Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation

Transcription

1 Range of validit and intermittent dnamics of the phase of oscillators with nonlinear self-ecitation D.V. Strnin 1 and M.G. Mohammed Comptational Engineering and Science Research Centre, Faclt of Health, Engineering and Sciences, Universit of Sothern Qeensland, Toowoomba, QLD 43, Astralia Abstract A range of active sstems, particlarl of chemical natre, are known to perform self-ecited oscillations copled b diffsion. The role of the diffsion is not trivial so that the differences in the phase of the oscillations throgh space ma persist, depending on the vales of the controlling parameters of the sstem. Firstl, we analse a 6th-order nonlinear partial differential eqation describing sch dnamics. We evalate the range of the parameters leading to different finite versions of the eqation, specificall a version based on nonlinear ecitation and a version based on linear ecitation. In the second part of the work we solve the eqation in two spatial dimensions b finite-difference discretization in space and sbseqent nmerical integration of a sstem of ordinar differential eqations in time. A forced variant of the eqation is derived and selected eact soltions are presented. The are also sed to verif the nmerical code. For the nforced eqation, irreglar dnamics intermitting with periods of slow evoltion are recorded and discssed. Ke words: Active dissipative sstem, nonlinear ecitation, parametric space, irreglar dnamics 1 Introdction In this work we analse oscillators representing a class of dissipative active sstems with selfecitation. As eamples of sch sstems mention fronts of gasless combstion [1, 2], certain 1 corresponding athor, strnin@sq.ed.a, tel.: , fa:

2 tpe of reaction-diffsion sstems [3, 4, ] and seismic waves in flid-satrated rocks [6]. Intitivel, diffsion (or thermal condctivit) shold smooth ot the differences of the phase of the oscillations in space. However, as is now well-nderstood [1, 3], the combined effect of the diffsion and self-ecitation ma prodce complicated dnamics, dring which the difference ma persist as time goes. Earlier we derived [7, 8, 9] a form of sch an eqation based on nonlinear self-ecitation, t = A 2 ( ) 2 + B( ) 4 + C 6, (1) where A > and C >. The former condition garantees that the term, A 2 ( ) 2, acts as a nonlinear ecitation (the anti-diffsion, D 2, with the nonlinear diffsion coefficient, D = A( ) 2 ), and the latter condition ensres dissipative effect of the term C 6. We will refer to (1) as the nonlinearl ecited phase (nep) eqation. Clearl, (1) has the trivial, spatiall-niform soltion = const. It is stable to small pertrbations since the linearized form of the eqation is dissipative to all wavelengths, t = C 6. Therefore, in order to kick-start self-sstaining dnamics one needs a sfficientl strong initial pertrbation. If, at some stage of the dnamics, the srface (,, t) flattens ot to significant etent, the motion will deca forever. Previosl we solved eqation (1) in 1D nmericall, sing Galerkin method, nder periodic bondar conditions [7]. A settled regime was obtained, in which a kink-shaped wave moves along a spiral trajector on the srface of a clinder (see graph (b)). Sch a regime profondl resembles an eperimentall observed spinning combstion wave [2], illstrated on the photo. It is important that it is not jst the correct shape a kink of the wave that is reprodced bt also the dnamic mechanism behind it. In a laborator, the spinning combstion occrs when the reacting chemical composition is dilted with some netral admitre. As a conseqence, it becomes difficlt for the combstion process to maintain itself (it actall occrs on the brink of collapse). As a reslt, the front has to form a cavit 2

3 (a) A post-combstion trace left b the (b) A rnning spinning wave soltion of Eq. (1) spinning solid flame on a hollow clinder; (five sccessive shapes) evolved from a cortes of A.G. Strnina (see also similar randoml chosen initial condition [7]. photos and discssions in [2, ]). Figre 1: Comparison of the eperiment and simlation. (kink) of certain size and moving with certain velocit, where the cold nbrned composition is srronded from two sides b the hot reaction prodcts. This is the onl wa the combstion can srvive nder the difficlt conditions. For the 1D topolog we did not investigate whether, apart from the periodic waves, there ma also realise irreglar regimes. Perhaps, those are possible at large diameters of the clinder, when there is enogh space for a nmber of kinks to co-eist interact with each other in a complicated wa. For the 2D topolog, that is with an etra dimension available, we have a strong anticipation that complicated, possibl chaotic, dnamics ma indeed form and self-sstain provided the size of the space domain is sfficientl large. Another area of application of the eqation (1) are reaction-diffsion sstems, in particlar those ehibiting nonlocal effects. For sch sstems eqation (1) was derived [9] b a rigoros procedre as a trncated version of the following infinite (also referred to as generalized) phase diffsion eqation, t = a a 2 ( ) 2 + b b b 3 ( 2 ) 2 + b 4 2 ( ) 2 + b ( ) 4 + (2) g g 2 + g g 4 ( 3 ) 2 + g 4 ( )

4 Here denotes the phase of oscillations, and a n, b n, g n, e n,... are constant coefficients. The right-hand side of (2) is virtall a Talor series in small parameter 2 (1/L) 2, with L standing for the tpical length scale, presmed large. There ma be different kinds of balance between the terms in (2) depending on the magnitdes and signs of the coefficients. For eample, when a 1 > and the initial field (,, ) is sfficientl smooth, the diffsion term dominates dring the entire period of evoltion. The eqation is effectivel redced to the linear diffsion eqation t = a 1 2. (3) In the case a 1 = ε < (assmed small) and, b the order of magnitde, a 2 = 1, b 1 = 1, eqation (2) redces to the Kramoto Sivashinsk (KS) eqation [1, 3], t = ε 2 + ( ) 2 4. (4) The KS eqation contains the linear anti-diffsion term, ε 2, which represents ecitation; it is conterbalanced b the dissipative term, 4. Taking into accont smallness of ε, it is straightforward to show that the scales of and L reslting from the balance are sch that the rest of the terms in (2) are negligible compared to the three balancing terms in (4). In a similar wa eqation (2) effectivel redces to the finite form (1) when b 4 = ε and, b the order, b = 1 and g 1 = 1 and, additionall, all the lower-order terms in the right-hand side of (2) are negligibl small. To meet this latter condition, the five coefficients a 1, a 2, b 1, b 2 and b 3 (alongside with the coefficient b 4 ) mst be small. Denoting the characteristic scale of the phase variations b U > we evalate in absolte vale: 2 ( ) 2 U 3 /L 4, ( ) 4 U 4 /L 4 and 6 U/L 6. The balance between the three terms of (1), εu 3 /L 4 U 4 /L 4 U/L 6, () governs the scales of the dissipative strctres, U ε, L (1/ε) 3/2. (6) The smallness condition for the five mentioned coefficients ma indeed realize in certain sstems and, as we show in the net section, is not a rare occasion. An eample is the following sstem [9], t X = f(x) + δ 2 X + k 1 g 1 (S 1 ) + k 2 g 2 (S 2 ), (7) 4

5 τ 1 t S 1 = S 1 + D 2 S 1 + h 1 (X), (8) τ 2 t S 2 = S 2 + D 2 S 2 + h 2 (X). (9) Here δ, k 1, k 2, τ 1, τ 2 and D are constants, and X, S 1 and S 2 represent the concentrations of reactants. This sstem is relevant to certain tpe of bio-sstems [4]. Eqations (7) (9) leads to a Ginzbrg-Landa (GL) eqation for the comple amplitde A measre of the concentrations with two nonlocal terms, +k 1 η 1 t A = µσa β A 2 A + δ 2 A dr G 1 (r r )A(r, t) + k 2 η 2 dr G 2 (r r )A(r, t), () where G n are copling fnctions reslting from the presence of chemicals S 1 and S 2. In one dimension, G n () = 1 2 (ζ n + iη n )e (ζn+iηn), n = 1, 2, (11) with ζ n = ( θ 2 n 2D ) 1/2, η n = ( θ 2 n 2D ) 1/2. (12) All the new parameters appearing in () (12) are constants. Rescaling () (we refer to [4] and [9] for details) leads to t A = A (1 + ic 2 ) A 2 A + (δ 1 + iδ 2 ) 2 A +K 1 (1 + ic 11 ) dr G 1 (r r )[A(r ) A(r)] +K 2 (1 + ic 12 ) dr G 2 (r r )[A(r ) A(r)]. (13) Eqation (13) contains 9 independent parameters: δ 1, δ 2, c 11, c 12, c 2, K 1, K 2, θ 1 and θ 2. The comple amplitde A is connected to the real-valed amplitde, a, and real-valed phase of the oscillations, ϕ, via A = ae iϕ. (14) Sbstitting (14) into the Ginzbrg-Landa eqation (13) and separating real and imaginar parts, we obtain t a = a a 3 + δ 1 2 a δ 1 a( ϕ) 2 + 2δ 2 a ϕ + δ 2 a 2 ϕ + Re I. (1)

6 where t ϕ = c 2 a 2 a ϕ + 2δ 1 + δ a ϕ δ 2 a a + δ 2( ϕ) 2 1 Im I, (16) a and I denotes the nonlocal terms in (13). I = I, (17) e iϕ The real amplitde, a, tends towards 1, driven b t a a a 3. Note that the terms following a a 3 in (1) are relativel small de to the slow variations in space. Despite being small, the make the amplitde a deviate from 1. At the same time the phase increases at an approimatel constant rate as ϕ = c 2 t, becase the amplitde is approimatel driven b t ϕ c 2 a 2 c 2. However, being pertrbed b the rest of the terms in (16), the phase deviates from c 2 t. We define the phase deviation,, via ϕ = c 2 t +. (18) For the slow spatial variations nder consideration, 1/L ε 1 is small. To provide consistenc with (6) we state ε 1 = ε 3/2. (19) This relation connects the small parameter ε (or choice), representing the coefficient b 4, with the size of the formed dissipative strctres, ε 1 (the conseqence of this choice). Ths, the variations of the amplitde and phase are slow becase the coefficient b 4 is small. We came to a tpical centre manifold sitation in which there is a fast variable, a, and a slow variable, ϕ. The centre manifold theor states that there eists a manifold to which the dnamics are attracted eponentiall qickl, a = a[ ϕ]. () Eqation () manifests a stiff connection between the amplitde and phase on the centre manifold. This link makes it possible to eliminate the amplitde from (16) and obtain a closed eqation for the phase. It is convenient to rescale the variables, t 1 = ε 2 1t and 1 = ε 1, so that We epand the amplitde into the series in ε 1, = ε 1 1, t = ε 2 1 t1. (21) a = 1 + ε 2 1a 2 + ε 4 1a 4 + ε 6 1a (22) 6

7 Using (21), (18) and (22) in the phase eqation (16) and the amplitde eqation (1), we derive, for the phase departre introdced in (18), ε 2 1 t1 = 2c 2 ε 2 1 a 2 + 2c 2 ε 4 1 a 4 + c 2 ε 4 1 a c 2 ε 6 1 a 6 +2δ 1 ε a δ 1 ε δ 2 ε a 2 δ 2 ε a 4 (23) +δ 2 ε 2 1 ( 1 ) 2 1 a Im I +... and, for the amplitde, ( ( ) ε 4 1 t1 a 2 + ε 6 1 t1 a 4 + = 2ε 2 1a 2 ε 4 1 2a4 + 3a2) 2 ε 6 1 2a6 + 6a 2 a 4 + a 3 2 ( +δ 1 ε 2 1 ε a 2 + ε a ) ( δ ε 2 1 a 2 + ε 4 1a ) ε 2 1( 1 ) 2 +2δ 2 ε 1 ( ε a 2 + ε a ) ε 1 1 (24) +δ 2 ( 1 + ε 2 1 a 2 + ε 4 1a ) ε Re I. Collecting terms ε 2 1 in (24) we obtain = 2a 2 δ 1 ( 1 ) 2 + δ (Re I) 2, (2) where (Re I) 2 denotes the coefficient at ε 2 1 in the ε 1 -series for Re I. After the necessar maniplations sing the epression I (we refer to [9] for details) we obtain a 2 in terms of 1 in the form a 2 = α α 2 ( 1 ) 2. Similarl, we find other a n in terms of 1, sbstitte them into (23) and, after doing necessar algebra, eventall arrive at the phase eqation of the form (2). We present it below in two parts. First we give the phase eqation following from the Ginzbrg-Landa eqation with onl one nonlocal term. Then we eplain how it shold be transformed in 7

8 order to represent the case with two nonlocal terms. t = 2 [2c 2 α 1 + δ 1 + K4 (ζ c 1η)C 2 K4 ] (c 1ζ + η)s 2 [ +( ) 2 2c 2 α 2 + δ 2 + K 4 (c 1ζ + η)c 2 K ] 4 (ζ c 1η)S 2 [ + 4 2c 2 β δ 2 α 1 + K 48 (η + c 1ζ)S 4 + K 48 (ζ c 1η)C 4 K 4 (η + c 1ζ)C 2 α 1 + K ] 4 (ζ c 1η)S 2 α 1 [ + 3 2c 2 β 2 + 2δ 1 α 1 + 2δ 2 α 2 K 12 (ζ c 1η)S 4 + K 12 (η + c 1ζ)C 4 K 2 (ζ c 1η)S 2 α 2 + K ] 2 (η + c 1ζ)C 2 α 2 + ( 2 ) [ 2 2c 2 β 1 + 2c 2 α δ 2 α 2 K 16 (ζ c 1η)S 4 + K 16 (η + c 1ζ)C 4 K 2 (ζ c 1η)S 2 α 2 + K ] 2 (η + c 1ζ)C 2 α ( ) 2 [ 2c 2 β 3 4c 2 α 1 α 2 4δ 1 α 2 K 8 (ζ c 1η)C 4 K 8 (η + c 1ζ)S 4 ] [ +( ) 4 2c 2 β 4 + 2c 2 α2 2 + K 48 (ζ c 1η)S 4 K ] 48 (η + c 1ζ)C 4 [ ( c 2 2 δ 1β + K 8 (ζ c 1η)C 2 β + K 4 4! (ζ c 1η)C 4 α 1 K 4 6! (ζ c 1η)S 6 + K 4 6! (c 1ζ + η)c 6 + K 8 (c 1ζ + η)s 2 β + K 4 4! (c 1ζ + η)s 4 α 1 ) βδ 2 K 2 (ζ c 1η) ( 16! C 6 12 S 2β 14! ) S 4α 1 K ( 1 2 (c 1ζ + η) 2 C 2β + 1 4! C 4α 1 1 )] 6! S , (26) where 2Γ(n + 1) C n (ζ, η) = cos [(n + 1) arctan(η/ζ)], (ζ 2 + η 2 (n+1)/2 ) 2Γ(n + 1) S n (ζ, η) = sin [(n + 1) arctan(η/ζ)], (ζ 2 + η 2 (n+1)/2 ) (27) 8

9 and α 1 = δ 2 2 K 8 S 2(ζ c 1 η) + K 8 C 2(c 1 ζ + η), α 2 = δ K 8 C 2(ζ c 1 η) + K 8 S 2(c 1 ζ + η), (28) β = α 1 δ K 8 (ζ c 1η)C 2 α 1 + K 8 (η + c 1ζ)S 2 α 1 K 4 24 (ζ c 1η)S 4 + K 4 24 (η + c 1ζ)C 4, β 1 = 3 2 α2 1 δ 1 α 2 + δ 2 2 α 1 + K 32 (ζ c 1η)C 4 K 32 (η + c 1ζ)S 4 K 4 (ζ c 1η)C 2 α 2 K 4 (η + c 1ζ)S 2 α 2 K 8 (ζ c 1η)S 2 α 1 + K 8 (η + c 1ζ)C 2 α 1, β 2 = δ 1 α 2 + δ 2 α 1 K 24 (ζ c 1η)C 4 K 24 (η + c 1ζ)S 4 (29) K 4 (ζ c 1η)C 2 α 2 K 4 (η + c 1ζ)S 2 α 2, β 3 = 3α 1 α 2 δ 1 2 α 1 δ 2 2 α 2 K 8 (ζ c 1η)C 2 α 1 K 8 (η + c 1ζ)S 2 α 1 + K 16 (ζ c 1η)S 4 K 16 (η + c 1ζ)C 4 + K 8 (ζ c 1η)S 2 α 2 K 8 (η + c 1ζ)C 2 α 2, β 4 = 3 2 α2 2 + K 4 24 (ζ c 1η)C 4 + K 4 24 (η + c 1ζ)S 4 + K 8 (ζ c 1η)C 2 α 2 + K 8 (η + c 1ζ)S 2 α 2 + δ 1 2 α 2. Observe that the coefficients a 1, a 2, b 1, b 2, b 3, b 4, b and g 1 are combinations of the independent parameters c 1, c 2, K, δ 1, δ 2 and θ (the latter is presented via ζ and η, see (12)). For the GL eqation with two nonlocal terms, (13), which is in or focs, the phase eqation is obtained b replacing K(ζ c 1 η)s n K 1 (ζ 1 c 11 η 1 )S n1 + K 2 (ζ 2 c 12 η 2 )S n2, 9 ()

10 K(η + c 1 ζ)s n K 1 (η 1 + c 11 ζ 1 )S n1 + K 2 (η 2 + c 12 ζ 2 )S n2, K(ζ c 1 η)c n K 1 (ζ 1 c 11 η 1 )C n1 + K 2 (ζ 2 c 12 η 2 )C n2, K(η + c 1 ζ)c n K 1 (η 1 + c 11 ζ 1 )C n1 + K 2 (η 2 + c 12 ζ 2 )C n2 (n = 2, 4, 6) in (26) and also in the epressions for α 1 and α 2, (28), and for β, β 1, β 2, β 3 and β 4, (29) (). It was shown [9, 4] that the parameters K 1 and K 2 mst satisf the restriction K 1 + K 2 < 1. (31) Another important condition that mst be met is dissipative natre of the term g 1 6 ψ, hence g 1 >. (32) As we noted, eqation (1) becomes a valid redced form of (2) when b 4 = ε, b = 1 and g 1 = 1. In addition, all the lower-order terms preceding these three in the right-hand side of (2) mst be negligible, which is achieved b assming that the coefficients a 1, a 2, b 1, b 2 and b 3 are sfficientl small or even zero. Ths, these coefficients alongside with the coefficient b 4 mst satisf the conditions of smallness a total of 6 conditions. We want to find ot how narrow/wide is the area in parametric space of independent parameters within which (2) redces to (1). We aim to determine a possibl larger piece of this area and, ot of criosit, compare it with the area leading to the KS eqation. This is done in Section 2. Or second aim is to solve eqation (1) nmericall in two spatial dimensions in order to see whether complicated/irreglar regimes ma settle with time and what strctre the have. We perform this in Section 3. Section 4 contains the conclsions. 2 Range of validit: nmerical reslts As an initial step we will demonstrate the eistence of the vales of the 9 independent parameters, θ 1, θ 2, c 12, c 11, c 2, K 1, K 2, δ 1 and δ 2, sch that the 6 conditions of smallness are satisfied, namel that b 4 = ε (small nmber) and also a 1, a 2, b 1, b 2 and b 3 are sfficientl small or zero [9]. Additionall, the restrictions g 1 >, δ 1 >, K 1 +K 2 < 1 mst be satisfied.

11 Or nmerical program, written in redce, solves the eqations stating that the coefficients a 1, a 2, b 1, b 2 and b 3 are eqal to zero eactl. It is eas to verif that the softer conditions of the coefficients being slightl awa from zero can alwas be achieved via tin variations of the independent parameters from their vales giving eactl zero vales to the coefficients. Or program also comptes the vale of the parameter g 1 to see whether it satisfies the reqirement g 1 >. The program works better when a small vale b 4 = ε is arranged in the form of a prodct of one of the nknown parameters, for eample K 1, and a small nmber: b 4 = K 1.1. For the inpt vales θ 1 = 1, θ 2 = 2, c 12 = 1 the compted otpt (in ronded form) is δ 1 = 27, K 1 = 49, K 2 = 89 and g 1 = 33. See that all the restrictions are met, namel g 1 >, δ 1 >, K 1 + K 2 < 1. (33) Clearl we can make b 4 as close to zero as we wish. Ths, for the above vales of the independent parameters the nep eqation is a valid trncation of the phase eqation (2). Obviosl, once there eists one valid point, it mst be srronded b a clod of other valid points. We aim to determine a possibl wider range of the vales of the parameters making the nep eqation valid. Similarl to the nmerical eample above, we eecte the following procedre. We arbitraril assign vales to the 3 independent parameters θ 1, θ 2 and c 12. Using the program, we compte the vales of the other 6 parameters from the list of 9. Finall, we inspect whether all the restrictions are met, and, based on the otcome, make a conclsion abot whether or not a particlar point in the 3D space (θ 1, θ 2, c 12 ) is a one where the nep eqation is valid. If at least one of restrictions (33) is violated, the nep eqation is not valid. Approimatel 8769 points in the 3D space were analsed and the reslt is shown in Fig. 1. All the points fall into three grops: the points where the nep eqation is valid, displaed as o, the points where the eqation is not valid, displaed as, and the points, displaed as, for which we were nable to make a conclsion abot their validit becase 11

12 c 12 θ 1 8 θ 2 Figre 2: The area of validit of the nep eqation. 12

13 6 4 C θ θ 2 1 Figre 3: Valid and invalid points for the nep eqation along selected horizontal lines. the comptation took too long to finish. For brevit we will call the circles valid points and the stars invalid points. As is seen, the volme made of the valid points has a complicated shape. The circles o ma seem to form a continos shape, however, at places the actall intermit with the stars. To show this fact, we scattered a small nmber of stars over the space, however, did not show all of them in order not to block the view of the circles. Bear in mind that the empt space arond the circles is meant to be filled with stars (and sqares). Note that the tall colmns in Fig. 1 etend to infinit. It is interesting to compare the validit areas for the nep and Kramoto-Sivashinsk eqations. The KS eqation (4) is relevant to man phsical sstems and can be dedced from (7) (9) as well. Its validit area can be readil eplored sing or program after a simple adjstment. The KS eqation is valid when onl one condition on the coefficients is imposed, namel a 1 = ε <, (34) 13

14 6 4 C θ 1 θ 2 Figre 4: Valid and invalid points for the KS eqation along selected horizontal lines. and, additionall, the following restrictions are satisfied, b 1 <, δ 1 > K 1 + K 2 < 1. (3) Ot of the 9 independent parameters we have freedom to choose 8 so that the remaining 1 parameter is to be compted from eqation (34). We choose this compted parameter to be δ 1. In analog to the nep case, we consider the 3D space (θ 1, θ 2, c 12 ). Let s select an arbitrar path in this space a straight line. As we move from point to point along the line, the vales of the 3 parameters change. For each point we assign vales to the other free parameters at or disposal, namel c 11, c 2, K 1, K 2 and δ 2. There is plent of freedom in doing so, and or choice is to borrow these vales from the nep comptational eperiments eected at the same vales of θ 1, θ 2 and c 12. The last step is to inspect signs of the compted parameters: if conditions (3) hold then the KS eqation is valid at a given point. If at least one of these restrictions is not satisfied, then the KS eqation is not valid at the point. We opted to ct off a bo in the 3D space (θ 1, θ 2, c 12 ) to captre considerable amont 14

15 8 C 12 θ 1 θ 2 1 Figre : Valid and invalid points for the nep eqation along selected vertical lines. of valid nep points revealed b Fig. 1. Then we pierced the bo b 2 straight lines forming a grid. The lines niforml fill the volme of the bo. At all the point of the lines validit of the KS and nep eqations was tested. In the similar wa we set p a grid of 2 vertical lines and analsed validit along them. The comptational reslts are displaed in Fig. 2. For a qantitative comparison between the KS and nep cases, we smmed p the distances occpied b the KS valid points and, separatel, nep valid points, over all 2 horizontal lines and calclated the ratio ν =total nep distance/total KS distance. The same was done for the vertical lines. The horizontal and vertical lines were analsed separatel in this calclation in order not to mi the distances measred in nits of c 12 and those measred in nits of θ 2. In both cases the ratio trned ot to be less than one: ν vert =.19, ν horiz =.8. This shows that even in a bo deliberatel chosen to contain man valid nep points, its total amont (the distance) is less than that for the KS eqation. This seems to be a natral otcome since the KS eqation imposes softer restriction on the coefficients of the phase eqation. We note that the performed comparison is b no means ehastive 1

16 C 12 θ 1 1 θ 2 Figre 6: Valid and invalid points for the KS eqation along selected vertical lines. considering the man-dimensional (9D) space we deal with. We realize that (a) the KS area ma be bonded or ma spread to infinit in a given direction; and (b) man valid KS points ma be located far awa from the validit area of the nep eqation. 3 Forced and nforced dnamics In this section we seek nmerical and analtical soltions of eqation (1) and its forced version, in two-dimensional space. In 2D eqation (1) is written as t = A ( ) [ ( ) 2 + ( ) 2] + B [ ( ) 4 + 2( ) 2 ( ) 2 + ( ) 4] (36) + C ( ). 16

17 Consider a sqare domain,, L. On the bondaries we stiplate zero vale of the first, second, and third derivatives normal to the bondar, =, 2 =, 3 = at = and = L, =, 2 =, 3 = at = and = L. 3.1 Derivation of the forced eqation So far we assmed that the rate c 2 was constant, both in time and space. Obviosl this is an ideal sitation while an real phsical sstem is not perfectl niform. It is interesting to eplore the case when the rate c 2 varies in space; this assmption wold represent a nonniform distribtion of the kinetics of the reacting sstem in space. In the ideal case of a constant rate c 2 the phase satisfies eqation (16), t ϕ = c 2 a 2 + = c 2 (1 + ε 2 1a 2 + ε 4 1a 4 + ε 6 1a ) , (37) where the amplitde a is represented b the series (22) in small parameter ε 1 and the second set of dots denotes the terms which do not contain c 2. Now assme that c 2 is not constant is space (bt still constant in time), c 2 = c () 2 + ε 2 c (1) 2 (, ), (38) with c () 2 being a constant and ε 2 being a new small parameter. Similarl to (18) we look for the phase in the form Then eqation (37) becomes ϕ(,, t) = c () 2 t + (,, t). [ ] [ ] c () 2 + ε 2 1 t1 = c () 2 + ε 2 c (1) 2 (1 + ε 2 1a ) 2 + = c () 2 + ε 2 c (1) 2 (1 + 2ε 2 1a ) +... } = c () 2 + ε 2 c (1) 2 + {[ ] c () 2 + ε 2 c (1) 2 2ε 2 1a (39) The amplitde components a n are separatel epressed in terms of 1 from the amplitde eqation (24) as we eplained in the previos section. Sbstitting the epressions for a n into (39), cancelling c () 2 in both sides and performing some maniplations leads to {[ ] } ε 2 1 t1 = ε 2 c (1) 2 ε ( 1 ) 2 c () 2 + ε 2 c (1) 2 2β {[ ] } {[ ] } +ε 4 1( 1 ) 4 c () 2 + ε 2 c (1) 2 2β ε c () 2 + ε 2 c (1) 2 δ 1 β , 17 ()

18 where δ 1, β n and β are the parameters composed of the original parameters of the phsical sstem. Note that the lower-order terms in 1, carried b a 2 and a 4, vanished becase the reslting factors in front of them are made zero. This is done b the appropriate choice of the magnitdes of the original parameters, as we mentioned in the previos section. To evalate the order of the small parameter ε 2 we need to take into consideration that all the terms in () mst have the same order of magnitde. Therefore, evalating qantities b their absolte vale, we have ε 2 c (1) 2 b 4 2 ( ) 2. (41) B the definition of ε and in view of the connection (19), b 4 = ε = ε 2/3 1. The scales of the length, L, and phase departre, U, are fond in (6), conseqentl, U ε = ε 2/3 1, L 1 ε 1, b 4 2 ( ) 2 b 4 U 3 L 4 ε/3 1. Since the parameter ε 2 is introdced presming c (1) 2 1, then, sing (41), we determine ε 2 = ε /3 1. Finall, neglecting ε 2 c (1) 2 inside the sqare brackets in () in comparison to c () 2 and retrning to the nscaled operators = ε 1 1 and t = ε 2 1 t1 we arrive at the forced eqation t = A 2 ( ) 2 + B( ) 4 + C 6 + f(, ), (42) where the force term, f(, ), is in fact the scaled rate ε 2 c (1) 2 (, ) from (). Or interest in the forced eqation (42) is three-fold: (1) eplore the effects of slow variations of the kinetics in space (the nonniformit of the reacting medim), (2) constrct eact soltions of the forced eqation and nmericall investigate their stabilit, (3) test the nmerical code b comparing the nmerical soltions with the eact soltions. 18

19 3.2 Eact soltions of the forced eqation. Testing the nmerical code We wrote a Matlab nmerical code to solve eqation (42). The spatial part of the eqation is discretized sing central finite differences and the reslting sstem of ordinar differential eqations is integrated in time b the dae2 solver [11]. The solver ensres a good accrac, nevertheless we carried ot or own tests as described frther in this section. The idea for the tests was to adopt the force fnction f(,, t) in special forms in order to ensre that the forced eqation (42) allows eact (a) stationar or (b) oscillating soltion of or choice. It was important that the soltions were stable otherwise it wold not be possible to obtain them in the corse of or nonstationar nmerical eperiments. Here is a simple eample where a desired soltion is nstable. Consider the ordinar differential eqation df/dt = F. and let s wish that the forced eqation df/dt = F + F 1 (43) has the eact soltion F = 1. According to (43) we choose F 1 = df /dt F = 1. It is eas to see, however, that the soltion F = 1 is nstable. For eqation (42) we wold not know apriori whether or chosen soltion is stable or not, bt if it is stable, or nmerical code shold be capable of reprodcing it. If it proves nstable, we wold need to choose a different soltion. With this in mind, we condcted two tests. For the first test we desired, as in the above simple eample, that the forced eqation had a stationar soltion. Of corse the soltion needs to be nontrivial (non-constant) is space and mst satisf the bondar conditions. For the second test we created a nonstationar soltion b mltipling the stationar one b an oscillating fnction of time. The two cases are represented b the single formla v(,, t) = M[1 + k sin(ωt)] 4 ( L) 4 4 ( L) 4, (44) where k = gives the stationar soltion and k the nonstationar one. Write eqation (42) as t = RHS + f(,, t), (4) 19

20 where RHS stands for the right-hand side of (1). Sbstitting (44) into (4) we get f(,, t) = t v(,, t) RHS[v(,, t)]. (46) giving 11 f(,) Figre 7: The eact soltion (44) (k = ) [left] and the force fnction (46) [right]. M = , L =.4. The aes show grid points.

21 Figre 8: Evoltion of the nmerical soltion of (42) towards the stationar regime. t = , 1.24, 1.73,.,.. The eqation coefficients A =, B = C = 1. The domain size L =.4. 21

22 7 6 ma t 9 Figre 9: Settling of the maimm of in the -grid eperiment Figre : The eact soltion (44) and settled nmerical soltion (circles) at t =. for, and 9 9-grid eperiments ((, ) at =.2 (grid points, and 4 respectivel)). 22

23 ma t Figre 11: The eact and nmerical (circles) soltions for the oscillator regime. f(,, t) = Mkω cos(ωt) 4 ( L) 4 4 ( L) 4 cm[1 + k sin(ωt)] {[ 43( L) ( L) ] 4 ( L) [ 24( L) ( L) ( L) ( L) ] [ 12 2 ( L) ( L) ( L) 2] + 3 [ 24( L) ( L) ( L) ( L) ] [ 12 2 ( L) ( L) ( L) 2] + [ 43( L) ( L) ] 4 ( L) 4} + gm 3 [1 + k sin(ωt)] {[ ( L) ( L) ( L) 2] 4 ( L) 4 + [ 12 2 ( L) ( L) ( L) 2] 4 ( L) 4} { [4 3 ( L) ( L) 3] 2 8 ( L) 8 + [ 4 3 ( L) ( L) 3] } 2 8 ( L) 8 23

24 bm 4 [1 + k sin(ωt)] 4 { [4 3 ( L) ( L) 3] 4 16 ( L) [ 4 3 ( L) ( L) 3] 2 8 ( L) 8 [ 4 3 ( L) ( L) 3] 2 8 ( L) 8 + [ 4 3 ( L) ( L) 3] } 4 16 ( L) 16. (47) Consider the stationar case first (k = ). The shapes of the eact soltion (44) and the corresponding force fnction are shown in Fig. 7. The nmerical soltion, as it evolves from the initial condition, is given in Fig. 8. At the initial moment the fnction was chosen to have a similar shape to (44) (with k = ), onl with half the height. Note the large time interval between the second and third snapshots. The time step t = 13. See that the nmerical soltion gradall increases in height and adjsts its peripheral parts. We see a clear resemblance between the nmerical and eact soltions. A more precise evidence of their closeness is given b Fig. 9; it shows the evoltion of the maimm of, that is the vale of the fnction in the centre of the domain. Stabilit of the soltion is proved b the nonstationar eperiment. There is a slight vertical drift of the nearl horizontal crve becase the stationar soltion is onl netrall stable. As a reslt, eqation (4) allows a famil of soltions = v(,, t) + C with different constants C. The inevitable nmerical mismatch between the eact force f(,, t) and discretized RHS in (4) cases the slow drift from one stationar soltion to another. Fig. compares the settled nmerical and eact profiles along the centre of the domain, =.2, demonstrating a good agreement ecept near the edges. Epectedl, the agreement improves for finer grids. Or second test reprodced the oscillating soltion (44) with k =.2, ω = 11 and the same vales for the other parameters (the time step t = 2 14 ). Fig. 11 displas the evoltion of the maimm of. See that the nmerical and eact soltions are close and, again, there is a slow drift de to the netral natre of stabilit. 3.3 Intermittent and irreglar dnamics nder the nforced eqation In this section we consider the original nforced eqation (1). We present jst one nmerical eperiment, which took a long time (months) to rn in real terms. The reason was the 24

25 large size of the domain that we chose aiming to provide enogh space for the dissipative strctres to form and interact with each other. An inevitable price for the choice was that man gridpoints were reqired to spatiall resolve the dissipative strctres. Hopefll this wold lead (and actall did) to the complicated irreglar dnamics. Small domains are less interesting as the wold reslt in a flat motionless -field. To investigate the onset of chaos at transitional domain sizes from small to large is an interesting task bt at this stage we leave it for the ftre becase of the compter limitations. We present a series of snapshots showing the nmerical eperiment with the domain size L = 2. A grid is chosen as a compromise between accrac and speed of comptation. The snapshots are shown in pairs: the fnction itself at a specified moment of time and, net to it, the corresponding plot of the ecitation term, A 2 ( ) 2. Recall that it is this term that is responsible for the energ release in the sstem, hence it indicates the area where the action is actall occrring at a given moment. The initial condition was a single narrow peak sitting on the platea (top Fig. 12). The snapshots are complemented b the graph of the integral ecitation, Fig. 17, L L A 2 ( ) 2 dd (48) as a measre of the intensit of the dnamics. After an initial period of violent dnamics (not shown) the srface becomes smooth as shown b the second-from-the-top snapshot in Fig. 12. Gradall, after the relativel calm evoltion dring t < 4 9 the dnamics bild p and remain intensive for 4 9 < t < 9 9. Then the motion slows down before picking p again arond t 1 9. It appears we have recorded a ccle of the dnamics, slow motion brst slow motion, and the beginning of the net ccle. Irreglarit of the dnamics within the ccle is demonstrated b Fig. 18. It shows the positions of the absolte maimm of the ecitation term on the (, )-plane. In addition, Fig. 19 also shows the positions of the second-largest maimm of the ecitation. The wa the shape of the fnction (,, t) looks and moves is remarkabl different to a tpical behavior of a linearl ecited sstem, sch as the Kramoto-Sivashinsk eqation and Nikolaevskii eqation. Under the bondar and initial conditions similar to those of or eperiments, the linearl ecited models prodce dissipative strctres simltaneosl emerging (with certain dominant wavelengths) all over the available spatial domain. This 2

26 8 ecitation 8 8 ecitation ecitation ecitation Figre 12: t = t, 9 9, 1 9,

27 ecitation ecitation ecitation ecitation Figre 13: t = 462 9, 9, 41. 9,

28 ecitation ecitation ecitation ecitation Figre 14: t = 8 9, 7 9, 88 9,

29 ecitation ecitation ecitation ecitation Figre 1: t = 61 9, , 738 9,

30 ecitation ecitation ecitation ecitation Figre 16: t = 822 9, , ,

31 8 7 integral ecitation, 9 8 t, 1. 9 Figre 17: Integral of the absolte ecitation. Figre 18: The positions of the global maimm of the ecitation at different moments. 31

32 Figre 19: The positions of the second-largest maimm of the ecitation at different moments. is becase an flat fragment of the -field qickl transforms into a lmp shape b the linear instabilit mechanism. B contrast, in or case, characterised b the linear stabilit, significant portions of the -field remain almost flat ntil the signal arrives to them from other locations. The maimm of the ecitation, which is concentrated on the steepest slopes, moves like a worm digging its wa in predominantl lateral direction. Having travelled over the domain, the worm sooner or later retrns to its earlier (, )-location. Qalitativel this is the same tpe of behavior as the spinning combstion waves, onl irreglar. 4 Conclsions The work consists of two parts. In the first part we evalated the area of validit of the nonlinearl ecited phase eqation in parametric space for a sstem of nonlocall copled oscillators. We compared the validit areas for the nep and KS eqations within the same sstem. The 8D sbspace of the 9D space of independent parameters was pierced b a set of lines. The 8 parameters spanning the sbspace were chosen to be θ 1, θ 2, c 12, c 11, c 2, K 1, 32

33 K 2 and δ 2. It is onl possible to graphicall displa three (or fewer) dimensions, for eample θ 1, θ 2 and c 12, which was done. The lines in the 8D sbspace are identical for both the nep and KS eqations. At each point on the lines each eqation is either valid or invalid and sometimes or reslt was inconclsive. We fond that the nep eqation has a sizable range of validit, et narrower than that for the KS eqation. This can be eplained b the stricter conditions reqired b the nep eqation. We derived the forced version of the nep eqation addressing nonniformit of the reacting medim in space. Selected eact soltions of this version were constrcted. Stabilit of the soltions was demonstrated b the nmerical eperiments. Comparison of the eact and nmerical soltions demonstrates satisfactor performance of the nmerical code. For the nforced version of the eqation, we revealed periods of fast irreglar dnamics intermitting with stages of slow evoltion. References [1] G.I. Sivashinsk, Nonlinear analsis of hdrodnamical instabilit in laminar flames, Acta Astronat. 4 (1977) [2] A.G. Merzhanov and E.N. Rmanov, Phsics of reaction waves, Rev. Mod. Phs. 71 (1999) [3] Y. Kramoto and T. Tszki, Persistent propagation of concentration waves in dissipative media far from thermal eqilibrim, Progr. Theor. Phs. (1976) [4] D. Tanaka and Y. Kramoto, Comple Ginzbrg-Landa eqation with nonlocal copling, Phs. Rev. E 68 (3) [] D. Tanaka, Chemical trblence eqivalent to Nikolaevskii trblence, Phs. Rev. E 7 (4) 12(R). [6] V.N. Nikolaevskii, in Recent Advances in Engineering Science, edited b S.L. Koh, C.G. Speciale, Lectre Notes in Engineering, 39 Berlin: Springer, 1989, 2. [7] D.V. Strnin, Atosoliton model of the spinning fronts of reaction, IMA J. Appl. Math. 63 (1999)

34 [8] D.V. Strnin, Nonlinear instabilit in generalized nonlinear phase diffsion eqation, Progr. Theor. Phs. Sppl. N 1 (3) [9] D.V. Strnin, Phase eqation with nonlinear ecitation for nonlocall copled oscillators, Phsica D: Nonlinear Phenomena 238 (9) [] D.V. Strnin, A.G. Strnina, E.N. Rmanov and A.G. Merzhanov, Chaotic reaction waves with fast diffsion of activator, Phs. Lett. A 192 (1994) [11] 34