stability of cellular states of the kuramoto-sivashinsky equation

Stability of Cellular States of the Kuramoto-Sivashinsky EquationAuthor(s): John N. Elgin and Xuesong WuSource: SIAM Journal on Applied Mathematics, Vol. 56, No. 6 (Dec., 1996), pp. 1621-1638Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2102590 .

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SIAM J. APPL MATH. (?) 1996 Society for Industrial and Applied Mathematics Vol. 56, No. 6, pp. 1621-1638, December 1996 004

STABILITY OF CELLULAR STATES OF THE KURAMOTO-SIVASHINSKY EQUATION*

JOHN N. ELGINt AND XUESONG WUt

Abstract. This paper is concerned with the instability property of a particular type of solution to the Kuramoto-Sivashinsky equation, namely, the symmetric, time-independent cellular states. Attention is focused on the dimension of their unstable manifolds. We show that as a control parameter varies, the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation. Further change in dimension is shown to be related to Hopf bifurcations from the cellular states. An asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur.

Key words. Kuramoto-Sivashinsky equation, stability, cellular states

AMS subject classifications. 34C, 58F

1. Introduction. The one-dimensional Kuramoto-Sivashinsky (KS) equation

(1.1) Ut + uux + Uxx + Uxxxx = 0

has been studied extensively as a paradigm in the effort to understand the complex dynamics in nonlinear partial differential equations (PDEs). It has been shown to have an inertial manifold a low-dimensional attractor which effectively makes it equivalent to a finite-dimensional system of ordinary differential equations (ODEs) [10, 25]. Solutions to (1.1) are often sought for the case of periodic boundary conditions, i.e., u(x + L, t) = u(x, t) for any t, with the system size L acting as a control parameter. As L is increased, numerical studies reveal a rich variety of spatial and temporal behaviours, including chaotic motion on the attractor [7, 13, 14, 19, 26]. Simple behaviours can be predicted, of course, such as travelling-wave solutions of the form u(x - vt). A special class of such solutions are the symmetric, time-independent cellular states. These states consist of a primary branch, which has one cell in the length L, and higher branches, which are the N-fold (N = 1, 2,...) replicas of the primary branch and hence have N cells in the length L; hereafter, we shall refer to these as N-cell cellular states. In general, these states are not stable solutions of (1.1). How- ever, they can play an important role in organising the long-time behaviour of the solution to (1.1), as fixed points and limit cycles do in an ODE. For instance, certain time-dependent behaviours are associated with the homoclinic connection of two cellular states, one of which is a translation of the other [19], or with the heteroclinic connection of two cellular states belonging to different branches [13, 14].

The objective of this article is to investigate the stability of such N-cell cellular states. In the limit where the amplitude of the N-cell state tends to zero (i.e., it approaches the quiescent state u = 0), the dimension of its unstable manifold is 2(N - 1). As the amplitude grows, the dimension of the unstable manifold is found to change in an ordered manner. In particular, instability of the cellular state is linked with the properties of an associated ODE, (1.5) below. One purpose of this article is to examine this connection. Unfortunately, this association does not account for all observed changes in the dimension of the unstable manifold. However, a complementary analysis using long-wavelength perturbations allows us to complete the picture.

* Received by the editors February 25, 1994; accepted for publication (in revised form) October 11, 1995. The research of the second author was supported by SERC grant GR/G57048.

tDepartment of Mathematics, Imperial College, 180, Queens Gate, London SW7 2BZ, UK.

1621



1622 JOHN N. ELGIN AND XUESONG WU

The main result of the analysis, the diffusion instability, which is manifest in the numerics as a succession of Hopf bifurcations, further alters the dimension of thie unstable manifold. In this way, we have a self-consistent picture which correctly accounts for the observed changes in the dimension of the unstable manifold of the symmetric cellular states.

Apart from being an interesting problem in its own right, the study of the instability of the periodic solutions is relevant to characterising the "structure" of the inertial manifold of the KS equation. For low-dimensional systems exhibiting chaotic behaviour, such as the Lorenz equations, it is suggested that their strange attractors are composed of the closure of the unstable manifold of all fixed points and limit cycles [9]. A similar conjecture has been proven for a class of nonlinear parabolic equations [4] and recently has been conjectured to apply also to the KS equation [19]. Since the basic "building blocks" are all the possible periodic orbits, it is thus appropriate first to classify them and then to determine their stability properties.

For (1.1) with imposed periodic conditions u(x, t) = u(x+L, t) for any t, it is often convenient to scale (1.1) so that u is 2ir-periodic by the following variable changes:

x - ?ax, t -? a 2t, u-? U/a,

where a = 2ir/L. Equation (1.1) then reads

(1.2) ut + uuX + uXX + a 2uxxxx = ?

and has the periodic condition

u(x, t) = u(x + 2ir, t).

Equation (1.1) has two families of continuous symmetries: (i) translation symmetry, i.e.,

(1.3) x-?x+xo, t-?t, u-?u,

and (ii) Galilean symmetry, i.e.,

(1.4) x- >x-vt, t-+t, u-u+uv.

Equation (1.1) is invariant under either symmetry operation. The above properties will be used in ?3.2 to discuss the stability of the travelling-wave solutions to long- wavelength perturbations.

In view of the Galilean symmetry, any travelling-wave solution can be trivially brought to rest. Travelling-wave solutions of (1.1) are then periodic solutions of the ODE

(1.5) 1u 2+ u/+/ C2

where the prime denotes d/dx and c is an integration constant. Evidently,

(1.6) (2 ( 2) 1= u 2(x)dx, 2 2Lj

where (.) denotes the average over the period L. For the rescaled equation (1.2), the corresponding constant is c-2 = c2 /a2.



STABILITY OF CELLULAR STATES 1623

The results of the asymptotic analysis in ?3 reveal that for each N-cell state (N > 2), there is a range of values of a, a$1) < a < a(N), within which the N-cell state is stable. This range is bounded by the following two inequalities:

(1.7) 0-c2< 0,

(1.8) aLC~~~2 a 2 (2)a2 < 0. (1.8) c2 = 2 &(c2/a)< &L &L

The criterion (1.7) has been proven by Frisch, She, and Thual [11], who termed it the elastic condition. The criterion (1.8) has not been proposed before, although we note that it can be inferred from the numerical results in [11]. We shall offer an indirect "proof" and also provide strong numerical evidence to support its correctness in ?3. We further determine the parameter values at which the inequality (1.8) is violated, giving rise to the so-called diffusion instability (see, e.g., [11] and [28]); these are

19min -NaN = 0.768/N.

Since the diffusion instability is found to correspond to a sequence of Hopf bifurcations, we have effectively estimated the critical parameter values for the onset of such Hopf bifurcations (at least for sufficiently large N); see ?3.2.

Although this paper concentrates on the symmetric cellular states, the same considerations apply to any family of periodic solutions. The symmetric cellular states are the simplest to handle because they bifurcate from the quiescent state, unlike the rotation waves, which are born by secondary bifurcations from the symmetric cellular states.

The rest of the article is organised as follows. In ?2 the properties of the periodic solutions of (1.5) are discussed, and their stability as a solution of (1.1) are then studied by using numerical methods. In particular, the importance of the elliptic regions is illustrated. In ?3, we consider the instability of high branches of the N-cell cellular states (N > 1), using an asymptotic analysis and phase dynamics approach based on N > 1. The criteria (1.7) and (1.8) are proven, and correspondence is established with the results of ?2. Some final comments are made in ?4.

2. Bifurcation of cellular states and their instability.

2.1. Elliptic regions and m/k-bifurcations. In order to study the instability of the cellular states, it is necessary first to examine their bifurcation properties. Travelling-wave solutions of (1.1) correspond to periodic orbits of (1.5). The latter is usefully expressed as the "dynamical system"

X' = Y, y/ = Zi

(2.1) Z'=c --X2-Y 2

where X _ u, and the prime denotes d/dx as before. Many authors have studied (2.1), and their main properties are now well documented; see, e.g., [5, 6, 15-18, 20, 23, 27]. Much of the interesting behaviour of (2.1) stems from the fact that (i) the flow is measure preserving and (ii) the symmetry transformation X -? -X, Y - Y,




Z + -Z, x -* -x leaves the equations unchanged. We begin by summarising the main results of system (2.1).

Consider Figure 1. It shows the primary branch of periodic solution of (2.1) together with the main bifurcation points. Any point on this curve corresponds to a symmetric periodic orbit with period L = L/(2ir) and an amplitude given by (1.6). The primary branch has a curious structure referred to elsewhere as a noose bifurcation [15, 16, 18] (see also [27]), because the two branches of the period-doubling bifurcation at C2 evolve so as to annihilate each other in the saddle-node at Cmax. At the period-doubling points similar symmetric orbits are born, while at the symmetry- breaking points, asymmetric orbits are born in pairs in accordance with the symmetry requirements. The primary branch bifurcates from the origin, c = 0. For small c <K 1, the (symmetric) orbit is approximated by

c2 (2.2) X=2csinQx- - sin2Qx?+ ,

6

where Q = 1-c2/12 + ?.- Since (2.1) is measure preserving, the product of the two nontrivial Floquet

multipliers say, p,i and At2-for a periodic orbit is unity, i.e., /u1/'2 = 1. For small c, the orbit is elliptic (pi1 and bt2 are a complex conjugate pair on the unit circle) until C2 0.31939, where a period-doubling bifurcation takes place (Al1 = -1). Along the lower branch, the orbit is hyperbolic (,tl < 0, ,u2 = 1/pi) until another period-doubling at CA 1.26595, then briefly becomes elliptic up to the saddle-node at Cmax 1.26623, at which p, = A2 = 1. Along the upper branch, starting from Cmax, there is a hyperbolic region (p,t > 0, '2 = 1/pl) up to a symmetry-breaking bifurcation at CB 0.5984, where ll= 2 = 1. Between CB and the period-doubling point cc 0.58086 there is a short elliptic region, after which the orbit is hyperbolic up to the period-doubling point at CD 0.34010, then elliptic up to c2, at which point it rejoins the primary branch. We are particularly interested in the four elliptic

7

5

3 -~~~~~~~~~~~ 3

L el 2 CA \e2

0.5 10 C

FIG. 1. Bifurcation diagram of orbit period L against the parameter c, illustrating the primary branch, which emanates from the point L = 1 and ends with the region e4. The main secondary bifurcation points are shown. In this and the following figures L L




regions, which are now labelled el, e2, e3, and e4 as shown in Figure 1. In these regions the Floquet multipliers are a conjugate pair lying on the unit circle.

Of particular interest are those values of the parameter c at which the Floquet multipliers [iL and ,u2 take the value

(2.3) (') = jU2 = ,U(m)-kexp(2irim/k), k > 2, m < [k/2],

where [ ] denotes "the integer part of." As ,Ui passes through [(m) for k > 2 orbits with a period k times that of the parent are born, in accordance with the mechanism described by Meyer [22] and MacKay [21]. We shall refer to the above bifurcations as m/k-bifurcations. For k > 4, these orbits emanate directly from the parent as shown in Figure 2. For k = 2, this is a period-doubling bifurcation. For k 3 and as a second alternative for k = 4, the situation is slightly different; daughter orbits are now born in a saddle-node bifurcation, one branch of which coalesces with the parent at the appropriate value of pl, i.e., either [f4l) or [(1). This is illustrated in Figure 2, and discussed elsewhere [18]. Clearly, a special case of m/k-bifurcations is m = 1 and is termed k-bifurcations by Kent and Elgin [18]. They discussed such a k-bifurcation branch where the orbit born in the period-doubling bifurcation at CA is linked in a nontrivial way to similar orbits born in a set of bifurcations from the elliptic region el when the multiplier passes through {(l) (k = 3,4,5,...); see Figures 3 and 5 of [18]. Similar conclusions were reached by Scovel, Kevrekidis, and Nicolaenko in [27] by exploiting the scaling laws of (1.5) and by Tsvelodub, Yu, and Trifonov in [29]. Kent also located a 2/5-bifurcation from the elliptic region el, where two orbits born with period five times that of the parent were followed until they attained homoclinic status [15]. These orbits are independent of those born at the 1/5-bifurcation point, which were part of the k-bifurcation branch.

The 2/5-bifurcation located by Kent occurs when [tl takes the value [L(2), and such m/k-bifurcations are a common feature of all elliptic regions. For example, in any elliptic region, one expects to see orbits of period seven times that of the parent emanating from the bifurcation points 1/7, 2/7, and 3/7. Moreover, daughter orbits thus born in any one of the elliptic regions may connect with similar orbits in any other (or the same) in a manner similar to that described for the k-bifurcation branch. For example, the orbit born in the period-doubling bifurcation at CD (the boundary of the elliptic region e4) is connected to one of the orbits born in the 1/3-bifurcation emanating from the elliptic region e2, as shown in Figure 1. The continuation and the connections of these m/k-bifurcations are the subject of study in progress.

ke L k;~4

L e

ki. L k =314 L

FIG. 2. Schematic illustration of the m/k-bifurcations, where e and h denote elltptic and hyperbolic orbits, respectively.




In each of the elliptic regions, the nontrivial Floquet multiplier-say, ,A (the other one being ,*) depends continuously on c. For any c and corresponding ,u, we can always choose suitable m and k such that p(m) is arbitrarily close to ,p. Suppose that this m/k-bifurcation occurs at Cmk. The continuous dependence of , on c implies that Cmk is arbitrarily close to c; i.e., the set of parameter values of c at which the m/k-bifurcations occur is dense in each of the elliptic regions. This denseness of the bifurcation set is a consequence of the fact that (2.1) is three dimensional and measure preserving. These two properties ensure that the Floquet multipliers can move only onto and off the unit circle at (?1, 0). This is the reason why we have (continuous) elliptic regions rather than a few discrete elliptic points. In passing we note that the existence of the m/k-bifurcation may also be established by exploring the symmetries associated with the infinite-dimensional system (2.7) and (2.8) using group theoretic methods (see, e.g., [2]). A related consideration to a system of coupled nonlinear oscillators is discussed by Aronson, Golubitsky, and Krupa [1].

In Figure 3, we show some bifurcations together with replicas of the primary branch. The latter are labelled I, II, III, and so on, where I is the primary branch. Similar replicas exist of course for all orbits. The replicas of the primary branch are the symmetric N-cell cellular states, whose stability properties are the main concern of this article. Note that the m/k-bifurcated orbits touch the k-fold replicas and may

c

C~~~~~~~~ I

R~~~~I I : ;

labelled~~~ ~~~ II III ...

e' I '

loo~~ ~~I~I I I

0,5- ~ ~ ~ : , : '~ '

4

labelled II, III, ....~ ~ ~ ~ ~ ~~~~I '




be considered to have bifurcated from the replicas. It is precisely such bifurcations which change the dimension of the unstable manifold of the k-fold replicas (the N-cell cellular states) and thus affect the stability of the latter in a manner which we now turn to discuss.

2.2. Stability of the cellular states. We now consider the stability of the cellular states of (1.1), represented by branches I, II, and so on. Recall that u(x, t) has the periodic L. For the N branch, the interval L contains N primitive cells say, X(x) with a wavelength 1 = L/N, i.e., X(x) = X(x + 1). It is convenient to rescale (1.1) so that the primitive cell length is 2ir by introducing y = kox, where

ko = 2ir/l = Na,

and a was introduced previously. Equation (1.1) then reads

(2.4) Ut + kouuqy + k 2Uyy + k4uyyyy = O,

while the periodic condition becomes u(y, t) = u(y + 2irN, t). The solution takes the form

+00

(2.5) U = Ao(t) + [Ap(t) cos py + Bp(t) sin ]. P=1

The coefficients Ap and Bp satisfy

(2.6) dt =?

(2.7) dv N {pAp-- E [Am (Bm+p - Bmp) - Bm(Am+p-Am-p)] I m=l

dB ko 1 00

(2.8) dtv Po{ 1pBp + - E [Am (Am+p + Am_p) + Bm(Bm+p + Bm? p)] } m=l

where following Demekhin, Gennadii, and Shkadov [7] we introduce the convention A-k = Ak, B-k -Bk, Bo = 0, and

VP = pa(l - p2a2).

Equation (2.6) indicates that Ao is a constant for any t. With an appropriate choice of the origin, the cellular states have odd symmetry. For such states, Ap = 0 for all p, while Bp _ bp (p = 1, 2,...) satisfy

00

(2.9) vpbp + - E bm(bm+p + bm_p) = 0. m=1

Equation (2.9) was truncated at a suitable order and solved using a Newton iteration method. For the primary branch, the truncation is at order 32. In order to ensure that sufficient Fourier modes are included, we also doubled the order of truncation to 64. The solutions from these two truncations showed no appreciable difference. The results were also checked against the periodic orbit obtained from the AUTO package [8]. The high branches, which bifurcate from a = 1/N, are obtained by rescaling the solution for the primary branch.




To investigate the stability of these cellular states, we introduce the small perturbation

(2.10) Ap(t) = ApeAt, Bp (t) = bp + BpeAt .

Note that while the primitive cells have a wavelength 1, the perturbation has a wavelength Nl. Substitution of the above into (2.7) and (2.8) and linearising in the usual way lead to the required eigenvalue problem, which is solved by using NAG subrou- tines. Instability results whenever there exists an eigenvalue with a positive real part. We now discuss the results of our study of the eigenvalue problem.

In the limiting case where the cellular states approach the quiescent state, i.e., bp -*0 O, the eigenvalue problem can be solved to give

(2.11) A = p2a 2(1 - p2a 2).

In this limit, ao - 1/N since the N-cell state bifurcates from the quiescent state at a = 1/N. Equation (2.11) then indicates that there are (N - 1) pairs of eigenvalues with ReA > 0, corresponding to p = ?1, ?2,.. ., ?(N - 1). For small but finite amplitudes, i.e., c < 1, our numerical calculation shows the number of unstable eigenvalues remains 2(N - 1), although one pair quickly splits into two distinct eigenvalues for the even-numbered cellular states. As c increases, the dimension of this unstable manifold first decreases. Eventually in the vicinity of the maximum value of (X2) (or c2), the cellular states acquire stability. This persists over a small range of values of the parameter c before the laminar states become unstable once again. We observe that the change in dimension of the unstable manifold occurs in a rather ordered way. In the following we give a description of this process for low values of the number N before generalising it to the arbitrary N-cell state.

I-state. This is too tightly constrained to be part of the general scheme outlined below. The quiescent state is stable, and the basic noose remains stable until the symmetry-breaking bifurcation at point CB, where it eventually loses stability; see, e.g., [3, 7, 12, 19] for further discussion.

II-state. The schematics are shown in Figure 4(a). Points a1-a4 are the period- doubled orbits (which can be viewed as being born through 1/2-bifurcations) from c2, CA, Cc, and CD in the elliptic regions e1-e2, respectively. Point a5 denotes the twofold replica of the symmetry-breaking bifurcation at CB. A numerical study indicates that the following proposition always holds: on crossing period-doubling points a1-a4, the replicas of these points, or any replica of the symmetry-breaking point (such as the twofold replica at a5), the dimension of the unstable manifold changes by one. For the quiescent state, there is a pair of degenerate unstable eigenvalues, which lose degeneracy to become two distinct eigenvalues once c is increased from zero. On crossing the point a1, one of the eigenvalues crosses into the left-hand plane, leaving a one-dimensional unstable manifold for the II-state between points a1 and a2. On crossing a2, the remaining unstable eigenvalue moves into the left-hand plane so that the cellular II-state becomes stable. Stability persists until point ae, where a Hopf bifurcation results in a conjugate pair of eigenvalues crossing to the right-hand plane, i.e., ReA > 0. In keeping with our conjecture, the dimension of the unstable manifold further increases by crossing each of the points a5 and a3, then finally decreases by one on crossing a4, leaving a cellular II-state with an unstable manifold of dimension three in the region a4-a6. The latter is consistent with the change in dimension of the IV-state on passing the point a6 (see below).




(a) (b) c c

a2 O2

x

1,0 x5 10 0 J

0,5-~ ~ ~~~~a a305

0,5-.O 4 2

xx

FIG.~~~~~~~~ b. Scemti 2lutaino /-iuctosadtecag ftedmnino h

30 6

2 4

(c) 1(d)

1,0 \ ?2x2 LVI 1,0 2 04

03 0~~~~~~~~~~~~~O -

unstable manifold of the branches II, III, IV, and V. The symbol 03n denotes n degenerate pairs of unstable eigenvalues and Xm denotes m unstable single eigenvalues. For clarity, some bifurcation points have been displaced slightly from their true positions. For branch V, f, and gj denote bifurcation points for 1/5- and 2/5-bifurcations, respectively.




The important point to note here is how secondary bifurcations from the noose (i.e., period-doubling and symmetry-breaking bifurcations) affect the dimension of the unstable manifold of a solution to the PDE.

III-state. The schematics are shown in Figure 4(b), where we use 0 to denote a degenerate pair of unstable eigenvalues and x to denote a single unstable eigenvalue; the integer suffix indicates the number of each type present so that 02 indicates two pairs of unstable eigenvalues. Points b1-b4 denote the points of emergence of period- tripled orbits from each of the elliptic regions e1-e4, respectively. Recall from our previous discussion and Figure 2 that there are actually two such period-tripled orbits from each point bi, one elliptic and one hyperbolic.1 Point a5 is the threefold replica of the symmetry-breaking bifurcation at point CB, and ao again denotes a Hopf bifurcation point. A numerical investigation shows that the following proposition always holds: on crossing m/k-bifurcation points, such as 1/3-bifurcation points b,-b4, the dimension of the unstable manifold always changes by two. One may consider this as a change of "one" for each of the pair of orbits produced at such points. On leaving the quiescent state, there are now two pairs of degenerate eigenvalues, indicating a four-dimensional unstable manifold. There is no loss of degeneracy in either pair as c increases from zero; this is a common feature of all odd branches of the cellular states. On passing point b1, one degenerate pair of eigenvalues moves into the left-hand plane, leaving the cellular states with a two-dimensional unstable manifold. This persists until point b2, when again consistent with our conjecture, the remaining pair move into the left-hand plane. The III-state thus becomes stable. The stability persists until the point ae, where a degenerate conjugate pair of eigenvalues crosses into the right-hand plane. Note the degenerate nature of the Hopf bifurcation. The dimension of the unstable manifold further increases by one at point a5 and by two at point b3, implying a seven-dimensional unstable manifold, before finally decreasing by two on passing point b4. The final decrease, similar to what happens in the I-branch and indeed to all branches, makes the five-dimensional unstable manifold compati- ble with the change in dimension of the VI-state on passing point b4 as discussed below.

Note again how secondary bifurcations from the noose alter the dimension of the unstable manifold of the cellular states.

So the procedure outlined above continues. For the IV-branch, we have 1/4- bifurcations from each elliptic region, together with twofold replicas of each of the original period-doubling bifurcations, leading to the situation shown in Figure 4(c). Here di (i = 1, 2, 3, 4) denote the 1/4-bifurcation points associated with the elliptic regions ei respectively, and ai (i = 1, 2, 3, 4) are twofold replicas of the same points appearing in Figure 4(a) while a5 is a fourfold replica of the symmetry-breaking bifurcation. Points ao and a, denote Hopf bifurcation points. The quiescent state has three degenerate pairs of unstable eigenvalues, one of which quickly loses degeneracy on leaving the quiescent state. The change in dimension of the unstable manifold then proceeds as shown in Figure 4(c) and is consistent with the above rules. The only difference from- previous cases is that there are now two Hopf bifurcation points, ao and a,i; at ao, a degenerate conjugate pair of eigenvalues moves into the right-hand plane, and at a1 a further conjugate pair moves into the right-hand plane, thus giving a six-dimensional unstable manifold beyond this point.

'The actual mechanism for 1/3-bifurcation is shown in Figure 2 but the details are not resolved in our figure. Nevertheless it is clear that one elliptic and one hyperbolic orbit emanate from such bifurcations.




V-branch. For the V-branch, there exist 1/5- and 2/5-bifurcations in each elliptic region. A 2/5-bifurcation from the elliptic region e1 is discussed in Kent's thesis [15]. For any odd branch (here N = 5), there are no replicas of the period-doubling bifurcations, but there is an N-fold replica of the symmetry-breaking bifurcation. The situation is depicted in Figure 4(d). The points a1 and a2 denote Hopf bifurcations; at each point a degenerate conjugate pair of eigenvalues cross into the right-hand plane.

Based on these observations, the rules regarding the bifurcations from an arbitrary N-branch as well as the change in dimension of the unstable manifold can be generalised as follows.

For odd N, there are the m/N (m = 1,2,... ,(N - 1)/2) independent m/k- bifurcations in each elliptic region. There are no replicas of the primary period- doubling bifurcations, but there is an N-fold replica of the symmetry-breaking bifurcation in the elliptic region e3. The quiescent state has (N - 1) pairs of degenerate unstable eigenvalues. As c increases, the upward-sloping branch acquires stability by "losing" (N - 1)/2 pairs of unstable degenerate eigenvalues in each of the elliptic regions e1 and e2, in the manner discussed above. After this, cellular states are stable. They then lose stability by a sequence of degenerate Hopf bifurcations, which occur at the points ak, k = 1, 2,... (N - 1)/2. On passing each of the bifurcation points, the dimension of the unstable manifold is increased by 4, so that once the sequence is finished the dimension is 4 (N - 1)/2 = 2(N - 1). This is then increased to 3(N - 1) + 1 on passing through the elliptic region e3, where the extra 1 is due to the N-fold replica of the symmetry-breaking bifurcation, before finally being reduced back to 2(N - 1) + 1, i.e., (2N - 1), in the elliptic region e4, the end of which attaches to the branch 2N.

When N is even, in each elliptic region, there are (N - 2)/2 m/k-bifurcations, each of which changes the dimension of the unstable manifold by 2, together with one N/2-fold replica of the primary period-doubling bifurcations from the noose, each of which changes the dimension of the unstable manifold by 1. On leaving the quiescent state, the cellular states have (N - 2) degenerate unstable eigenvalues, together with two single unstable eigenvalues (one pair losing degeneracy), so that the unstable manifold has dimension 2(N - 2) + 2 = 2(N - 1). In each of the elliptic regions e1 and e2, the cellular states lose (N - 2)/2 double and one single unstable eigenvalue so that in each region the dimension of the unstable manifold is reduced by (N - 1). After passing all m/k-bifurcations and the replicas in the elliptic regions e1 and e2, all unstable eigenvalue are lost and the cellular states become stable. This persists until point ao, which marks the first of a sequence of Hopf bifurcations. The cellular states lose stability by a degenerate Hopf bifurcation at ao, which results in an unstable manifold of dimension 4. This is followed by a sequence of Hopf bifurcations at point

,, ... *, aN-3, each of which increases the dimension by 2. Hence, once this sequence is completed, the unstable manifold has dimension 2(N - 1). This then increases to 2(N - 1) + (N - 1) + 1 = 3N - 2 in region e3, where the extra 1 is due to the N-fold replica of the symmetry-breaking bifurcation. Finally just before the boundary of region e4, the dimension of the unstable manifold decreases to (2N - 1).

We next consider the joining of N to the 2N branch, as shown in Figure 5. Let -y denote the bifurcation point. In the immediate vicinity of -y, the unstable manifolds have the dimensions as indicated, which results are readily deduced from our discussion above. The change in stability (or, rather, change in dimension) is consistent with standard consideration.




2N-1 2 N-1

Yr /2N

b/

FIG. 5. Schematic illustration of the joining of the N branch to the 2N branch, where 'y is the bifurcation point and (2N - 1) and 2N are the dimensions of the respective unstable manifolds.

As N increases, the boundary of the first instability region moves closer and closer to the curve maximum, and in the limit N -* oo, the entire upward-sloping branch is unstable. This conclusion can be drawn by noting that for any N-branch, the upper boundary of the instability is determined by p(l), which tends to 1 as N -* oo. This indicates that the upper boundary approaches asymptotically to the boundary of the elliptic region e2, i.e., to the the saddle-node Cmax. On passing Cmax, the cellular states gain stability. They then lose stability through a Hopf bifurcation. The point at which this occurs also exhibits a universal feature as N -* oo. Therefore we turn to the large N limit.

3. Large N limit. In this section, we consider a complementary approach to the large N limit and show that the boundaries of stability of the cellular states are determined by the inequalities (1.7) and (1.8). The first of these indicates that in a plot c2 (or (X2)) against period L (or equivalently 1), any upward-sloping regions are always unstable, as we found in the previous section. The second inequality is related to the Hopf bifurcations. Using an asymptotic analysis based on N > 1, we shall show that the instability corresponds to a negative diffusion in a slow (phase) variable in the framework of phase dynamics (see, e.g., [28]).

3.1. Elasticity condition. Again consider (2.4) with imposed periodicity

u(y, t) = u(y + 27rN, t).

The travelling wave whose stability is to be examined consists of N cells represented by the symmetry noose X(y). It satisfies (cf. (2.4))

(3.1) koXXy + k XYY + k 4XYYYY = 0.

Here X is 27r-periodic in y. We now perturb around this in the usual way, i.e.,

(3.2) u(y, t) = X(y) + eAtf (y).

The linearised form of (2.4) becomes

(3.3) Af + k4fyyyy + k2fyy + ko(Xf)y = 0.

According to Floquet theory, f takes the form

(3.4) f (y) = F(y) exp(iqy),




where F is 27r-periodic, and to comply with the imposed periodicity on u, we require q = m/N (m = 1, 2, ..., [N/2]). Substituting (3.4) into (3.3) and regrouping terms give

(3.5) AF + LoF + (iq)LIF + (iq)2L2F + (iq)3L3F + (iq)4L4F = 0,

where

(3.6) Lo= 04d + 02d +ko (X.),

(3.7) Lp+l = (p + 1)-i [Lp, y],

with [ ] denoting a commutator. Equation (3.5) was first derived by Nepomnyashchy [24], who sought a perturbation solution in the long-wavelength limit by treating q = m/N as a small parameter. As in [24], the solution is then expanded as

(3.8) F(y) = Fo + qF + q2F2+..., A=Ao+qA?+q2A2?+".-

Substituting (3.8) into (3.5) and equating coefficients of qn to zero give the following results.

At order qo, we have

(3.9) AoFo + LoFo = 0,

which has the solution Fo = X', A0 = 0. At O(q), we obtain

(3.10) A1Fo + LoF1 + iL1Fo = 0,

which has a solution

(3.11) F1 = iko&k0X- .

To obtain the expression for A1, which contains information about the stability of the cellular states to leading order, we need to consider the expansion at order q2, which is

(3.12) A1F1 + A2X' + LoF2 + iL1F1 - L2FO = 0-.

Since all Fn are 27r-periodic in y, it follows that

(3.13) A1 (F1) + i(Li F1) = 0,

where (.) denotes an average over the periodic 27r (cf. (1.6)). Substituting (3.7) and (3.11) into (3.13), we obtain

(3.14) A1 =-2 k0(X2)

The cellular states are therefore unstable when the right-hand side of (3.14) is positive, i.e., when &k0(X2) < 0, or &1(X2) > 0 since ko = 27r/l. Using (1.6), we can write (3.14) in the form

(3.15) A = 2 42 9C2




Equation (3.15) is consistent with a result of Frisch, She, and Thual [11], who termed the statement &c2/0l < 0 the "elastic condition."

A conclusion to be drawn from (3.15) is that sufficiently high branches of the cellular states with the positive slope, i.e., &C2/&l > 0, are unstable. However, when C 2/11 < 0, the stability is determined by A2, to solve for which we need to proceed

with the expansion to O(q3). This gives

iko X A2 = (XF2)+k2+?ko(F2). A1

Unfortunately, it does not seem possible to find a closed form expression for F2 from (3.12). Nevertheless, we are able to show that branches with negative slopes eventually lose stability using a slightly indirect approach as follows.

3.2. The diffusion condition and phase dynamics. As stated previously, along the N-branch of the cellular states, the interval L contains N primitive cells say, ul(x) with a wavelength 1 = L/N, i.e., ul(x) = ul(x + 1). Therefore the perturbation wavelength, L, is much longer than 1 when N > 1. We thus can follow an argument of Shraiman [28] to study long-wavelength perturbations to these states. Shraiman observes that equation (1.1) possesses translation and Galilean symmetries; see (1.3) and (1.4). This suggests that the appropriate perturbation takes the form

(3.16) u(x, t) = ul(x + ?(x, t)) + ?(x, t),

where q is the phase mode and ( is the Galilean mode; both vary slowly on space and time. More precisely, qt, Ox e << 1, c 6, and (t, (x -2. Shraiman then seeks evolution equations for qt and (t in the form of polynomials in Ox, (, and their derivatives with respect to x. The possible form of the equations is restricted by the requirement that they respect reflection, translation, and Galilean symmetries which (1.1) possesses so that to Q(E3), q and ( satisfy

(3.17) qt + q + (+x + zqxx + _yxx + !OXqqxx = 0,

(3.18) (t + (( + >(xx + o5xx + 4ufx/xx = 0.

The coefficients iz, -y, a, and so on are functionals of the cellular state ul(x) and therefore functions of the cellular state wavelength 1. The relation between them can be established by considering the case when q contains a contribution corresponding to a global dilation of the cellular pattern, i.e., q -4 X + &kox, where 6k0 << 1. Then the appropriate form for u(x, t) can be written in two equivalent ways:

u(x, t) = u1 (x + q(x, t) + 6kox) + ((x, t) (3.19) = uT(x + (1 - 6ko)>(x, t)) + ((x, t),

where 1 + 61 = I1 - 27r6ko/k 2. Each form leads to the set of equations (3.17) and (3.18): in the first case q is replaced by X + Skox and the coefficients iz and -y are evaluated at wavelength 1, while in the second case q is replaced by (1 - 6ko)q and the coefficients are now evaluated at the I so that i'(l) = i'(l) + 61,'. Equivalence between the two sets of equations then imposes the following set of constraints on the coefficients:

(3.20) lby'=y l = -r l/ =-r.

(3.21) v= 0, la' +? = -0,7 lo/p + 203 = 0,




where the prime denotes the derivative with respect to 1, d/dl. These equations have the solutions

(3.22) v= a, =a21, 3, = a3/i,

(3.23) = a4/l, a = a3/P + a5/l, i = a4/l+ a6,

where ai (i = 1, . . . , 6) are arbitrary constants (independent of 1), and for later con- venience we have introduced 1 = 1/27r.

We now establish a connection between the parameters v, -y, and so on and the Ai introduced in the previous subsection. A linear stability analysis of equations (3.17) and (3.18), to perturbation proportional to exp(At + iqkox), gives

(3.24) A = (, + v)k 2q2 ? iyV'koIq(

Here q = m/N, with m being an integer and m << N. To leading order, we have instability when a- < 0. When a > 0, the leading-order contribution to A is purely imaginary, and instability at second order occurs whenever (iz + v) > 0; U is then related to the frequency of these modes.

A useful comparison can be made between (3.24) and results from the previous subsection. Evidently, ko2a corresponds to -A 2 given by (3.14) so that we have

(3.25) ~ ~~~~~~_1-&(X2) (3.25) - -1 X = a3/P + as /l 2 &l

This is easily integrated. Using the requirement that (X2) = 0 when I = 1 to eliminate one of the constants, we obtain

(3.26) (X2) = - (a7 + 2a5)/i + 2a5/1 + a7.

This has a turning point at I= (a7 + 2a5)/a5. We choose the free parameters a5 and a7 so that (X2) takes its maximum recorded value 2c2ax at im. The relevant part of (3.26) is shown in Figure 6 together with the actual noose; the agreement on the upward-sloping section is rather good.

Consider now the criterion (1.8). Here the relevant quality is (r, + k)ko, which, from (3.23), is

(3.27) ko2Q + v) = a4/? +(a, + a6)/P.

The choice of the parameter values a, + a6 =-2a7 and a4 = -2a5 permits this to be written in the form

(3.28) k z + V) = - i dl( X2))*

Instability results whenever d (l2 (X2)) < 0, i.e., d (c2/a2) < 0, which is just (1.8). Along the primary branch, the quality c2/a2 has a maximum at a 0.768. Therefore along the N branch (N > 1), the Hopf bifurcation occurs at

(3.29) =0.768

In Figure 7, we compare our numerical results for Hopf bifurcation points and the above asymptotic approximation for N = 2,3,4,5,6. Note that although (3.29) is




2,0

1.0-

t 1,0 T 2,0

FIG. 6. A comparison of the result given by (3.26) (the dotted line) and the actual noose (the solid line).

0,4

N 0,3 -

0,2-

0,1 0 o

1 2 3 4 5 6 N

FIG. 7. The critical values of the parameter a for the branches N = 2,3,4,5,6. The solid line represents the asymptotic approximation (3.29), and the circles represent the (exact) numerical result.

valid only for sufficiently large N, it is surprisingly accurate even for N = 2, with an error within 5%. Clearly, as N increases, the agreement improves.

While the discussion above is not a rigorous proof, it certainly supports the conjecture that the second criterion for the instability is given by (1.8). The result (3.28) requires only an appropriate choice of a4 and (a, + a6). Any attempt to fit k2 (i + v) to lq diP(X2)) necessarily requiresp= 2, which is the essence of our conjecture.

In order to see clearly the role played by a and 2 (,s + v) in the stability of the cellular states, we now "diagonalise" (3.17) and (3.18) using the transformation

(3.30) +

(3.31) ? = -

( + + AGx)




where 4 = Xx and j = 779. To the required order, the evolution equations are transformed to

(3.32) (t + VJr 2 (xx= -24xr + 4 )

(3-33) t V7x + 2 (x -26(x + 4 [ )

where X = (a - /3)/a. Higher-order terms have been omitted. The roles of a and 2(z + v) are now clear. When a > 0, W can be regarded as the wave velocity, and ( and ( evolve along different characteristics; when a < 0 the characteristics are purely imaginary, indicating instability. The coefficient - 2 (iz + v) can be interpreted as a diffusion coefficient. When a > 0 and (iz + v) > 0, the system becomes linearly unstable through negative diffusion of the slow-phase variables. For this reason, we term (1.8) the diffusion condition; cf. [11, 28].

4. Conclusion and final comments. In conclusion, we have investigated the bifurcations and the stability of a class of cellular states of the KS equation. We show that the four elliptic regions of the primary branch of the associated ODE play an important role. Namely, in each of these regions there exists a dense subset of parameter values at which m/k-bifurcations occur. These bifurcations, together with the replicas of the primary period-doubling and symmetry-breaking bifurcations, change the dimension of the unstable manifold of the cellular states in an ordered way. A general rule for this is summarised. We observe that the instability of high branches of cellular states exhibits certain universal features and can be explained by asymptotic analysis and/or phase dynamics. Using these approaches, we show that they lose stability either through negative elasticity or through negative diffusion. The latter instability corresponds to the Hopf bifurcations and gives rise to a set of oscillatory waves. The critical value of the control parameter at which this instability occurs is estimated rather accurately.

Although we have considered only the special case of the symmetric states, the same considerations may carry over to other families of travelling-wave solutions of the KS equation. In other words, the elliptic regions of the corresponding periodic solutions is likely to change the dimension of the unstable manifold of these waves in the manner outlined in ?2.2, and an asymptotic analysis of the slow-phase variable may provide complementary information, although the asymptotic results will be different from (1.7) and (1.8).

Acknowledgements. The authors would like to thank Dr. P. Kent for helpful discussion. Thanks are also extended to the referees for their constructive comments and suggestions, particularly to one of the referees who kindly pointed out to us that some of our observations may be formally proven by using group theoretic techniques.

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stability of cellular states of the kuramoto-sivashinsky equation

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