LOW-DIMENSIONAL MODELS OF THEGINZBURG–LANDAU EQUATION∗

FRANK KWASNIOK†

SIAM J. APPL. MATH. c© 2001 Society for Industrial and Applied MathematicsVol. 61, No. 6, pp. 2063–2079

Abstract. Low-dimensional numerical approximations for two boundary value problems of thecomplex Ginzburg–Landau equation in a chaotic regime are constructed. Spatial structures calledprincipal interaction patterns are extracted from the system according to a nonlinear variationalprinciple and used as basis functions in a Galerkin approximation. The dynamical description interms of principal interaction patterns requires fewer modes than more conventional approachesusing Fourier modes or Karhunen–Loeve modes as basis functions.

Key words. low-dimensional numerical approximations, optimal basis functions, nonlinearvariational principle, Ginzburg–Landau equation

AMS subject classifications. 35A40, 65M60, 65K10, 35Q35

PII. S0036139900368212

1. Introduction. Given the well-known fact that solutions to dissipative partialdifferential equations are often attracted to manifolds of relatively low dimension, theconstruction of minimal dynamical approximations capturing the principal propertiesof the long-term behavior of the complete system is an interesting task. A class offrequently used numerical approximation schemes is given by the Galerkin methods.The efficiency of such Galerkin schemes, i.e., the number of degrees of freedom re-quired to capture the long-term dynamics of the partial differential equation, dependscrucially on a proper choice of the basis functions. Traditionally, the basis functionsare eigenfunctions of a suitably chosen linear differential operator, commonly Fouriermodes. However, such Fourier–Galerkin methods cannot be expected to be very effi-cient since Fourier modes are completely general. A dynamical description based onmodes adapted to the particular system under consideration should be more adequatewhen searching for a minimal model.

The Karhunen–Loeve (KL) decomposition (also referred to as principal compo-nent analysis, empirical orthogonal function analysis, or proper orthogonal decompo-sition) has occasionally been used as a tool to arrive at low-dimensional dynamicalapproximations of partial differential equations [16, 3, 15, 10, 11, 12, 13]. The KLmodes allow for an optimal representation of an attractor in phase space in a meanleast-squares sense with a given number of modes. They are easily obtained as theeigenfunctions of the covariance tensor of the system. The KL approach has beenextended to the Sobolev eigenfunctions [7]. However, the optimality criterion definingthe KL modes and Sobolev eigenfunctions, respectively, does not refer to the timeevolution of the truncated system obtained when projecting the partial differentialequation onto these modes. As has been pointed out by Hasselmann [6], a methodol-ogy referring simultaneously to both spatial and temporal features of the system bytaking into account the dynamics of the reduced model in order to define the basisfunctions may be even more efficient. The proposition of Hasselmann has been illus-trated and applied to a geophysical fluid system by Kwasniok [8]. It has also been

∗Received by the editors February 16, 2000; accepted for publication (in revised form) November15, 2000; published electronically April 24, 2001.

http://www.siam.org/journals/siap/61-6/36821.html†Leibniz-Institut fur Atmospharenphysik, Schlossstraße 6, 18225 Kuhlungsborn, Germany

(kwasniok@iap-kborn.de).

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2064 FRANK KWASNIOK

used in [2, 1]. The methodology of Hasselmann has been extended by Kwasniok usingan adjoint technique and applied to a periodic solution of the Kuramoto–Sivashinskyequation [9].

In the present paper the method derived in [9] is applied to two boundary valueproblems of the complex Ginzburg–Landau equation with chaotic dynamics of differ-ent degree of complexity. The paper is organized as follows. In section 2, the modelsystems considered in the present study are introduced. Then the methodology of con-structing minimal dynamical approximations to these systems is described in detail.In section 4, the results are given and discussed.

2. The Ginzburg–Landau equation. The Ginzburg–Landau equation for thecomplex-valued amplitude A(x, t) on the domain 0 ≤ x ≤ π reads

∂A

∂t− q2(i+ c0)

∂2A

∂x2 − ρA− (i− ρ)A|A|2 = 0,(1)

with q, c0, and ρ real and positive constants. The Ginzburg–Landau equation appearsas an envelope equation in the weakly nonlinear treatment of various fluid systemssuch as plane Poiseuille flow, Rayleigh–Benard convection, and Taylor–Couette flowas well as reaction-diffusion systems. It describes the nonlinear time evolution ofthe amplitude of linearly unstable modes close to bifurcations including the physicalmechanisms of linear growth, diffusion, dispersion, and nonlinearity. In this study,the Ginzburg–Landau equation is regarded as interesting in its own right as a modelsystem for the investigation of low-dimensional nonlinear dynamics. The parametersetting c0 = ρ = 0.25 is used (as in [16, 10, 17]) which does not correspond to anyparticular physical application.

Two boundary value problems for the Ginzburg–Landau equation are considered.The first one is specified by homogenous Neumann boundary conditions

∂A

∂x(0, t) =

∂A

∂x(π, t) = 0 ∀ t(2)

and a parameter value q = 0.95. The second one is specified by homogenous Dirichletboundary conditions

A(0, t) = A(π, t) = 0 ∀ t(3)

and q = 0.14. These two boundary value problems have been introduced in [16, 17]and have been used as model systems for the construction of low-dimensional modelsbased on KL modes in [16, 15, 10]. It can be shown that all solutions to the twoboundary value problems are pointwise bounded for all times. Moreover, with anysolution A also A exp(iφ) for an arbitrary real constant φ is a solution and satisfiesthe respective boundary condition. Therefore the ensemble mean vanishes:

〈A〉 = 0.(4)

For both boundary value problems the scalar product

[f, g] =1

π

∫ π

0

f∗(x) g(x) dx(5)

for two sufficiently smooth functions f and g on the interval 0 ≤ x ≤ π is introduced.The asterisk denotes the complex conjugate.

LOW-DIMENSIONAL MODELS 2065

3. Finite-dimensional approximations. The construction of a minimal dy-namical model proceeds in two steps: First, the Ginzburg–Landau equation is castinto a finite-dimensional dynamical description using a standard Fourier–Galerkin pro-cedure. Second, a variational principle is applied in this finite- but high-dimensionalphase space in order to identify a low-dimensional subspace optimally suited for theconstruction of a reduced model.

3.1. Spectral basis. The suitable set of basis functions for a Galerkin approx-imation of the Neumann problem is

{ϕα;α = 1, . . . , N} = {1} ∪{√2 cosnx;n = 1, . . . , n�

}.(6)

The basis contains N = n�+1 modes corresponding to 2n�+2 real degrees of freedomwhen truncating at wavenumber n�. For the Dirichlet case the basis is

{ϕα;α = 1, . . . , N} ={√2 sinnx;n = 1, . . . , n�

}.(7)

It contains N = n� modes corresponding to 2n� real degrees of freedom when trun-cating at wavenumber n�. In both cases the modes form an orthonormal set:

[ϕα, ϕβ ] = δαβ , α, β = 1, . . . , N.(8)

Considering the linear subspace spanned by the N modes

F = span{ϕ1, . . . , ϕN},(9)

the dynamical field A is approximated by the finite series

AF =N∑

α=1

Aα ϕα(10)

with complex expansion coefficients Aα. Insertion of (10) into the Ginzburg–Landauequation and projection onto the basis functions yields the spectral system

Aα =[ρ− q2(i+ c0)n

2α

]Aα + (i− ρ)

[ϕα, AF |AF |2

].(11)

nα denotes the wavenumber of the mode ϕα. The nonlinear term in (11) is evaluatedusing a dealiased spectral transform method [4] employing a discrete cosine transformin the Neumann case and a discrete sine transform in the Dirichlet case, respectively,of length 4n�.

All solutions to (11) are bounded for arbitrary truncations n� as can be seen from

the dynamics of the energy function [AF , AF ] =∑N

α=1 |Aα|2. Moreover, with AF ,AF exp(iφ) for arbitrary real φ also is a solution to (11). Therefore one has

〈AF 〉F = 〈AF 〉 = 0,(12)

where 〈·〉F denotes the ensemble mean with respect to the invariant measure inducedby the dynamical system of (11).

Test calculations reveal that for the Neumann problem with n� = 6 (14 realdegrees of freedom) and for the Dirichlet problem with n� = 32 (64 real degrees offreedom) the long-term dynamics monitored by second moments, temporal Fourier

2066 FRANK KWASNIOK

spectra, Lyapunov exponents and the Lyapunov dimension are captured with suffi-cient accuracy. All properties of the spectral approximation will be identified withthe corresponding properties of the full Ginzburg–Landau equation. Especially, 〈·〉F isidentified with 〈·〉; the subscript F will be dropped from now on in order to simplify thenotation. Both boundary value problems exhibit chaotic behavior; the Lyapunov di-mension of the attractor on which the asymptotic motion resides as obtained from theLyapunov spectrum according to the Kaplan–Yorke formula is 3.049 in the Neumanncase and 9.02 in the Dirichlet case, respectively. Figures 1 and 2 give an impressionof the character of the solution. In the Neumann problem only the lowest harmon-ics are excited; the temporal behavior of the solution is dominated by one frequency(cf. Figures 3 and 4). The Dirichlet problem is considerably more complex. Featureson small spatial scales occur; the temporal spectrum is rather broad (cf. Figures 10and 11).

3.2. Principal interaction patterns. We now consider an L-dimensional sub-space P ⊂ F spanned by only a limited number of spatial modes pj , which will becalled principal interaction patterns (PIPs):

P = span{p1, . . . , pL}, L < N.(13)

In order to simplify the notation, from now on Greek indices always run from 1 toN , Latin ones from 1 to L, if not explicitly indicated otherwise. Each PIP is a linearcombination of the Fourier modes:

pj =∑α

Pαj ϕα.(14)

P is the matrix with the vectors of Fourier coefficients of the PIPs as its columns. Ais expanded into a series of PIPs:

AP =∑j

zj pj .(15)

Without loss of generality [8, 9] the patterns are assumed to form an orthonormal setand their amplitudes are assumed to be uncorrelated:

[pj , pk] =∑α

P ∗αjPαk = δjk,(16)

〈zjz∗k〉 =∑α,β

ΓαβP∗αjPβk = δjkλ

PIPj .(17)

Γ denotes the covariance matrix of the Fourier coefficients:

Γαβ = 〈AαA∗β〉.(18)

The Galerkin projection of the Ginzburg–Landau equation onto the PIP modes yieldsthe dynamical system for the expansion coefficients

zj =∑k,l,m

ajklmzkzlz∗m +

∑k

bjkzk,(19)

LOW-DIMENSIONAL MODELS 2067

0

1

2

3x

0

5

10

15

20

25

t-1

0

1

Re@AHx,tLD

0

1

2x

0

1

2

3x

0

5

10

15

20

25

t

-1

-0.5

0

0.5

1

Im@AHx,tLD

0

1

2

x

Fig. 1. Real part (top) and imaginary part (bottom) of the solution to the Neumann problem.

where the interaction coefficients are given by

ajklm = (i− ρ) [pj , pk pl p∗m] ,(20)

bjk = δjk ρ− q2(i+ c0)∑α

n2αP

∗αjPαk.(21)

The following symmetry relations among the nonlinear interaction coefficients hold:

ajklm = ajlkm = amklj =ρ− i

ρ+ ia∗kjml.(22)

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0

1

2

3x

0

5

10

15

20

25

t

-1

0

1

Re@AHx,tLD

LOW-DIMENSIONAL MODELS 2069

construction the correct mean state. The boundedness of solutions for the PIP modelcan be seen from the time evolution of the energy function [AP , AP ] =

∑j |zj |2.

3.3. The variational principle. Consider an initial condition A0F =

∑αA

0α ϕα

characterized by the Fourier coefficients A0α at initial time t = 0. Let A

τF =

∑αA

τα ϕα

be the state obtained when integrating the Fourier–Galerkin approximation forwardin time from t = 0 to t = τ with initial condition A0

F . Consider now the projectionof the initial condition onto PIP space A0

P =∑

j z0j pj with z0

j = [pj , A0F ]. Let

AτP =

∑j z

τj pj be the state obtained when integrating the reduced model forward

in time from t = 0 to t = τ with initial condition A0P . Aτ

P is different from theprojection of the complete system onto the PIP space

∑j [pj , A

τF ] pj . A cost function

is introduced that measures the spatially and temporally integrated squared errorbetween the state given by the full model and that given by the reduced model:

Q(A0F , P ; τ�) =

1

2τ�

∫ τ�

0

[AτP −Aτ

F , AτP −Aτ

F ] dτ.(24)

Taking the ensemble average over all initial conditions on the attractor yields a costfunction that depends only on the pattern set and the free parameter τ�:

χ2(P ; τ�) = 〈Q(A0F , P ; τ�)〉.(25)

The optimal set of patterns is determined by minimizing the error function χ2. Inpractice the ensemble average is replaced by the average over a finite number ofrealizations of Q with initial conditions taken at uncorrelated times from a long timeseries generated by an integration of the system of (11) assuming ergodicity of theflow. Introducing the projection of the error at time τ onto the PIPs,

ετj = [pj , AτP −Aτ

F ] = zτj − [pj , AτF ],(26)

the cost function can be split into two parts:

χ2 = χ20 + χ2

d(27)

with

χ20(P ) =

1

2

∑

α

Γαα −∑j

∑α,β

ΓαβP∗αjPβj

(28)

and

χ2d(P ; τ�) =

1

2τ�

∫ τ�

0

∑j

〈|ετj |2〉 dτ.(29)

χ20 measures the mean squared error due to the projection onto PIP space and doesnot depend on the dynamical behavior of the reduced system; χ2

d represents the errorin PIP space and refers to the dynamics of the PIP model. The temporal integral in(29) is approximated by a finite sum using the trapezoidal quadrature rule:

Qd =1

2τ�

∫ τ�

0

∑j

|ετj |2 dτ =1

2

K∑k=0

∑j

wk|ετkj |2.(30)

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τk are equally spaced mesh points on the interval 0 ≤ τ ≤ τ� (τk =kK τ�; k = 0, . . . ,K)

and the weights wk are given by

w0 =1

2K,(31)

wk =1

K, k = 1, . . . ,K − 1,(32)

wK =1

2K.(33)

The variational principle poses a nonlinear minimization task with the NL complexelements of P as variables which has to be solved numerically by iterative techniques.

The PIP approach may be viewed as a nonlinear extension of the KL decom-position. In the limit τ� → 0 the term χ2

d vanishes and the PIPs coincide with theKL modes. With increasing τ� more and more information on the dynamics of thereduced model is included and the PIPs then may deviate from the KL modes.

3.4. Minimization of the cost function. The minimization of the cost func-tion is performed numerically using a limited memory quasi-Newton algorithm withthe Broyden–Fletcher–Goldfarb–Shanno update of the approximated Hessian matrix[5]. In view of the coincidence of the PIPs with the KL modes in the limit τ� → 0,the KL modes are used as a first guess in the minimization. The algorithm requiresexact evaluation of the cost function and its gradient for arbitrary sets of patterns.The first order variation of χ2

d with respect to the patterns yields

∂χ2d

∂Pαr=

⟨∂Qd

∂Pαr

⟩

=

⟨(∂Qd

∂z0r

)∗A0

α

⟩+

K∑k=0

wk

⟨(ετkr )

∗Ak

α

⟩

+∑j,k,l

πrjkl Λαjkl +∑j

(⟨∂Qd

∂brj

⟩∗Υαj +

⟨∂Qd

∂bjr

⟩Ξ∗jα

)(34)

with the complex tensors

πrjkl =

⟨∂Qd

∂arjkl

⟩∗+

⟨∂Qd

∂aljkr

⟩∗+ρ+ i

ρ− i

(⟨∂Qd

∂ajrlk

⟩+

⟨∂Qd

∂akrlj

⟩)(35)

= πrkjl = πljkr =ρ− i

ρ+ iπ∗jrlk,

Λαjkl = (i− ρ) [ϕα, pj pk p∗l ] = Λαkjl,(36)

Υαj =[ρ− q2(i+ c0)n

2α

]Pαj ,(37)

Ξjα =[ρ− q2(i+ c0)n

2α

]P ∗αj .(38)

The tensor Λ is evaluated by means of the spectral transform method. The expressions∂Qd

∂z0j

, πrjkl, and∂Qd

∂bjkare calculated efficiently using the method of adjoint equations

as described in detail in [9]. The set of complex interaction coefficients of the PIP

LOW-DIMENSIONAL MODELS 2071

model is {ajklm} ∪ {bjk}; the corresponding set of adjoint variables is introduced as{ωjklm} ∪ {ηjk}. The system of adjoint equations reads

yj = −2∑k,l,m

a∗ljkm z∗kzmyl −∑k,l,m

ajlmk zlzmy∗k −

∑k

b∗kj yk,(39)

ωjklm = −z∗kz∗l zmyj ,(40)

ηjk = −z∗kyj .(41)

The constraints of (16) and (17) do not restrict the solution of the minimizationproblem but are imposed only to remove the gauge freedom in the representationof the subspace P. They are accounted for by applying a Gram–Schmidt orthonor-malization and a diagonalization of the covariance matrix in PIP space at each stepof the minimization. The variational principle can then be formulated as an uncon-strained minimization problem. This procedure has been described in great detail inthe appendix of [9] and is not repeated here.

4. Results and discussion.

4.1. The Neumann problem. PIP models have been extracted from the sys-tem based on 10000 realizations of the cost function corresponding to initial conditionsseparated from one another by 5 time units. The integration time τ� is taken to be 4time units; the sampling interval of the data is 1 time unit corresponding to K = 4mesh points in the approximation of the cost function.

A reduced model with three complex patterns (six real degrees of freedom) turnsout to be capable of reproducing all essentials of the long-term behavior of the system.Figure 3 shows the variance as a function of the spatial coordinate obtained from asimulation with three PIPs in comparison to the converged solution based on sevenFourier modes. The agreement is almost perfect. In order to check the behavior ofthe reduced model in the time domain the Fourier spectrum of Re[A(0)] is plottedin Figure 4. Again the correspondence is nearly perfect. Figure 5 gives the spatialstructure of the three PIPs extracted from the system in comparison with the cor-responding KL modes. The PIPs and the KL modes are rotated in a way that thecomponent with the largest modulus is purely real and positive. In this case, the KLmodes are already very similar to the optimal modes. Considerable differences occuronly in the imaginary part of the third mode. Both the KL modes and the PIPsare quite similar to the lowest harmonics. The three KL modes and PIPs contain99.9980% and 99.9976%, respectively, of the variance of the system.

In Figures 6–9, several global quantities characterizing the performance of thereduced model at different truncations are given, namely, the spatially integratedvariance 〈[AP , AP ]〉P , the largest Lyapunov exponent (the only positive one), theforth Lyapunov exponent, and the Lyapunov dimension as given from the Lyapunovspectrum by the Kaplan–Yorke formula. The forth Lyapunov exponent is the firstnegative one since the Neumann problem and all its finite-dimensional Galerkin ap-proximations give rise to two zero exponents. The Lyapunov exponents have beencalculated using the standard method of Shimada and Nagashima [14]. The resultsfor the PIP models are compared to the models based on KL modes and Fouriermodes with the same number of degrees of freedom. In all situations where thereare significant differences between the models, the PIP model improves on the KLand Fourier models. For the highly truncated models using only two patterns, theimprovement is quite substantial in almost all quantities shown. With three modes,the KL approach is already almost perfect with respect to most of the quantities

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Fig. 3. Variance as a function of space for the Neumann problem in the complete model (solid)and in a simulation with three PIPs (dashed).

Fig. 4. Temporal power spectrum of Re[A(0)] for the Neumann problem in the complete model(solid) and in a simulation with three PIPs (dashed).

considered. The small difference between the three PIPs and KL modes (cf. Figure5) manifests itself most clearly in the negative Lyapunov exponent (Figure 8) whichreflects the dynamics of the dissipative components of the flow. The exact value of thefourth Lyapunov exponent appears to be quite sensitive to the small scale component(wavenumber four) visible in the imaginary part of the third mode.

In order to check to what extent the PIPs extracted at q = 0.95 (the value at whichthe Lyapunov dimension is largest [17]) also capture the dynamics of the Neumannproblem at other values of q, the various bifurcation points occurring in the interval0.6 ≤ q ≤ 1.3 [17] have been traced for the three-mode PIP approximation andcompared with the converged solution. The same has been done for the three-modeKL approximation in [15]. The agreement for the PIP approximation is extremelyclose. The deviation is under 1% for all bifurcation points. The same already holdsfor the KL approximation, but with one exception. In the exact solution, a period-doubling sequence starts at q = 0.827, versus q = 0.84 in the three-mode KL model.The three-mode PIP model yields q = 0.833.

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Fig. 5. First, second, and third (from top to bottom) PIP (dashed) and KL mode (solid).

Fig. 6. Spatially integrated variance for the Neumann problem as a function of the number ofmodes in simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

4.2. The Dirichlet problem. For the Dirichlet problem, an ensemble of 50000realizations has been used. The initial conditions are separated by 2 time units. Theintegration time is τ� = 2 time units; the sampling interval is 1 time unit correspondingto K = 2.

In contrast to the Neumann problem, no distinct minimal number of modes nec-essary in a reduced model can be determined. The approximation converges more

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Fig. 7. Largest Lyapunov exponent for the Neumann problem as a function of the number ofmodes in simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

Fig. 8. First negative Lyapunov exponent for the Neumann problem as a function of the numberof modes in simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

Fig. 9. Lyapunov dimension for the Neumann problem as a function of the number of modesin simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

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Fig. 10. Variance as a function of space for the Dirichlet problem in the complete model (solid)and in a simulation with 10 PIPs (dashed).

Fig. 11. Temporal power spectrum of Re[A(π2)] for the Dirichlet problem in the complete model

(solid) and in a simulation with 10 PIPs (dashed).

slowly when using progressively more modes. As an example, Figures 10 and 11 showthe variance as a function of space and the temporal Fourier spectrum of Re[A(π2 )], re-spectively, obtained from a long-term integration of a PIP model based on 10 complexpatterns (20 degrees of freedom) in comparison to the converged solution obtainedwith 32 Fourier modes. The accordance is quite good but a considerable error re-mains. One has to include as many as 14 patterns to get nearly perfect agreement.Figure 12 gives the spatial structure of the first, the second, the fifth, and the tenthmode in the PIP model with 10 modes in comparison with the corresponding KLmodes. The first two patterns are virtually identical; when going to the higher ordermodes with small spatial scales more and more differences occur. The 10 KL modesand PIPs account for 99.50% and 99.48%, respectively, of the variance of the system.

In Figures 13–16 the performance of reduced models based on Fourier modes, KLmodes, and PIPs, respectively, as a function of the truncation is characterized by thespatially integrated variance 〈[AP , AP ]〉P , the largest Lyapunov exponent, the Kol-mogorov entropy defined as the sum of the (four) positive exponents and the Lyapunovdimension as given from the Lyapunov spectrum by the Kaplan–Yorke formula. In allsituations with significant differences between the models, one observes an improve-

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Fig. 12. First, second, fifth, and tenth (from top to bottom) PIP (dashed) and KL mode (solid).

ment of the PIP model on the KL and Fourier models which is most substantial forthe highly truncated models with only 9 to 12 patterns. Interestingly, with respect tothe positive Lyapunov exponents, the highly truncated KL approximations performeven more poorly than the corresponding Fourier approximations.

The PIP approximation with 10 modes derived at q = 0.14 also reproduces allbifurcation points of the Dirichlet problem in the interval 0.08 ≤ q ≤ 1.0 very faithfullyas does the 10-mode KL approximation [15]. Moreover, long-term simulations atq = 0.08 have been performed. This is especially interesting since the Lyapunov

LOW-DIMENSIONAL MODELS 2077

Fig. 13. Spatially integrated variance for the Dirichlet problem as a function of the number ofmodes in simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

Fig. 14. Largest Lyapunov exponent for the Dirichlet problem as a function of the number ofmodes in simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

Fig. 15. Kolmogorov entropy for the Dirichlet problem as a function of the number of modesin simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

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Fig. 16. Lyapunov dimension for the Dirichlet problem as a function of the number of modesin simulations based on PIPs (dashed), KL modes (dotted), and Fourier modes (solid).

dimension of the Dirichlet problem continuously increases for q ≤ 0.14; at q = 0.08 ithas a value between 17 and 18 [17]. The dynamics at q = 0.08 are thus much morecomplex than at q = 0.14 where the modes have been extracted. Nevertheless, the PIPmodel with 10 modes is still able to reproduce the variance pattern and the Fourierspectrum, although not as well as at q = 0.14, as one would expect. It performs clearlybetter than the 10-mode KL model. In particular, the positive Lyapunov exponentsare far overestimated by the KL model, as is already the case at q = 0.14.

When comparing the PIP modes obtained with τ� = 1 and τ� = 2 it is observedthat the modes have not yet fully converged with an integration time τ� = 2. Pre-sumably, there is potential for some further improvement on the results reported herewhen using even larger values of τ�; but the minimization procedure is already com-putationally very costly with τ� = 2 in view of the large ensemble necessary due tothe high dimension of the underlying attractor.

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