[Abstract]: Generalized nonlinear phase diffusion equation describes oscillators weakly coupled by diffusion. The equation generally contains infinite number of terms and allows a variety of dynamic balances between them. We consider a truncated version of the equation in which nonlinear excitation drives the dynamics. A group of active systems leading to this truncation is modelled by reaction-diffusion equations with effective nonlocal coupling. We formulate the conditions on the parameters resulting in the truncation and discuss numerical experiments showing complex spatio-temporal behaviour

Strunin, Dmitry V.

English

University of Southern Queensland ePrints

Proceedings of the 5th Asian Mathematical Conference, Malaysia 2009A NEW CASE OF TRUNCATED PHASEEQUATION FOR COUPLED OSCILLATORS1 D.V. Strunin1 Department of Mathematics and Computing,University of Southern Queensland, Toowoomba, QLD 4350, Australiae-mail: strunin@usq.edu.auAbstract. Generalized nonlinear phase diffusion equation describes oscillators weakly coupledby diffusion. The equation generally contains infinite number of terms and allows a varietyof dynamic balances between them. We consider a truncated version of the equation in whichnonlinear excitation drives the dynamics. A group of active systems leading to this truncationis modelled by reaction-diffusion equations with effective nonlocal coupling. We formulate theconditions on the parameters resulting in the truncation and discuss numerical experimentsshowing complex spatio-temporal behaviour.1 IntroductionUnder certain conditions reaction-diffusion systems can be reduced to oscillators weakly coupled bydiffusion. For the phase of oscillations Kuramoto and Tsuzuki [1, 2] derived the equation∂tψ = a1∇2ψ + a2(∇ψ)2+b1∇4ψ + b2∇3ψ∇ψ + b3(∇2ψ)2 + b4∇2ψ(∇ψ)2 + b5(∇ψ)4+e1∇6ψ + . . . ,(1)where an, bn, en, . . . are constant coefficients. The right-hand side of (1) can be regarded as a powerseries in small parameter ∇2 ∼ (1/L)2, where L is the large spatial scale of variation of ψ. Equation (1)can be truncated to finite forms based on different balances between the terms. In this paper we outline aprocedure by which a new kind of truncation is obtained for the reaction-diffusion systems with nonlocalcoupling.Tanaka and Kuramoto [3] analysed the system with effective nonlocal coupling in the form∂tX = f(X) + δˆ∇2X+ kg(S) , (2)τ∂tS = −S +D∇2S + h(X) . (3)Here X is the vector representing concentrations of reactants, δˆ is a diagonal matrix responsible fordiffusion, f and g are the vector functions, k, τ and D are constants. Implicitly system (2)–(3) involves abifurcation parameter µ which, when changing from µ < 0 to µ > 0, brings about oscillatory instability.Note that equation (3) is linear; this allows to eliminate S by expressing it in terms of X and thensubstituting into (2). Provided that the parameter k is small, k ∼ O(|µ|), this results in the complexGinzburg-Landau equation with effective nonlocal coupling [3]∂tA = µσA− β|A|2A+ δ∇2A+ kη ′∫dr′G(r− r′)A(r′, t) , (4)where A is (loosely speaking) proportional to X; σ, β, δ and η′ are parameters. It is assumed thatReσ > 0 . (5)The coupling function G satisfies the normalization condition∫G(r) dr = 1 . (6)The complex parameter δ is responsible for diffusion and, therefore, has positive real part,Re δ > 0 . (7)Using (6), it is convenient to write (4) in the form∂tA = µσ′A− β|A|2A+ δ∇2A+ kη ′∫dr′G(r − r′)[A(r′, t)−A(r, t)] , (8)whereσ′ = σ + kη ′/µ . (9)It is assumed that the system is supercritical, that isReσ′ > 0 . (10)Some parameters in (8) can be eliminated by changing variables. The imaginary part of the coefficientµσ′ vanishes after transforming A→ A exp[iµ Imσ′t] and the diffusion coefficient D as well as Re β andµReσ′ become unity by rescaling A, t and r. Eventually (8) is transformed to∂tA = A− (1 + ic2)|A|2A+ (δ1 + iδ2)∇2A+K(1 + ic1)∫dr′G(r− r′)[A(r′, t)−A(r, t)] ,(11)where c1, c2, δ1, δ2 and K are real constants. Condition (7) is equivalent toδ1 > 0 . (12)The complex amplitude A is connected to the real phase of oscillations, ϕ, viaA = ae−iϕ , (13)where a is the real amplitude. In one-dimensional caseG(x) =12(ζ + iη)e−(ζ+iη)|x| , (14)whereζ =(1 +√1 + θ22)1/2, η =(−1 +√1 + θ22)1/2. (15)Here θ = ω0τ is the parameter proportional to the basic frequency of oscillations, ω0, and characteristictime τ . As for the phase, it is convenient to analyse its departure from c2t, defined asϕ = c2t+ ψ . (16)It can be shown that the departure ψ satisfies equation (1). It is derived from (11) via the phase reductionprocedure using (13). In this procedure the integral is decomposed in a power series in ∇. Each coefficientin equation (1) turns out to be a combination of the independent parameters δ1, δ2, c1, c2, K and θ.It is important to note that the parameter K is limited in magnitude. When transferring from (8) to(11), the factor K(1 + ic1) appears as the combinationK(1 + ic1) =kη ′/µReσ′.Taking real part and using (5), (9) and (10), we haveK =Re(kη ′/µ)Reσ′=Reσ′ − ReσReσ′= 1− ReσReσ′< 1 . (17)All the results above are obtained by Tanaka and Kuramoto [3]. Tanaka also showed [4] that, if the valuesof the independent parameters are properly chosen, equation (11) reduces to the Nikolaevskii equation [5]∂tψ = a1∇2ψ + b1∇4ψ + e1∇6ψ + a2(∇ψ)2 , (18)with b1 > 0 ensuring that the fourth derivative provides excitation. Observe that the excitation term islinear.In this paper we show that it is possible to truncate (1) to a form where the excitation is nonlinear.Previously we designed such kind of equation using phenomenological arguments in order to simulateunstable combustion fronts [6],∂tψ = −ε(∇ψ)2∇2ψ + b5(∇ψ)4 + e1∇6ψ . (19)In [7] we briefly discussed (19) in the context of the diffusion-coupled oscillators. In (19) the excitationis provided the first term in the right-hand side. The term can be regarded as the negative diffusion,−∇2ψ, with the positive nonlinear coefficient ε(∇ψ)2.The model (19) becomes a truncation of (1) if the following six conditions are met,a1 = a2 = b1 = 0 ,b2 = b3 = 0 , b4 = −ε .(20)It is easy to show that, given ε is small, the balance between the terms of (19) leads to smallness of ψand of ∇ ∼ 1/L. Consequently, the rest of the terms in (1) shown by dots are negligible because theyare of higher order either in ψ or ∇ or both.As we mentioned, equation (11) contains the six independent parameters and one may expect that,for some of their combinations, the six conditions (20) can be met. However, our recent research indicatesthat more independent parameters may be necessary to satisfy (20). Therefore, we add another reactantinto (2)–(3),∂tX = f(X) + δˆ∇2X+ k1g1(S1) + k2g2(S2) , (21)τ1∂tS1 = −S1 +D∇2S1 + h1(X) , (22)τ2∂tS2 = −S2 +D∇2S2 + h2(X) , (23)where k1 ∼ k2 ∼ O(|µ|). The respective nonlocal Ginzburg-Landau equation has the form∂tA = µσA− β|A|2A+ δ∇2A+k1η1′∫dr′G1(r− r′)A(r′, t) + k2η2 ′∫dr′G2(r− r′)A(r′, t) , (24)where each Gn carries its own θn, n = 1, 2. Now we have the 9 independent parameters at our disposal,namely δ1, δ2, c1, c2, c3, K1, K2, θ1 and θ2. The coupling functions satisfy the normalization conditions∫G1(r) dr =∫G2(r) dr = 1 .Using these conditions we modify (24) to∂tA = µσ′A− β|A|2A+ δ∇2A+k1η1′∫dr′G1(r− r′)[A(r′, t)−A(r, t)]+k2η2′∫dr′G2(r− r′)[A(r′, t)−A(r, t)] ,(25)whereσ′ = σ + k1η1′/µ+ k2η2′/µ . (26)Rescaling (25) in the similar way to (8), we obtain∂tA = A− (1 + ic2)|A|2A+ (δ1 + iδ2)∇2A+K1(1 + ic1)∫dr′G1(r− r′)[A(r′)−A(r)]+K2(1 + ic3)∫dr′G2(r− r′)[A(r′)−A(r)] ,(27)whereK1(1 + ic1) =k1η1′/µReσ′, K2(1 + ic3) =k2η2′/µReσ′. (28)Now we can formulate restrictive conditions on K1 and K2 to replace (17). Taking real parts in(28) and summing up, we getK1 +K2 =Re(k1η1′/µ) + Re(k2η2′/µ)Reσ′and, using (26), (5) and (10),K1 +K2 =Reσ′ − ReσReσ′= 1− ReσReσ′< 1 . (29)Another restriction to be met is positiveness of the coefficient at ∇6ψ in (19). It is needed to ensure thatthe term is dissipative,e1 > 0 . (30)Having the 9 independent parameters we have been able to satisfy the 6 conditions (20) subject to therestrictions (12), (29) and (30). We intend to report full details of this work elsewhere.2 Numerical experimentsIn this section we present some numerical simulations with the two-dimensional version of (19). Thecomputations were performed in the square region, 0 < x < 2, 0 < y < 2 with b4 = −10, b5 = e1 = 1. InCartesian coordinates∂tψ =b4 (∂2xψ + ∂2yψ)[(∂xψ)2 + (∂yψ)2]+b5[(∂xψ)4 + 2(∂xψ)2 (∂yψ)2 + (∂yψ)4]+e1(∂6xψ + 3 ∂4x∂2yψ + 3 ∂2x∂4yψ + ∂6yψ).(31)The boundary conditions were chosen arbitrarily,∂xψ = 0 , ∂2xψ = 0 , ∂3xψ = 0 at x = 0 and x = 2 ,∂yψ = 0 , ∂2yψ = 0 , ∂3yψ = 0 at y = 0 and y = 2 .(32)Equation (31) was discretized in space on a uniform grid with the step 0.025 using 2nd-order accuratecentral differences and a 7×7 point stencil. Boundary conditions (32) were discretized using a fictitiouspoint approach with 3 additional fictitious layers of discretization points introduced outside of each of theboundaries. Then formally the physical boundary points are treated as interior. The boundary conditions(32) are then approximated using 2nd-order central finite differences and the resulting equations are usedto eliminate the fictitious points in favour of the interior points adjacent to the boundary. An explicit1st-order accurate forward Euler method was used to discretize in time. Comparing the nonstationaryterm and dissipation, we see that the time step must be as small as κ∆x6, where ∆x is the spatial stepand κ is some factor. The experiments showed that κ should not exceed 0.003.As an initial condition we placed a narrow hump in the corner (x, y) = (2, 2). Snapshots of thesolution are presented in Fig. 1. The phase field is curved at all times and exhibits seemingly irregularbehaviour. Periods of slow evolution dominated by the dissipation intermit with surges of activity drivenby the nonlinear excitation. The computations were performed up to T ≈ 5.9 × 10−5. A closer look atthe topography reveals more or less discernible step-like structures which propagate in normal directionto the average motion of the field (upwards in the Figure).It is interesting to compare characteristic time scales: the discretization step, ∆t, the typical timescale of the structures, ∆t1, and the total duration of the experiment, T . As we mentioned above,∆t = 0.003∆x6 = 0.003 · 0.0256 ∼ 7 · 10−13. For the step-like structures roughly ∆t1 ∼ ℓ 6/e1, where ℓis their typical length and e1 = 1. Observe from the snapshot at t = 1.383 · 10−5, that a step occupiesabout 5 to 6 grid cells, giving ℓ ∼ (5 to 6) · 0.0256. Thus, ∆t1 ∼ ℓ 6/c ∼ (0.38 to 1.1) · 10−5 and we have∆t ∼ 10−12 , ∆t1 ∼ (0.38 to 1.1) · 10−5 , T ∼ 5.9 · 10−5 .The time ∆t1 exceeds the numerical step ∆t by several orders, and the experiment duration T is largerFigure 1: The phase field at different momentsstill by approximately an order. Therefore, the numerical time step was sufficiently short to resolve themotion of the structures and the experiment was sufficiently long to embrace this motion. The repeatingbursts of the phase activity hint that the dynamics would last infinitely long because of the persistentaction of the source.3 ConclusionsA set of oscillators weakly coupled by diffusion are generally described by the infinite nonlinear phaseequation (1). Omitting full technical details in this short paper we outlined a procedure by which we haveobtained a truncated version of this equation with nonlinear excitation. Numerical experiments revealseemingly random dynamics.Acknowledgement The author is grateful to S. Suslov for the help with the numerical simulations.References[1] Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative mediafar from thermal equilibrium, Prog. Theor. Phys., Vol. 55, 1976, p. 356.[2] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Berlin: Springer–Verlag, 1984.[3] D. Tanaka and Y. Kuramoto, Complex Ginzburg-Landau equation with nonlocal coupling, Phys.Rev. E, Vol. 68, 2003, 026219.[4] D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence, Phys. Rev. E, Vol. 70, 2004,015202(R).[5] V.N. Nikolaevskii, in Recent Advances in Engineering Science, edited by S.L. Koh and C.G. Speciale,Lecture Notes in Engineering, Vol. 39 (Berlin: Springer, 1989), p. 210.[6] D.V. Strunin, Autosoliton model of the spinning fronts of reaction. IMA J. Appl. Math., Vol. 63,1999, p. 163.[7] D.V. Strunin, Nonlinear instability in generalized nonlinear phase diffusion equation. Progr. Theor.Phys. Suppl., No. 150, 2003, p. 444.

Autosoliton model of the spinning fronts of reaction.

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Chemical turbulence equivalent to Nikolaevskii turbulence,

Complex Ginzburg-Landau equation with nonlocal coupling,

in Recent Advances in Engineering Science, edited by S.L.

Nonlinear instability in generalized nonlinear phase diﬀusion equation.

Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,

A new case of truncated phase equation for coupled oscillators

http://eprints.usq.edu.au/7169/1/Strunin_AMC2009_AV.pdf

A new case of truncated phase equation for coupled oscillators

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