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Rostock. Math. Kolloq. 49, 163{184 (1995)

Subject Classi cation (AMS)

Primary 76E30; Secondary 58F12

Peter Takac

Dynamics on the Attractor for the Complex Ginzburg-Landau Equation1

Dedicated to the professors of mathematics L. Berg, W. Engel, G. Pazderski, and H.- W. Stolle.

ABSTRACT. We present a numerical study of the large-time asymptotic behavior of solutions to the one-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. Our parameters belong to the Benjamin-Feir unstable region. Our solutions start near a pure-mode rotating wave that is stable under sideband perturbations for the Reynolds number R ranging over an interval (Rsub ; Rsup ). We nd sub- and super-critical bifurcations from this stable rotating wave to a stable 2-torus as the parameter R is decreased or increased past the critical value Rsub or Rsup . As R > Rsup further increases, we observe a variety of dynamical phenomena, such as a local attractor consisting of three unstable manifolds of periodic orbits or 2-tori cyclically connected by manifolds of connection orbits. We compare our numerical simulations to both rigorous mathematical results and experimental observations for binary uid mixtures. KEY WORDS. Periodic orbit; 2- and 3-tori; stability; local attractor; psedo-spectral method; Fourier modes

1 Introduction

An important tool in the theory of phase transitions and instability waves is the timedependent complex Ginzburg-Landau equation (CGL, for short)

2 @tA = (1 + i )@xxA + (R ? (1 + i )jAj2)A; ?1 < x < 1; t 0;

1

(1)

This work was supported in part by the Applied Mathematical Sciences subprogram of the O ce of Energy Research, United States Department of Energy, under Contract W-31-109-Eng-38 through Argonne National Laboratory. The author expresses his appreciation to the Mathematics and Computer Science Division at Argonne National Laboratory for the hospitality during his 1990{91 academic leave.

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where x and t are the spatial and temporal variables, respectively, and the unknown complexvalued function A(x; t) represents an order parameter or a wave function. Here, , and R 2 (?1; 1) are parameters. Originally discovered by Ginzburg and Landau 6] for a phase transition in superconductivity, this equation was subsequently derived for instability waves in hydrodynamics such as the nonlinear growth of Rayleigh-Benard convective rolls (Newell and Whitehead 17]), the appearance of Taylor vortices in the Couette ow between counter-rotating cylinders (Stuart and Di Prima 21]), and the development of Tollmien-Schlichting waves in plane Poiseuille ows (Blennerhassett 1]). Also instability waves for perturbation concentration in chemically reacting and di using systems are described by the CGL equation (Kuramoto and Tsuzuki 12]). In these applications, Eq. (1) describes the small and slowly varying (in space and time) amplitude and phase of a mode that bifurcates via an oscillatory instability from a homogeneous basic state (Newell 16]). The parameter R corresponds to a Reynolds number; we use it as the bifurcation (or control) parameter. We impose periodic boundary conditions

A(x + 1; t) = A(x; t); ?1 < x < 1; t 0:

(2)

For example, periodic boundary conditions are appropriate for experiments with RayleighBenard convective rolls in a binary uid mixture contained in a cell of annular geometry, where the complex amplitude A(x; t) describes the wave moving along the boundary between two concentric convective rolls, cf. Janiaud et al. 8, 9]. The oscillatory instability of the two convective rolls develops as the control parameter R increases and crosses a critical value R0 > 0. At R = R0 the boundary between the two concentric rolls is a circle. A generic point on this circle is determined by its azimuthal angle , 0 < 2 . Hence x = =2 in Eqs. (1, 2). For R > R0 near R0, after a transient, a wave rotating either clockwise or counter-clockwise settles along the circular boundary between the two rolls. The complex amplitude A(x; t) of this rotating wave is a solution of Eqs. (1, 2) given by

A(x; t) = anei(knx?!n t); n = 0; 1; 2;

;

(3)

where kn = 2n is the wavenumber of the annular spatial pattern with jnj wavelengths, an is 2 2 a complex constant satisfying janj2 = R ? kn , and !n = R + ( ? )kn . The real constants and are determined by measuring the angular frequency !n for several values of R. As R further increases from R0 R0;n and crosses a critical value R1 R1;n > R0, the spatial pattern of the rotating wave starts to exhibit an amplitude modulation of the form

jA(x; t)j = a(x ? ct); a 6 const;

(4)

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165

with a second frequency c c(R) near cn = c(R1) 6= 0 for R > R1 near R1. This temporally biperiodic pattern is characterized by the complex wave function A(x; t) = (1 + U (x ? ct))anei kn(x?ct)?!t] V (x ? ct)e?i!t; (5) where U is a small relative perturbation of the rotating wave anei kn(x?ct)?!t] and ! is a small perturbation of !n , for R ? R1 > 0 small (cf. Takac 22]). In our numerical simulations we x the constants = ?1 and = 5, and thus, by the Newell criterion 15], we are in the Benjamin-Feir unstable region 1 + < 0. We focus on the large-time asymptotic behavior of the solutions A(x; t) to Eqs. (1, 2) starting from an initial distribution A(x; 0) near the rotating wave (3) for n = 1. In particular, we investigate the sideband instability of this rotating wave, cf. Eckhaus 5]. We have performed numerical simulations for a large range of values of the bifurcation parameter R 2 (0; 1). We report here only those cases which clearly exhibit local attractors of special interest. For particular values of R, we have obtained stable rotating waves, 2-tori, and periodic orbits that have a temporally constant modulus which becomes time-periodic as R passes a critical value. We determine several critical values of R at which a bifurcation from a stable periodic orbit to a stable 2-torus takes place. We have also obtained local attractors consisting of two or three unstable manifolds of periodic orbits or 2-tori cyclically connected by manifolds of connection orbits, where the manifolds of periodic orbits or 2tori have distinct dimensions. We compare these periodic orbits and 2-tori with those ones previously obtained by rigorous analysis and/or numerical simulations, cf. Newton and Sirovich 18, 19], Sirovich and Newton 20], and Takac 22]. This comparison provides some analytic background for our numerical simulations. We emphasize that our simulations using periodic boundary conditions (2) are di erent from those performed in Moon et al. 14], Keefe 10], and Doelman 3] with Neumann boundary conditions. Since we are also interested in the stability of the solutions to Eq. (1) in a suitable Banach or Hilbert space of smooth 1-periodic functions f : R ! C , we do not presuppose any special form of these solutions. The numerical simulations presented in this article are the rst ones studying the stability of 2-tori for the full system (1, 2). As usual, we write R = (?1; 1) and C = R iR to denote the real line and the complex plane, respectively. Our numerical integration of the evolution equation (1) uses a pseudo-spectral method (PSM, for short) for su ciently long time (from t = 0 up to t = 1000). This PSM consists of an approximation by Fourier series in space (with ?N th through N th Fourier modes, N = 15, 31 and 63) combined with a fourth-order Runge-Kutta discretization in time (with the time step h = 0:0001, 0:0002 and 0:0005). We have implemented this method on an Ardent Titan Supercomputer with four parallel processors using a FORTRAN code with both vector and parallel optimizations. All computations have been carried out in double precision complex arithmetics.

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This article is organized as follows. In Section 2 we study bifurcations from rotating waves (3) to the 2-tori (5) together with their stability. In Section 3 we observe certain periodic orbits that have a temporally constant modulus which becomes time-periodic as R passes a critical value. In Section 4 we obtain local attractors consisting of two or three unstable manifolds of periodic orbits or 2-tori cyclically connected by manifolds of connection orbits. Finally, Section 5 contains a discussion.

2 Bifurcations from Rotating Waves

In this section we investigate bifurcations from stable rotating waves of the form (3) to the 2-tori (5) together with the exchange of stability, for n = 1. We nd a stable 2-torus (5) for 137:9 R 139:7 which bifurcates supercritically from a stable rotating wave (3). A perturbation of the rotating wave (3) can be written as

A(x; t) = (1 + B (x; t))anei(knx?!nt);

(6)

where B is small enough. The sideband instability analysis is carried out by inserting Eq. (6) into (1) and retaining only the terms of the rst order in B (cf. Doering et al. 4] or Takac 22]):

2 @tB = (1 + i )@xxB + 2i(1 + i )kn @xB ? (1 + i )janj2(B + B );

(7)

2 where janj2 = R ? kn. The asterisk denotes the complex conjugate. The sideband stability of the zero solution of Eq. (7) to perturbations of a discrete wavenumber km = 2 m, m = 1; 2; , is determined by writing

B (x; t) = b+ (t)eikmx + (b?(t)) e?ikm x:

The evolution equations for the complex amplitudes b+ (t) and b? (t) have the form

(8)

C+ (1 + i )janj2 d b+ = ? dt b? (1 ? i )janj2 (C ?)

where

!

b+ b?

(9) (10)

2 C = (1 + i )km 2(1 + i )kmkn + (1 + i )janj2:

2 The neutral stability curves (R; kn ) for the zero solution of Eq. (9) satisfy the relation (cf. 4, 22])

"

4k2 1 +

2 km + (R ? k2) 2 km + R ? k 2

2#

2 = 2(1 + )(R ? k2) + (1 + 2)km

(11)

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167

Figure 1: Neutral stability curves in the (R1=2; k)-plane for the parameter values ( ; ) = (?1; 5). where the continuous variable k replaces the discrete wavenumber k = kn . For ( ; ) = p (?1; 5) these curves are plotted in the ( R; k)-plane in Fig. 1. Each curve corresponds to a xed value of jmj = 1; 2; . For a xed value of R > 0, the rotating wave (3) is stable to sideband perturbations (8) if and only if exactly one solution k of Eq. (11) satis es k > jknj. In particular, the rotating wave for n = 1 is stable to all sideband perturbations if and only if the Reynolds number R belongs to an interval (Rsub ; Rsup ). From a magni cation of the graph in Fig. 1 (cf. Takac 22]) for m = n = 1, we have found the numerical values (precise up to computer round-o errors)

Rsub = 84:96 and Rsup = 137:90:

(12)

The existence and uniqueness (up to shifts in space and time) of the 2-tori having the form (5) has been proved by Takac 22] using bifurcation theory for Rsub ?R > 0 and R?Rsup > 0 small enough. The second frequency c = cn for the critical values R = Rsub and R = Rsup can be computed in a similar way as the neutral stability curves. We insert Eq. (5) into (1) and retain only the terms of rst order in U (cf. Takac 22]) (1 + i )U 00 + (cn + 2i(1 + i )kn )U 0 ? (1 + i )janj2(U + U ) = 0;

2 where janj2 = R ? kn . This equation has a nonzero solution of the form

(13) (14) (15)

U (x) = b+eikm x + (b?) e?ikmx

if and only if the linear system ! C + ? ikmcn (1 + i )janj2 b+ = 0 (1 ? i )janj2 (C ?) ? ikm cn b?

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has a nonzero solution, where C is de ned by Eq. (10). This is the case if and only if R satis es Eq. (11) and

2 c = 2k( ? ) k2 R ? k k2 : m+R?

(16)

The numerical values (12) for m = n = 1 yield the respective frequencies

csub = 40:36 and csup = 53:81:

(17)

To study the stability of the 2-tori from Eq. (5) we have used numerical integration of the evolution equation (1). Applying our pseudo-spectral method (PSM, for short, described in Section 1) for su ciently long time (from t = 0 up to t = 1000), we have obtained the following numerical results for = ?1 and = 5:

2.1 A supercritical bifurcation at R = Rsup

The numerical value of Rsup obtained by PSM coincides with its value from (12) within the indicated precision, Rnum = 137:9. For R = Rnum the numerical solution of (1) slowly looses sup sup all Fourier modes except for the rst one, thus converging towards the pure-mode rotating wave (3) with n = 1. Various initial values at t = 0 led to a fast transition before t = 1 into a state with the rst Fourier mode more than 103-times larger than the remaining ones which also decayed exponentially with the increasing wavenumber. The decay of these sideband Fourier modes (n 6= 1) then slowly continued throughout the entire evolution (0 t 1000). On the other hand, for R = 138:0 a similar transition was observed, but after t = 1 the decaying sideband modes started to settle away from zero until t = 100 when their decay ceased completely. The rst mode was on the order of 105-times larger than the zeroth and second modes. From the graph of the amplitude modulation (4) the frequency c was found to be cnum = 1=0:0186 = 53:76. Increasing R up to the value R = 139:7 we observe sup increasing sideband modes. The amplitudes of these modes are constant in time with nearly linear logarithmic decay (Fig. 2(a)). The graph of the amplitude modulation (4) in Fig. 2(b) gives the frequency c = 1=0:0194 = 51:55 at R = 139:7. For 137:9 R 139:7 the stability of this temporally biperiodic motion (2-torus) was tested by various small perturbations. At R = 139:8 this 2-torus becomes unstable. The zeroth Fourier mode slowly increases and becomes the dominant one. Eventually the amplitudes of all modes become constant in time with nearly linear logarithmic decay again (Fig. 3(a)). The graph of the amplitude modulation (4) in Fig. 3(b) is time-independent (c = 0). During the transition from a 2-torus to a limit-cycle, the amplitude (4) decreases to zero at one point where a phase slip occurs.

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169

Figure 2: For R = 139:7, (a) logarithmic decay of Fourier modes; (b) amplitute modulation.

Figure 3: For R = 139:8, (a) logarithmic decay of Fourier modes; (b) amplitude modulation.

2.2 A subcritical bifurcation at R = Rsub

The numerical value of Rsup obtained by PSM di ers from its value from (12), Rnum = 84:99. sub num , for R = Rnum the numerical solution of (1) slowly looses Similarly to the case R = Rsup sub all Fourier modes except for the rst one, thus converging towards the pure-mode rotating wave (3) with n = 1. For R = 84:98 a stable 2-torus (5) was observed with cnum = 1=0:0248 = sub 40:32. At R = 84:97 this 2-torus becomes unstable. Again, the zeroth Fourier mode slowly increases and becomes the dominant one with the remaining Fourier modes approaching zero, and thus the rotating wave (3) with n = 0 is obtained. To summarize the results of this section, we have found a stable 2-torus (5) for 137:9 R 139:7 which bifurcates supercritically from a stable rotating wave (3) for n = 1. Although several previous articles 2, 8, 9, 11, 13] studied solutions similar to (5), their stability under the temporal evolution was not known.

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3 Bifurcations from Some Periodic Orbits

In this section we are concerned with primary and secondary bifurcations from the Stokes wave A(x; t) = a0 e?i!0 t which is the special case of a rotating wave (3) for n = 0. In the next section we will need these bifurcations in order to be able to explain the dynamics on some local attractors for the CGL equation. Some analytic and computational background for these simulations can be found in Newton and Sirovich 18, 19] and Sirovich and Newton 20]. Therefore we present here only those results that are either new or necessary for a proper understanding of the next section. The reader is referred to Guckenheimer and Holmes 7] for general facts about dynamical systems and bifurcations.

3.1 Separable periodic orbits

We begin with the stability of the Stokes wave for ( ; ) = (?1; 5). Similarly as in the previous section, the Stokes wave (n = 0) is stable to all sideband perturbations if and only if the Reynolds number R belongs to an interval (0; RI ), where : RI = ?2 2(1 + 2)=(1 + ) = 128:3049 (18) is the value of R satisfying Eq. (11) for k = 0 and m = 1, see Fig. 1. According to Newton and Sirovich 18], as the parameter R crosses RI , the Stokes wave bifurcates to another periodic orbit of the separable form

A(x; t) = F (x)e?i t:

Here is a real number, and F : R ! C is a 1-periodic function satisfying (1 + i )F 00 + (R + i ? (1 + i )jF j2)F = 0; ?1 < x < 1;

(19) (20)

by (1, 2) and (19). Our numerical simulations show that the periodic orbit (19) is stable for all R from an interval (RI ; RII ), where we have observed the value

RII = 162:4:

(21)

In particular, we have F 6 const. As R crosses RII , the periodic orbit (19) loses its stability and bifurcates to a stable 2-torus. An analytic approximation of the value RII is presented in Newton and Sirovich 19]. More precisely, we have obtained a local attractor A of real dimension 2 consisting of all functions f : R ! C such that

f (x) = F (x ? x0)ei

t0

for some x0; t0 2 R:

(22)

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171

Hence, A is a 2-torus. Furthermore, for all R 2 (139:7; RII ) = (139:7; 162:4) (see x2.1 and (21) above), our simulations show that the solutions A(x; t) to Eqs. (1, 2) starting from an initial distribution A(x; 0) near the rotating wave (3) for n = 1 are attracted to A.

3.2 2-tori with periodic modulus

As R increases past the critical value RII , the separable periodic orbit (19) bifurcates supercritically to a 2-torus. Our simulations show that also exchange of stability takes place. The corresponding stability analysis for this bifurcation has been carried out in Sirovich and Newton 20]. The 2-torus has the form A(x; t) = (F (x) + G(x; t))e?i t (x; t)e?i t; (23) where the perturbation function G : R R ! C takes small values for R ? RII > 0 small, and it is 1-periodic in x and -periodic in t, for some > 0. The half-period =2 coincides with the time-period of the zeroth Fourier mode of the square modulus jA(x; t)j2 = j (x; t)j2. As this mode is always real and nonnegative, can easily be determined from the graph of this mode versus the time t. For all R 2 (RII ; RII + 0:1) = (162:4; 162:5); we have observed approximately the same value of , (24) II = 1:23: The 2-torus (23) then remains stable for all values R 2 (RII ; 180:0]. More precisely, we have obtained a local attractor A of real dimension 3 consisting of all functions f : R ! C such that f (x) = (x ? x0; ?t1)ei t0 for some x0; t0; t1 2 R: (25) Hence, A is a 3-torus. Furthermore, for all R 2 (RII ; 180:0] = (162:4; 180:0]; our simulations show that the solutions A(x; t) to Eqs. (1, 2) starting from an initial distribution A(x; 0) near the rotating wave (3) for n = 1 are attracted to A. It is remarkable that the angular frequency in both cases (19) and (23) stays near the angular frequency of the Stokes wave 2 !0 = R + ( ? )k0 = ?R with a relative error of less than 1:5%, for all R 2 (RI ; 180:0] = (128:3; 180:0].

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3.3 A 2-torus with periodic modulus for R = 180:0

Now we will closer examine a transparent case of a 2-torus (23) for R = 180:0. We write

A(x; t) = r(x; t)ei

(x;t)?!0t];

(26)

where r(x; t) = jA(x; t)j, (x; t) 2 R, and !0 = ?180:0 is the angular frequency of the Stokes wave. In our gures below we limit the values of the phase angle (x; t) to the interval (? ; ], thus admitting jumps of size 2 in the values of (x; t). Given a real function f (x; t), we make its 3-dimensional plot as follows. The origin of our coordinate system is in the middle of the left side of our window, the axis x (axis t, respectively) runs from the origin towards the right lower (right upper) corner, and the values of f are plotted on the vertical axis.

Figure 4: For R = 180:0, the modulus jA(x; t)j for (a) 0 t 0:116; (b) 0:058 t 0:174.

Figure 5: For R = 180:0 , the negative phase angle ? (x; t) for (a) 0 (b) 0:058 t 0:174.

t

0:116 ;

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173

By (23), the modulus r(x; t) is 1-periodic in x and -periodic in t. The graphs of r(x; t) for (x; t) 2 0; 1] 0; ] and (x; t) 2 0; 1] =2; 3 =2], respectively, are plotted in Fig. 4(a,b). From magni cations of these plots we have obtained the period = 0:116. Analogously, the graphs of the negative phase angle ? (x; t) are plotted in Fig. 5(a,b). Again, from magni cations of these plots we have deduced that the relative error j ? !0j = j (x; t + ) ? (x; t)j j!0j j!0j is less than 1:5%. We will derive this estimate also from the following study of the complex Fourier modes cn (t) of the function A(x; t) of x 2 0; 1]. It follows from (23) that cn(t)ei t is a -periodic function of t. In Fig. 6(a,b) we plot the trajectories of the functions cn (t)ei!0t (0 t < 25 ) in the complex plane for each n = 0 and n = 1, respectively. The phase angle of the function

cn (t)ei t=cn (t)ei!0t = ei(

?!0 )t

is equal to ( ? !0)t. By (an animation leading to) Fig. 6(b), it takes approximately 21 time-periods of length for this angle to increase or decrease from 0 to 2 . Thus, the relative di erence between and !0 can be estimated by 2 = 2 j ? !0 j j!0j 21 j!0j 21 0:116 180:0 < 0:015: : : Consequently, the period 2 =j j = 2 =180 = 0:035 corresponding to is more than 3-times smaller than = 0:116.

Figure 6: For R = 180:0 , the complex plots of cn(t)ei!0t (0 t < 2:9) for (a) n = 0 ; (b) n = 1. To summarize the results of this section, we have found a stable 2-torus (23) for 162:4 < R 180:0 which bifurcates supercritically from a stable periodic orbit (19) at RII = 162:4.

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4 A Few Complicated Local Attractors

We have observed that the 2-torus (23) collapses as R is increased from 180:0 to 181:0. We have not detected any signs of a bifurcation from this stable 2-torus to a stable 3-torus for R 2 (180:0; 181:0). What we have detected for a number of values of R 2 181:0; 185:0] is an interesting local attractor which we analyze below.

4.1 A local attractor for R = 185:0

We start our simulations from an initial distribution A(x; t0) near the rotating wave (3), where n = 1, k1 = 2 , : ja1j = (185:0 ? (2 )2)1=2 = 12:0632; (27) : and !1 = ?185:0 + 6 (2 )2 = 51:8701. For a later comparison, the period 1 corresponding to the angular frequency !1 has the value : (28) 1 = 2 =!1 = 0:1211: After a transient time interval of 2 units we set our time variable t to t = 0 (i.e. t0 = ?2) and begin our observations. Our simulations show that the modulus jA(x; t)j remains both spatially and temporally constant for a relatively long time,

jA(x; t)j = 12:063 for all x 2 0; 1] and t 2 0; 0:37]:

Examining also the phase angle (x; t) from (26), we conclude that A(x; t) is very close to the rotating wave (3) with n = ?1, for all t 2 0; 0:37], see (27). As the time t increases past 0:37, the instability of the rotating wave causes both spatial and temporal deviations of the modulus jA(x; t)j from the value 12:063. By Fig. 7(a,b), these deviations have the form of an increasing travelling wave with a sine-like shape moving from the right to the left with the velocity c?1 = ?61 (approximately). Although the present parameter R = 185:0 is far beyond the bifurcation value Rsup = 137:90 from (12), formula (16) from our instability analysis in Section 2 yields a comparable value of the velocity (for m = 1, n = ?1) 0 ? (2 2 : c?1 = 2 (?2 ) 6 185:185:0 ) = ?59:31: ^ The deviations continue to increase up to t = 0:744 when a phase slip occurs at x = 0:40 where A(x; t) = 0, see Fig. 8(a,b). After this moment, A(x; t) starts approaching a 2-torus apparently having the form (23). This 2-torus can clearly be seen for 0:80 t 0:95 in

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175

Figure 7: For R = 185:0 , the deviations jA(x; t)j ? 12:063 for (a) 0:49 (b) 0:70 t 0:75.

t

0:50 ;

Figure 8: The phase slip for R = 185:0, (a) modulus; (b) negative phase angle. Fig. 9(a,b). It is very similar to the 2-torus in Fig. 4(a,b), but its period = 0:06 is about twice smaller than in the latter case ( = 0:116). As the time t increases past 0:95, the instability of the 2-torus becomes visible. For 0:95 t 1:06 we observe a connection orbit from this 2-torus towards a periodic orbit having the form (19), see Fig. 10(a,b) for the modulus jA(x; t)j. This modulus remains temporally constant for all t 2 1:06; 1:29] (Fig. 11(a)). It has a shape close to that of the function j sin k1(x ? x0)j, where k1 = 2 and x0 = 0:31. Inspecting also the negative phase angle ? (x; t) de ned in (26) (Fig. 11(b)), we conclude that A(x; t) has a form close to

A1(x; t) = "e?i t sin k1(x ? x0);

(29)

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Figure 9: For R = 185:0, the modulus jA(x; t)j for (a) 0:80 t 0:90; (b) 0:90 t 1:00. where j"j = 14:0 and

: = !0 ? (x; t + TT) ? (x; t) = ?185:0 + 0:2 = 66:33: 025 : 0:095. It is worth a mention that this solution A(x; t) The corresponding period is 2 = = very well agrees with the following analytic solution to Eqs. (1, 2) obtained in Takac 22] by rigorous bifurcation analysis, 1 ei tA(x; t) = " (30) %j 2 %j 2 2 sin kn x ? 32"j2 sin 3kn x + 32"j2 (3 sin 3kn x + sin 5kn x) + O(j"j6); kn kn where " is a complex bifurcation parameter whose square modulus j"j2 > 0 measures the 2 smallness of R ? kn > 0, kn = 2n 6= 0, % 1+ii , and R + i is related to j"j2 by 1+ 2 R + i ? (1 + i )kn = 3 j"j2 1 ? % j"j2 + O(j"j4) : (31) 2 1+i 4 32kn Together with the rotating wave (3), the solution (30) bifurcates from the zero solution as 2 R increases past kn . As the time t increases past 1:29, also this separable periodic orbit shows its instability. For 1:29 t 1:43 we observe a connection orbit from this periodic orbit towards a rotating wave (3) with n = ?1, see Fig. 12(a,b). This rotating wave is identical (up to multiplication by a complex unit re ecting a phase shift) with the one we have started from at t = 0. Again, our simulations show

jA(x; t)j = 12:063 for all x 2 0; 1] and t 2 1:43; 1:91]:

From this point on the entire scenario repeats as we have just described it. This can also be seen from the complex plots of the zeroth and rst Fourier modes of the function jA(x; t)j2

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Figure 10: For R = 185:0, the modulus jA(x; t)j for (a) 1:00 t 1:05; (b) 1:05 t 1:10.

Figure 11: For R = 185:0, (a) modulus for 1:06 1:09 t 1:10.

t

1:29; (b) negative phase angle for

of x 2 0; 1], see Fig. 13(a,b). It takes a time interval of (approximate) length 1:55 units for this scenario to repeat. It is somewhat amazing that the scenario repeats after approximately the same time interval. This fact strongly suggests that our numerical simulations have produced a homoclinic periodic orbit (of period T = 1:55) which shadows three heteroclinic connection orbits, the rst one running from the rotating wave (3) (n = ?1) towards the 2-torus (23), the second one running from this 2-torus towards the periodic orbit (30), and the third one returning from this periodic orbit towards the rotating wave (3) (n = ?1). Furthermore, even if we have started our simulations from A(x; t0) near the rotating wave (3) with n = 1, the simulated shadowing periodic orbit has never returned to this rotating wave again. We are unable to explain why our approximating dynamical system \prefers" the rotating wave (3) with n = ?1 to the one with n = 1. Notice that the approximating dynamical system uses the same number of positive and negative complex Fourier modes eikn x, jnj N , for the

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Peter Takac

Figure 12: A connection orbit for R = 185:0 and 1:30 t 1:40, (a) modulus; (b) negative phase angle.

Figure 13: For R = 185:0 and 0 t 10:0, (a) zeroth Fourier mode (real, on vertical axis) versus time (on horizontal axis); (b) rst Fourier mode (complex plot) of the square modulus jA(x; t)j2. spatial approximation, i.e. A(x; t) is approximated by the truncated Fourier series

AN (x; t) =

X

jnj N

cn (t)eiknx:

(32)

In particular, the corresponding truncation of Eq. (1) yields a system of 2N + 1 ordinary di erential equations (ODE's, for short) for cn(t), t 0, cf. Doelman 3]. Thus, the only possible reason for n = ?1 being preferred to n = 1 we can think of are some round-o computer errors, despite of double precision complex arithmetics used in our computations. In any event, the local attractor A suggested by our simulations must be symmetric with respect to the re ection x 7! ?x (i.e. n 7! ?n) because it contains the periodic orbit (30) having this re ection symmetry, and this periodic orbit is connected to all, the 2-torus (23) and the rotating waves (3) with n = 1.

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More precisely, given the symmetries of the CGL equation, we have obtained a local attractor A consisting of (i) two 1D manifolds (circles) A01 and A0?1 of rotating waves (3) for n = 1, respectively; (ii) a 3D manifold (a 3-torus) A00 of 2-tori (23); (iii) a 2D manifold (a 2-torus) A000 of separable periodic orbits (30); and (iv) manifolds of connection orbits, C 0 1, C 00 and C 0001. Here, C 0 1 connects from A0 1 to A00 and has real dimension 3, C 00 connects from A00 to A000 and has 3D, and C 0001 connects from A000 to A0 1 and has 2D.

4.2 A local attractor for R = 250:0

This case is a simpli cation of the previous one from x4.1. Again, our simulations start from an initial distribution A(x; t0) near the rotating wave (3), where n = 1, k1 = 2 , : ja1j = (250:0 ? (2 )2)1=2 = 14:51; (33)

: and !1 = ?250:0+6 (2 )2 = ?13:13. After a transient time interval of about 2 units we set t to t = 0 and begin our observations. For 0 t 0:005, the modulus jA(x; t)j stays within the interval (14:4; 14:6) for all x 2 0; 1], see Fig. 14(a). Examining also the phase angle (x; t) from (26) in Fig. 14(b), we conclude that A(x; t) is very close (with relative precision < 1%) to the rotating wave (3) with n = ?1, for all t 2 0; 0:005], see (33). However, Fig. 14(a,b) suggest that an unstable 2-torus may be present. This 2-torus may be the reason why the computed orbit does not get closer to the rotating wave.

Figure 14: For R = 250:0 and 0 t 0:005, (a) modulus; (b) negative phase angle.

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Figure 15: For R = 250:0 , the deviations jA(x; t)j ? 14:5 for (a) 0 (b) 0:05 t 0:10.

t

0:05 ;

As the time t increases past 0:005, we observe both spatial and temporal deviations of jA(x; t)j ? 14:5 from 0 forming an increasing travelling wave with a sine-like shape moving from the right of the left with the velocity c1 = ?50 (approximately), see Fig. 15(a,b). The deviations continue to increase up to t = 0:085 when a phase slip occurs at x = 0:51 where A(x; t) = 0. In this case we observe no sign of A(x; t) approaching a 2-torus (23). Rather, after the phase slip occurs, A(x; t) starts approaching a periodic orbit having the form (30), see Fig. 16(a,b). The modulus jA(x; t)j remains temporally constant for all t 2 0:23; 0:25]. Moreover, in (29) we have j"j = 17 and

: = !0 ? (x; t + T ) ? (x; t) = ?250:0 + 2 = 1:33: T 0:025

Figure 16: For R = 250:0, (a) modulus for 0:23 0:24 t 0:25.

t

0:25; (b) negative phase angle for

As the time t increases past 0:25, also this separable periodic orbit shows its instability. For 0:25 t 0:315 we observe a connection orbit from this periodic orbit towards a rotating wave (3) with n = ?1, see Fig. 17(a,b). This rotating wave is identical (up to multiplication

Dynamics of the Ginzburg-Landau equation

181

Figure 17: A connection orbit for R = 250:0 and 0:25 t 0:30, (a) modulus; (b) negative phase angle. by a complex unit re ecting a phase shift) with the one we have started from at t = 0. Again, our simulations show

jA(x; t)j ? 14:5 < 0:1 for all x 2 0; 1] and t 2 0:315; 0:320]:

From this point on the entire scenario repeats as we have just described it. This can also be seen from the complex plots of the zeroth and rst Fourier modes of the function jA(x; t)j2 of x 2 0; 1]. It takes a time interval of (approximate) length 0:316 units for this scenario to repeat. Given the symmetries of the CGL equation, we have obtained a local attractor A consisting of (i) two 1D manifolds (circles) A01 and A0?1 of rotating waves (3) for n = 1, respectively; (ii) a 2D manifold (a 2-torus) A00 of separable periodic orbits (30); and (iii) four 2D manifolds of connection orbits, C 0 1 and C 00 1. Here, C 0 1 connects from A0 1 to A00, and C 00 1 connects from A00 to A0 1.

4.3 Separable periodic orbits for R = 300:0 and R = 350:0

Both these cases are analogous to the case R 2 (RI ; RII ) = (128:3049; 162:4) from x3.1. The separable periodic orbits (19) have the angular frequencies = ?161:30 for R = 300:0 and = ?257:87 for R = 350:0. The shape of their modulus jA(x)j and the phase angle j (x)j from (26) suggest that these periodic orbits should lie on the same bifurcation branch of separable periodic orbits as those in x3.1. To summarize the results of this section, for each R = 185:0 and 250:0, we have found an

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interesting local attractor A. For R = 185:0, A consists of two 1D manifolds (circles) of rotating waves (3) for n = 1, a 3D manifold (a 3-torus) of 2-tori (23), a 2D manifold (a 2-torus) of separable periodic orbits (30), and manifolds of connection orbits. For R = 250:0, A consists of two 1D manifolds (circles) of rotating waves (3) for n = 1, a 2D manifold (a 2-torus) of separable periodic orbits (30), and four 2D manifolds of connection orbits.

5 Discussion

For the supercritical bifurcation at R = Rsup = 137:90 we have obtained a stable 2-torus for 137:9 < R < 139:7. For the subcritical bifurcation at R = Rsub our results are qualitatively similar to the experimental and numerical results obtained by Janiaud et al. 8, 9] who observed a 2-torus given by Eq. (5) persisting for long time before it collapsed. We have observed its instability outside the tiny interval 84:97 < R < 84:99; the stability for R = 84:98 may be due to numerical errors, cf. (12). Finally, the numerical simulations studying the stability of the 2-tori having the forms (5) and (23) suggest that these 2-tori may bifurcate into (possibly unstable) 3-tori having the following form, A(x; t) = B (x ? ct; t= )e?i t; ?1 < x < 1; t 0; (34) where c 6= 0, 6= 0, > 0, and B (x; ) is 1-periodic in both x; 2 R. Namely, given the symmetries of the CGL equation, the 2-tori (23) form a 3D manifold which itself is a 3-torus (25) invariant under the semi ow generated by Eqs. (1, 2). Hence, it is very reasonable to expect the possibility of a bifurcation that preserves this 3-torus, but the 2-tori (23) bifurcate into a triperiodic motion (34). In particular, inserting (34) into (1) we obtain the following boundary value problem for the unknown (1; 1)-periodic complex-valued function B (x; ), ?1 @ B ? (1 + i )@ 2 B ? c@ B ? (R + i )B = ?(1 + i )jB j2B; x; 2 R: (35) x xx Here, suitable values for the real constants 6= 0, 6= 0, > 0, c 6= 0, R > 0, and 6= 0 are to be found so that Eq. (35) admits a (1; 1)-periodic solution B (x; t) such that its modulus jB (x; t)j is nonconstant in both x and t. A di erent 3-torus motion was obtained in Moon et al. 14] by numerical simulations.

References

1] Blennerhassett, P. J. : On the generation of waves by wind. Philos. Trans. Roy. Soc. London, Ser. A, 298, 451{494 (1980)

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2] Doelman, A. : Slow time-periodic solutions to the Ginzburg-Landau equation. Physica D 40(2), 156{172 (1989) 3] Doelman, A. : Finite-dimensional models of the Ginzburg-Landau equation. Nonlinearity 4(2), 231{250 (1991) 4] Doering, C. A., Gibbon, J. D., Holm, D. D. and Nicolaenko, B. : Lowdimensional behaviour in the complex Ginzburg-Landau equation. Nonlinearity 1(2), 279{309 (1988) 5] Eckhaus, W. : Studies in Nonlinear Stability Theory. Berlin 1965 6] Ginzburg, V. L. and Landau, L. D. : On the theory of superconductivity. Zh. Eksp. Teor. Fiz. (USSR) 20, 1064 (1950) English transl. in: Ter Haar, D. (ed.): Men of Physics: L. D. Landau. Vol. I, pp. 546{568. New York 1965 7] Guckenheimer, J. and Holmes, P. : Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York 1983 8] Janiaud, B., Guyon, E., Bensimon, D. and Croquette, V. : Sideband instability waves with periodic boundary conditions. In: Busse, F. H. and Kramer, L. (eds.): Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems. NATO ASI Series B; pp. 45{50. New York 1990 9] Janiaud, B., Pumir, A., Bensimon, D., Croquette, V., Richter, H. and Kramer, L. : The Eckhaus instability for travelling waves. Physica D 55(3&4), 269{ 286 (1992) 10] Keefe, L. R. : Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation. Stud. Appl. Math. 73, 91{153 (1985) 11] Kuramoto, Y. : Phase dynamics of weakly unstable periodic structures. Prog. Theor. Phys. 71(6), 1182{1196 (1984) 12] Kuramoto, Y. and Tsuzuki, T. : On the formation of dissipative structures in reaction-di usion systems. Prog. Theor. Phys. 54(3), 687{699 (1975) 13] Landman, M. J. : Solutions of the Ginzburg-Landau equation of interest in shear ow transition. Stud. Appl. Math. 76, 187{237 (1987) 14] Moon, H. T., Huerre, P. and Redekopp, L. G. : Transitions to chaos in the Ginzburg-Landau equation. Physica D 7(1-3), 135{150 (1983) 15] Newell, A. C. : Envelope equations. Lectures in Appl. Math. 15, 157{163 (1974)

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16] Newell, A. C. : The dynamics of patterns: A survey. In: Wesfreid, J. E., Brand, H. R., Manneville, P., Albinet, G. and Boccara, N.: Propagation in Systems Far from Equilibrium. pp. 122{155. Berlin 1988 17] Newell, A. C. and Whitehead, J. A. : Finite bandwidth, nite amplitude convection. J. Fluid Mech. 38(2), 279{303 (1969) 18] Newton, P. K. and Sirovich, L. : Instabilities in the Ginzburg-Landau equation: periodic solutions. Quart. Appl. Math. 44(1), 49{58 (1986) 19] Newton, P. K. and Sirovich, L. : Instabilities in the Ginzburg-Landau equation II. Secondary bifurcation. Quart. Appl. Math. 44(2), 367{374 (1986) 20] Sirovich, L. and Newton, P. K. : Periodic solutions of the Ginzburg-Landau equation. Physica D 21(1), 115{125 (1986) 21] Stuart, J. T. and Di Prima, R. C. : The Eckhaus and Benjamin-Feir resonance mechanisms. Proc. Roy. Soc. London Ser. A 362, 27{41 (1978) 22] Takac, P. : Invariant 2-tori in the time-dependent Ginzburg-Landau equation. Nonlinearity 5(2), 289{321 (1992)

received: September 6, 1995 Author:

Prof. Dr. Peter Takac, Ph.D. Fachbereich Mathematik Universitat Rostock Universitatsplatz 1 D{18055 Rostock, Germany e-mail: peter.takac@mathematik.uni-rostock.de

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