effect on thesystem it must then include frequencies that are closeto one or more of the low-order resonances of the sys-tem, i.e. its spectrum has to contain frequencies close tothe Hopf frequency ω h itself (1:1-forcing), close to 2 ω h (2:1-forcing), or close to 3 ω h (3:1-forcing). These strongresonances lead to additional terms in the weakly non-linear description and qualitatively affect the system [8].To wit, in the weakly nonlinear regime the complex oscil-lation amplitude A satisfies a complex Ginzburg-Landauequation of the form dAdt = ( µ + iν ) A + (1 + iβ )∆ A − (1 + iα ) A | A | ++ γ ¯ A + η ¯ A + ζ. (1)The forcing terms ζ , γ ¯ A , and η ¯ A express the effect offorcing the system at the frequencies ω , 2 ω , and 3 ω , re-spectively, with ω = ω h + ν . The parameter µ expressesthe distance from the Hopf bifurcation, which is shiftedby a O ( | η | ) compared to the unforced case. Here wewill focus on the case ζ = 0. To include the forcing nearthe 1 : 1-resonance one can eliminate the inhomogeneousterm ζ by shifting A by the fixed-point solution A sat-isfying ζ = − ( µ + iν ) A + (1 + iα ) A | A | − γ ¯ A − η ¯ A and use A instead of ζ as an external parameter [9]. As is apparent from (1), the forced Hopf bifurcation isdescribed by an equation that is very similar to a two-component reaction-diffusion equation. The only andsignificant difference is the term involving β which char-acterizes the dispersion of unforced traveling wave so-lutions, which would be absent in the reaction-diffusioncontext. It plays, however, an essential role in excit-ing patterns with a characteristic wavenumber [10] andcannot be omitted (cf. (2) below). Pattern selection ina general two-component reaction-diffusion system hasbeen studied in detail by Judd and Silber [11], who findthat in principle not only stripe and hexagon patternscan be stable in such systems, but also super-square andsuper-hexagon patterns. They also find that despite thelarge number of parameters characterizing these systemssurprisingly only few, very special combinations of theparameters enter the equations determining the patternselection. Amplitude Equations.
In this paper we will staybelow the Hopf bifurcation taking µ <
0. Thus, as inFaraday systems, in the absence of forcing, no oscillationsarise. To investigate the weakly nonlinear stable stand-ing wave patterns possible in (1) we derive amplitudeequations for spatially periodic planforms. The linearstability of the state A = 0 is easily obtained by splittingthe equation and the amplitude A into real and imagi-nary parts ( A ≡ A r + iA i ). The usual Fourier ansatz A r,i ∝ e ikx yields then the neutral stability curve γ n ( k )with the basic state being unstable for γ > γ n ( k ). Theminimum γ c ( k ) of the neutral curve is found to be at k c = µ + νβ β , γ c = ( ν − µβ ) (1 + β ) . (2)Since µ <
0, the condition k c > k are closer to resonance than homogeneous oscillationswith k = 0 [10]. A typical neutral curve is illustrated inFigure 1 for µ = − ν = 4, β = 3 and ζ = 0 . The weaklynonlinear analysis presented in this paper is valid for val-ues of γ near γ c . The range of validity is restricted by γ n ( k = 0) where spatially homogeneous oscillations areexcited by the forcing, which interact with the standing-wave modes with wavenumber k c . k γ ( γ c ,k c ) Figure 1: Neutral stability curve for (1) with µ = − β = 3, ν = 4 and ζ = 0. The critical point ( k c, γ c ) = “q , √ ” ismarked by a red circle. To determine the stability of the various planforms wefirst determine the amplitude equations for rectangle pat-terns, which are comprised of two modes separated by anangle θ in Fourier space. We expand ( A r , A i ) as (cid:18) A r A i (cid:19) = ǫ X j =1 ,θ Z j ( T ) e i k j · r + c.c. (cid:18) v v (cid:19) + O ( ǫ ) , (3)where 0 < ǫ ≪ Z ( T ) and Z θ ( T ) depend on the slow time T = ǫ t . The wavevectorsare given by k = ( k c ,
0) and k θ = ( k c cos( θ ) , k c sin( θ )).We also expand γ as γ = γ c + ǫ γ .The usual expansion leads to the amplitude equationsfor ( Z , Z θ ), dZ dT = λ ( γ − γ c ) Z − (cid:0) b | Z | + b ( θ ) | Z θ | (cid:1) Z , (4) dZ θ dT = λ ( γ − γ c ) Z θ − (cid:0) b ( θ ) | Z | + b | Z θ | (cid:1) Z θ . (5)If θ = nπ , n ∈ Z , the quadratic nonlinearity inducesa secular term and the expansion has to include threemodes rotate by 120 ◦ relative to each other, (cid:18) A r A i (cid:19) = ǫ X j =1 Z j ( T ) e i k j · r + c.c. (cid:18) v v (cid:19) + O ( ǫ ) . (6)The parameters can be chosen such that the quadraticsolvability condition is delayed to cubic order, yielding dZ dT = λ ( γ − γ c ) Z + σ ¯ Z ¯ Z − (7) − (cid:0) b | Z | + b (cid:0) | Z | + | Z | (cid:1)(cid:1) Z , with similar equations for Z and Z . More complex patterns can be described by combiningthese two analyses. For example, a super-hexagon pat-tern comprised of two hexagon patterns { Z , Z , Z } and { Z , Z , Z } that are rotated relative to each other by anangle θ SH is described by the amplitude equation dZ dT = λ ( γ − γ c ) Z + σ ¯ Z ¯ Z −− (cid:0) b | Z | + b (cid:0) | Z | + | Z | (cid:1)(cid:1) Z − X j =0 b ( θ SH + j π | Z j | Z and corresponding equations for Z j , j = 2 , . . . , η ≡ η r + iη i : λ = p β | β | , σ = 2 p β ( aη r + η i ) β √ a , (8) b = 3 ψ + 769 χ, (9) b ( θ ) = 6 ψ + 8 f ( θ ) χ, (10) b = 6 ψ + 10 χ + φ (11)with ψ = − a ( α − β ) p β β (1 + a ) , (12) χ = − β ( ν − µβ )2( µ + νβ ) σ , (13) φ = 4(1 + β ) η i [( a + β ) η r − ( aβ − η i ] β (1 + a )( ν − µβ ) (14)and the angle dependence given by f ( θ ) = 3 + 16 cos θ (4 cos θ − . Here a = p β + β . The scaling of the nonlinear coef-ficients is based on a normalized eigenvector v = (cid:18) v v (cid:19) . Pattern Selection.
As shown by Judd and Silber [11]for general two-component reaction-diffusion systems, atthe point of degeneracy at which the quadratic coefficient σ and with it the coefficient χ vanishes not only stripepatterns but also hexagon or triangle patterns can bestable. The conditions for hexagons (or triangles) to bestable are φ < − φ < ψ < − φ . (16)Whether hexagon or triangle patterns are stable dependson higher-order terms in the amplitude equations [12],which are not considered here. For 0 < ψ < − φ ,hexagons are unstable to stripes. Both patterns bifur-cate unstably for ψ < α and β .Specifically, at the degeneracy one has η i = − aη r , imply-ing φ ( σ = 0) = − aη r (1 + β ) β ( ν − µβ ) < . (17)Here we have made use of the fact that β ( ν − µβ ) > µ < k c > α leads to stable hexagons, which is determined by the α − dependence of ψ . A distinguishing feature of thesehexagon patterns is that both ‘up’- and ‘down’-hexagonsare simultaneously stable and are likely to form compet-ing domains. Fig.2 shows an example of the competitionbetween ‘up’- and ‘down’-hexagons in a numerical simu-lation of (1).Unfolding the degeneracy, i.e. taking 0 < | σ | ≪ γ − range given by γ c − σ λ ( b + 2 b ) < γ < γ c + σ (2 b + b ) λ ( b − b ) ≡ γ HS . Note that γ HS > γ c , even if (15,16) are not satisfied,since stripes do not exist for γ < γ c . The instabilityof hexagons at γ HS only arises if b > b , that is, if3 ψ +14 χ/ φ > . With σ = 0 the up-down symmetry ofthe amplitude equations (7) is broken and, depending onthe sign of σ , either up- or down-hexagons are preferred.Turning to other planforms, Judd and Silber foundthat rectangular planforms cannot be stable at or nearthe degeneracy point [11]. Interestingly, however, theyfind that while super-hexagons cannot be stable at thedegeneracy point, they can arise in a very small param-eter regime in its vicinity if the conditions φ > , (18) − φ < ψ < φ φ in terms of the deviation σ from the degen-eracy condition, φ = − β ) aβ ( ν − µβ ) (cid:18) p β η i − ( a + β ) η i √ a σ (cid:19) , shows that - for small | σ | - φ can be made positive onlyby making η i small as well ( η i = O ( σ )). Even then φ can only be slightly positive, φ = O ( σ ) , requiring that ψ = O ( σ ) in order to satisfy (19). Under these conditions all cubic coefficients in (9,10,11) would become of O ( σ )and without knowledge of the next-order coefficients nostability predictions can be made.Away from the degeneracy, for σ = O (1), the above ar-guments suggest that it should be possible to satisfy thestability conditions (18,19). For σ = O (1) they are, how-ever, not the correct stability conditions since they ignorethe angle dependence of the cubic coefficients, which is O ( σ ). We use (18,19) therefore only as a guide to locateparameter regimes in which super-hexagons may be ex-pected to be stable and then determine the full stabilityeigenvalues. One such case is given by the linear pa-rameters used in Figure 1 with the nonlinear parameters α = − η = e iπ/ .Moreover, for σ = O (1) the weakly nonlinear analysispredicts also stable rectangle patterns satisfying | b ( θ ) | | β | " − a a p β (cid:18) − f ( θ ) (cid:19) χ . Within the hexagon sub-space, for | b /b | > γ is increased beyond γ HS .Satisfied simultaneously, the two conditions | b /b | > | b ( θ ) | < b would yield a situation in whichhexagons are unstable to stripes, which in turn are un-stable to rectangle patterns.Using direct numerical simulations of the forced com-plex Ginzburg-Landau equation (1) we have studied towhat extent the predictions of the weakly nonlinear anal-ysis are borne out. In the degenerate case σ = 0 we find,as predicted, either stripes or hexagons to be stable de-pending on the value of α chosen. A typical hexagonalpattern obtained from random initial condition is shownin Fig.2. As expected, it exhibits competing domains ofup- and down-hexagons. Figure 2: Competing domains of up- and down-hexagon do-mains starting from random initial conditions with linear pa-rameters as in Fig. 1 and nonlinear parameters α = − η r = 0 . η i = 0 . √
10 + 3). η r and η i are chosen so that σ = 0 . |A| ( | A | )( d | A | d t ) Numerical Simulation DataWeakly Nonlinear PredictionFP n FP Figure 3: Comparing amplitude from numerical simulationwith amplitude predicted by weakly nonlinear analysis. Sim-ulation parameters: µ = − ,ν = 4, β = 3, α = − η = e iπ/ . F P n corresponds to the fixed point obtained from numericalsimulation, F P to that obtained from the weakly nonlinearcalculation to cubic order. Away from the degeneracy, σ = O (1), the validity ofthe weakly nonlinear analysis can be severely restrictedby the fact that the amplitudes of all stable branches are O (1), which formally suggests the significance of higher-order terms in the expansion. Indeed, in and near theparameter regimes for which the weakly nonlinear analy-sis predicts stable super-hexagon patterns we do not findany indication of their stability. To assess explicitly thesignificance of the higher-order terms in the amplitudeequations for σ = O (1) we extract them directly fromthe numerical simulation for the case of hexagon pat-terns. Fig. 3 shows the numerically determined depen-dence of | A | − d | A | /dt on | A | for γ = γ c . For very small | A | it agrees well with the result obtained by the third-order weakly nonlinear theory, which yields the straightdashed line. However, even for γ = γ c the amplitude | A | saturates only at a value of | A | ≈ .
14 for which the third-order theory deviates significantly from the full result. Afit of | A | − d | A | /dt to a higher-order polynomial showsthat the magnitude of the quintic term in the amplitudeequation reaches a value of 20% of the cubic term. Thefixed point obtained from weakly nonlinear analysis tocubic truncation deviates from the numercially obtainedfixed point by 30%. This supports out interpretationthat in this regime the cubic amplitude equation does not allow quantitative predictions.In summary, we have investigated the regular spatialplanforms that can be stably excited in a sytem under-going a Hopf bifurcation by applying a periodic forcingfunction that resonates with the second and third har-monic of the Hopf frequency. We have done so withinthe weakly nonlinear regime by deriving the appropri-ate amplitude equations describing the selection betweenvarious planforms. By tuning the phase of the forcingclose to 3 ω h one can always reach the point of degen-eracy where no quadratic terms arise in the amplitudeequations, despite the quadratic interaction in the un-derlying complex Ginzburg-Landau equation. Hexagonsarise then in a supercritical pitchfork bifurcation. De-pending on the parameters of the unforced system wefind that in this regime either the hexagon patterns orstripe patterns can be stable. In the former case com-peting domains of up- and down-hexagons are found innumerical simulations when starting from random initialconditions. Surprisingly, despite the extensive controlafforded by the two forcing terms, no square, rectangle,or super-hexagon patterns are stable in the vicinity ofthis degeneracy, irrespective of the parameters of the un-forced system. Only in the regime in which hexagonsarise in a strongly transcritical bifurcation the weaklynonlinear theory predicts the possibility of stable rectan-gles or super-hexagons. There, however, direct numer-ical simulations of the complex Ginzburg-Landau equa-tion indicate no such stability and we show that terms ofhigher order in the amplitudes are relevant.By introducing a further forcing frequency, which isalso close to the 2:1-resonance, the transcritical bifur-cation to hexagons can be avoided. As we show in aseparate publication, the corresponding, more elaborateweakly nonlinear theory then correctly predicts stablequasi-patterns comprised of four, five, or more modes[13].For larger values of the forcing the spatially periodicstanding wave modes interact with a spatially homoge-neous oscillation that is also excited by the forcing. Theinteraction between these two types of modes could leadto interesting patterns, which are, however, beyond thescope of this paper.We gratefully acknowledge support by NSF throughgrant DMS-0309657. [1] B. Christiansen, P. Alstrom, and M. T. Levinsen, Phys.Rev. Lett. , 2157 (1992).[2] W. S. Edwards and S. Fauve, Phys. Rev. E , R788(1993).[3] A. Kudrolli, B. Pier, and J. Gollub, Physica D , 99(1998).[4] H. Arbell and J. Fineberg, Phys. Rev. E , 036224(2002).[5] W. Zhang and J. Vinals, Phys. Rev. E , R4283 (1996). [6] M. Silber, C. M. Topaz, and A. C. Skeldon, Physica D , 205 (2000).[7] A. M. Rucklidge and M. Silber, Phys. Rev. E ,055203(R) (2007).[8] P. Coullet and K. Emilsson, Physica D , 119 (1992).[9] J. Conway and H. Riecke (unpublished).[10] P. Coullet, T. Frisch, and G. Sonnino, Phys. Rev. E ,2087 (1994).[11] S. L. Judd and M. Silber, Physica D , 45 (2000). [12] M. Silber and M.R.E. Proctor, Phys. Rev. Lett.81