STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS

2004,24B(3):321-336

..4atht1hdff:~cientia

1~~JIJ!~m STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS 1

*)

M a Shuqing ( ~ ~ Chang Qianshun ( 't it!l@i ) Institute of Applied Mathematics, the Chinese Academy of Sciences, Beijing 100080, China Abstract In this paper, the pseudospectral method to solve the dissipative Zakharov equations is used. Its convergence is proved by priori estimates. The existence of the global attractors and the estimates of dimension are presented. A class of steady state solutions is also disscussed. The numerical results show that if the steady state solutions satisfy some special conditions, they become unstable and limit cycles and strange attractors will occur for very small perturbations. The largest Lyapunov exponent and analysis of the linearized system are applied to explain these phenomena. Key words

Strange attractors, pseudospectral method, Zakharov equations

2000 MR Subject Classification

1

65P20, 76FXX

Introduction

Zakharov[8] has proved that the propagation of Langmuir waves in plasmas can be described by a system which is nowadays called Zakharov equations. It is interesting for this system and we want to know its behavior when time tends to infinity. Here we consider the dissipative Zakharov system with periodical boundary conditions, that is 1

).2

ntt

+ cai;

-

.6.(n

2

+ lEI)

iEt +.6.E - nE + hE

n(x,O) = no(x),

= f(x),

= g(x),

nt(x,O) = ndx),

(1.1) (1.2) (1.3)

E(x,O) = Eo(x),

(1.4)

n(x, t) = n(x + L, t),

(1.5)

+ L, t),

(1.6)

E(x, t) = E(x

x E

n,

tEl,

1 Received May 8, 2001; revised July 7, 2003. This research is supported in part by National Natural Science Foundation of China and in part by State Key Laboratory of Scientific and Engineering Computing, Academia Sinica.

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where n = [0, L), 1= [0, T] and L denotes the period in spatial variable. The complex function E(x, t) represents the envelop of the electric field, and n(x, t) the deviation of the ion density from its equilibrium value. The parameter A is proportional to the ion acoustic speed. Sulem and Sulem[3] have proved the existence of weak and strong solutions. Flahaut[l] has studied the long time behavior of the solutions of this system theoretically, where the author has proved the existence of a global attractor by establishing time uniform priori estimates and showed that the attractor has a finite fractal dimension. In [2], by applying finite difference methods, the discretized case is discussed and the existence of the global attractor is obtained. Both [1] and [2] show that this system has a global attractor. It is natural to consider that if the attractor is strange or if there are subattractors which are strange. In this paper, our interests are to analyse the existence of strange attractors and their behaviors by numerical methods. We study a class of special steady state solutions (no, Eo), where no and Eo are real and complex constants respectively. Let Eo = E; + iEi . The analysis of linearized system demonstrates that the steady state solution (no, Eo) will become more and more unstable with the growth of E i . Such is the largest Lyapunov exponent. The largest Lyapunov exponent /-lmax shows that there is a critical point E?, when E; > E? /-lmax will be nonnegative. This determines the occurrence of chaotic motion. The phase space portraits show that for small E, such that E, < E? the asymptotic state of (no, Eo) is also a steady state point. And for small positive values of E; - E? there arises a limit cycle. With the continuous growth of E; the limit cycle loses its stability and we obtain a trapping region, i.e., a bounded region of phase space to all sufficiently close trajectories from the so called basin of attraction are attracted asymptotically for long enough time. The paper is organized as follows. Section 2 contains a description of the numerical method. Here the spatial discretization is accomplished by a pseudospectral method. In Section 3 a priori estimates are obtained, and the convergence is discussed in Section 4. The existence of the global attractors is obtained in Section 5, the uniform upper bounds for the Hausdorff and fractal dimensions of attractors are proved. Section 6 is devoted to analysis of the linearized system. Here we get some relations among the no, Eo and the parameters of the equations. When these relations are satisfied there may be arise exponential growth. The pseudospectral method is applied to solve the Zakharov equations in Section 7. Many interesting results are obtained in this part. Section 8 presents an evaluation of the largest Lyapunov exponent and in Section 9 some concluding remarks are discussed. In this paper, C stands for a generic constant and at different place its value may be different.

2

Pseudospectral Method for the Equations

In this section we will discuss the Fourier pseudospectral method. First we introduce some notations. The periodical Sobolev space H;(n) is defined as follows: H;(n) = {u E Hl~c(R) I u(x

f!t

+ L)

= u(x)}.

Let Uj = u(Xj, t), Xj = jh, h = denote the step size of space and * stands for the exp (i21fjx/L), i = A. Then function w(x) of H;(n) can complex conjugate. Let c/>j =

Jr

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be expanded as the form of Fourier series: W(X)

=L

Wj¢j.

jEZ

Let SN = span{¢j}.;t:=.lJ¥-' P N the orthogonal project of H;(n) upon SN, WN(X) = PNw(x). And IN denotes the interpolation operator defined by IN! E SN and INw(xj) = w(Xj), Vw(x) E SN'

Let nt

= m(x, t) in (1.1) and (1.2), then we have the following system (2.1) mt

+ O,\2 m iEt

,\2 ~(n

+ ~E -

nE

+ IEI 2 )

+ irE =

= ,\2!,

(2.2)

g.

(2.3)

In Fourier pseudospectral method, approximate solutions satisfy (2.4) (mNt, ¢)h

+ o,\2(mN, ¢)h

- ,\2(~(nN

+ (~EN, ¢)h

- (nNE N, ¢)h

i(ENt, ¢)h

= ,\2 (IN!, ¢)h'

+ ir(EN, ¢)h = (lNg, ¢)h,

(2.5) (2.6)

= (no, ¢)h,

(2.7)

(mN(x, 0), ¢)h = (nl' ¢)h,

(2.8)

(EN(x, 0), ¢)h = (Eo, ¢)h,

(2.9)

(nN(x, 0), ¢)h

3

+ IEN!2), ¢)h

Priori Estimates In this section four theorems about the a priori estimates are proved. Lemma 3.1[7] For any u, v E C(O),

Lemma 3.2[6] Assume that W E H;(n), for any 8 2:: p, 2:: 0, there exists a positive constant C independent of N, such that

Lemma 3.3[7J Assume that W E H;(n), for any 8 2:: p, 2:: 0, there exists a positive constant C independent of N, such that

Lemma 3.4[4J such that

For any

W

E H;(n) (82:: 1), there exists a constant C independent of w,

From Lemma 3.1 it is easy to have

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Lemma 3.5

Equations (2.4)-(2.6) are equivalent to the following equations: (mN' ¢) = (nNt, ¢),

(mNt, ¢)

Vol.24 Ser.B

+ a)..2(mN, ¢) -

i(ENt, ¢)

V¢ E SN,

(3.1)

)..2(~nN, ¢) - )..2(~(IENI2), ¢)h = )..2(lN i. ¢),

+ (~EN, ¢) -

(nNEN' ¢)h

+ i')'(EN, ¢) =

(lNg, ¢).

(3.2) (3.3)

Theorem 3.1 Suppose that Eo E L~(n), 9 E L 2(n), then for the solution ofthe problem (2.4)-(2.9), there exists a constant C independent of N, such that IIEN(t)11 ~ C. Proof Choosing ¢ = EN and taking the imaginary part in (3.3) imply that

but Im(lNg, EN) ~ IIINgllllENl1

s 211ENII "(

2

1

2

+ 2"(llgllh'

where Lemma 3.2 is used. Thus

Using Gronwall inequality, we have (3.4) Theorem 3.2 Suppose that no E L~(n), ni E H;I(n), Eo E H;(n), f E H-I(n) and 9 E HI (n). Then for the solution of the problem (2.4)-(2.9), there exists a constant C independent of N such that IluNxl1 ~ C, IlnN11 ~ C and IIENxl1 ~ C, where UN is defined by (3.5) Proof

Choose ¢

= UN and use (3.5) to get

d 2 21 dtlluNxl1 + a).. 21 IUNx 112 +)..2 (nN,mN ) +)..2 (IENI2 ,mN)h = -).. 2 (lNf,UN),

(3.6)

which implies

Furthermore, choose ¢ 1d

= E Nt

2dtllENxil

Let ¢

2

+

in (3.3) and take the real part to get 1

2

2(nN,IENlt)h

= EN in (3.3) and take the

+ "(Im(EN,ENt)

= -Re(lNg,ENt),

(3.8)

real part to get (3.9)

Let ¢

= INg in (3.3) and take the real part to get '(3.10)

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Substitute (3.9) and (3.10) into (3.8), then 2

:t IIENx l1

+ (nN' IENlz)h + 2-yIIENxIl 2 + 2-y(nN'lEN 12 k

= 2Im(ENx,!Ngx) Let H(t) =

It follows from (3.7)

2Im(nNEN,!Ng)h.

lIuNxl1 2+ A? linN 11 2+ 2A211ENxll2 + 2A 2(nN, IENI 2)h.

+ 2A2 (3.11) that :tH(t) + D:A

s

(3.11)

~ IIcpll2

-

211uNx1l 2+ 4-YA 211ENx11 2 + 4A2-y(nN' IENI 2)h

(3.12)

4A 2Im(ENx, [Ngx) - 4A 2Im(n NEN, [Ng)h.

In view of Lemma 3.3 and Lemma 3.4, we obtain Im(ENx,!Ngx)

s CIIENxllllg,.1I s ~IIENxW + C 2~lIgll~,

Im(nNEN,!Ng)h ~·llnNIIII[NgllooIIENIi~ A21 1nNI12+ CllglI~IIENII2. Now we are to estimate the term (nN,IENI 2)h' Also by using Lemma 3.4, the following inequality holds

Thus (3.12) can be rewritten as d 21uNx112+ -YA2II ENx 11 2dtH(t) + D:A 1

This implies that

:t

H(t)

ClinNI1 2 ~ -allcpl12 + C. ~

~ CH(t) + C.

(3.13)

(3.14)

Then the following inequality is deduced H(t)

s C(T),

(3.15)

where Gronwall inequality is used. Using Lemma 4 we have

The definition of H(t) implies

•

2

H(t) ~ lIuNxll

A2 + Tlin NI1 2 + A211ENxW

2

H(t) ~ C(luNxll +

- C,

A2

T"n N 112 + A211ENxl1 2 + 1).

Finally we obtain (3.16)

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The proof is completed. Similar to Theorem 3.2, the following two theorems can be proved. Theorem 3.3 Suppose that nl E L;(n), no E H;(n), Eo E H;(n), f E L 2(n) and g E H1(n). Then for the solution of the problem (2.4)-(2.9), there exists a constant C independent of N, such that IlmNIl S; C, IlnNxl1 S; C and IIENxxl1 S;C. Theorem 3.4 Suppose that f E H1(n), 9 E H1(n), nl E H~(n), no E H;(n) and Eo E H;(n). Then for the solution of the problem (2.4)-(2.9), there exists a constant C independent of N, such that IlmNxl1 S; C, IlnNxxl1 S; C and IIENxxxll S;

4

c.

Convergence This section is devoted to the convergence of the pseudospectral method. Let ew

Thus ew =

=w -

WN,

~w

=W -

'T/w

PNW,

= PNW -

WN.

'W + 'T/w' Substracting (3.1)-(3.3) from (1.1)-(1.3) yields the error equations ('T/m, ¢)

('T/mt, ¢)

= ('T/nt, ¢),

+ aA 2('T/m , ¢) - A2(D.'T/n , ¢) - A2 (D.(IE I2

i('T/Et, ¢)

+ (D.'T/E' ¢) -

V¢ E SN, -

(4.1) 2

I N(IENI )), ¢) = (j - IN I. ¢),

(nE - IN (nNEN ), ¢)

+ i--Y('T/E' ¢) =

(g - lNg, ¢).

(4.2) (4.3)

Theorem 4.1 Suppose that f(x) E CS(n), g(x) E CS(n) and the solution of the equation (1.1)-(1.6) satisfies n(x, t) E C(I, H;(n)), E(x, t) E C(I, H;(n)) (8 > 1). Under the conditions of Theorem 3.3 or Theorem 3.4, there exist positive constants C(T) and M, such that for N ~ M: sup lin - nN11 S; CN 1 tEl

s

,

sup liE - ENlloo S; CN 1 tEl

s

•

where nN, EN are the solutions of equation (2.4)-(2.9). Proof Similar to the proof of Theorem 3.2, we choose ¢ = 'T/u in (4.2) and obtain (4.4) It follows from the definition of cp in Section 3 that

Furthermore, (4.1) implies that

So (4.4) can be rewritten as

~

:t

(11'T/uxI1

2

+ A211'T/nI1 2 ) + A2(IEI 2 -

2

I N ( IE N I ), 'T/nt ) S; CN- 2s .

(4.5)

Now let ¢ = 'T/Et in (4.3) and take the real part to get

~

:t

II'T/ExI1 2

+ Re(nE -

IN (nNEN ), 'T/Et)

+ --ylm('T/E, 'T/Et)

= Re(g -

lNg, 'T/Et).

(4.6)

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Let ¢ = 'TJE in (4.3) and take the real part to get

Moreover in (4.3) let ¢ = 9 - INg and take the real part to get

Substituting (4.7) and (4.8) into (4.6) implies that

d 2 ) 2"1 dtll'TJExll + Re(nE - IN ( nNEN ,'TJEt)

+'YII'TJExI1 2 + -yRe(nE - IN(nNEN),'TJE)

= -Im('TJEx, (g -

(4.9)

INg)x) - Im(nE - IN(nNEN),g - INg).

Furthermore, choosing ¢ = 'TJE and taking the imaginary part in (4.3) imply that

:t lI'TJE11 It follows from 2(4.5)

2

2Im(nE - IN (nNEN ), 'TJE)

-

+ 2-YII'TJEII 2 = (g - lNg, 'TJE)'

+ 4A2(4.9) + ,8(4.10) that

d 2 dt (lI'TJuxI1

+ A2 11'TJn112 + 2A211'TJEx1l 2 + ,811'TJEI1 2 )

+2A2(IEI 2

-

+aA

(4.10)

211'TJuxW

I N(IENI2 ) , 'TJnt)

+ 4A2Re(nE - IN (nNE N ), 'TJEt)

+ 4-yA 211'TJExI1 2 + 2-y,8I1'TJEI1 2

(4.11)

+4-YA 2Re(nE - IN(nNEN),'TJE) - 2,8Im(nE - IN(nNEN),'TJE) ::; CN- 2s

-

4A2Im('TJEx, (g - INg)x)

The terms of (4.11) are to be estimated gradually. It follows by simple computation that

where £1

£2

And

where

= -(lEI; -

I N(IEI 2 )t> 'TJn) ::; C(N- 2s

= (IEI2 -IENI 2 , 'TJnt)h = (2Re(PNE*'TJE) -

+ II'TJnln,

l'TJEI 2 , 'TJnt)h + 2Re(PNE~E' 'TJnt)h.

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Thus £2 + £4 leads to £2

+ £4

=

2 :t[2Re(PNEriE,71n)h + (nN, I71EI )h + 2RePNE~E,71n)h +2Re(n~E

+ PNE~n, 71Et)h] -

-2Re((PNE~E)t,71n)h

2Re(PNEt~E, 71n)h - (PNnt71E, 71E)h

(4.12)

- 2Re((n~E + PNE~n)t,71E)h'

Denote the last four terms of (4.12) by £5' It follows from Lemmas 3.2, 3.3, Theorems 3.3, 3.4 and the hypothesis that Thus (IEI 2 - IN (lEN 12), 71nt) + 2Re(nE - IN(nNE N), :"Et) d ~ dt [(IEI2 - IN(lEI 2), 71n) + 2Re(nE - IN(nE), 71E) 2 +2Re(PN E71E' 71n)h - (nN' I71EI )h +2RePNE~E,71n)h

+ 2Re(n~E + PNE~n,71E)h] + C(N- 2s + 117111 2 + I171EI1 2).

Use Lemma 3.3 and Lemma 3.4 then

and

-4..\2Im(nE - IN(nNEN),g - lNg) ~ CN- 2s + CII71EI1 2 + CII71nI1 2, /3(g - lNg, 71E) ~ C(N- 2s + I171EI1 2). Furthermore

Substituting the above inequalities into (4.11), we obtain

where

+ ..\21171nI1 2 + 2..\21171ExI1 2 + /31171EI1 2 + (IEI2 - IN(IEI 2), 71n) +2Re(nE - IN(nE),71E) + 2Re(PN E71E' 71n)h - (nN' I71EI 2)h +2Re(PNE~E' 71n)h + 2Re(n~E + PNE~n, 71E)h.

Q(t) = II71ux1l

2

Denote the last six terms of Q(t) by £6. It is easy to know that the following inequality is true

Then (4.13) implies that (4.14)

Ma & Chang: STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS

No.3

Thus

:t Q(t)

~ CN 2- 2s + CQ(t).

By the definition of ew , ~w and 1Jw,it is easy to know inequality, from (4.15), we obtain

329

(4.15)

II1Jw(O)11 = O.

So Q(O)

= O.

Q(t) ~ CN 2- 2s, '
Using Gronwall

(4.16)

Now we estimate the last six terms of Q(t) again: (4.17) Apply (4.17) to Q (t) again and choose the value of j3 big enough, then

So the solutions of equations (1.1)-(1.6) and solutions (2.4)-(2.9) satisfy:

II1Jux11 2 + II1Jn11 2 + II1JEI1 2 + II1JExI1 2 ~ CN 2- 2s. Ilewll

Because finally, from (4.18),

sup tEl

~ II~wll

+ II1Jwll

Ilenli ~ CN I - s,

~ CN-s

sup tEl

lIeElloo

(4.18)

+ II1Jwll,

s CN

I

-

s.

The proof is completed.

5

Global Attractors and· Dimensions

To obtain the existence of the global attractors a slight modification for (2.4) and (2.5) is necessary. Similar to [2], denote m(x, t) = ftn(x, t) + m(x, t), here E > 0 small enough. Hence (2.4)-(2.9) can be rewritten as (5.1)

+ (a..\.2 - E)(mN,¢)h - E(a:..\.2 -..\.2(A(nN + IEN I2 ) , ¢)h = ..\.2(lN l, ¢)h,

(mNt,¢)h

i(ENt, ¢)h

+ (AEN, ¢)h -

(nNE N, ¢)h

E)(nN,¢)h

+ i')'(E N, ¢)h = (lNg, ¢)h,

= (no, ¢)h, (mN(x, 0), ¢)h = (nl, ¢)h, (nN(x, 0), ¢)h

(EN(X, 0), ¢)h = (Eo, ¢)h,

(5.2) (5.3) (5.4) (5.5) (5.6)

In this section the results and their proofs all are similar to those of [2], so we only give out the results and omit the proofs. Theorem 5.1 Suppose that Eo E L~(!1), 9 E L 2(!1), then the solution of the problem (5.1)-(5.6) satisfies (5.7)

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Furthermore if the initial value satisfies IIEol1 ~ R, R > 0, there exist a constant B independent of N, t and the discrete functions such that IIENII ~ B. Denote co = H;l x L~(O) x H~(O), Cl = L~(O) x H~(O) x H;(O), C2 = H~(O) x H;(O) x H;(O). The discrete norms are defined as lI(mN,nN, EN) 11;0

= lIuNlli + IinNII2 + II ENlli,

IImNI1 2 + IInNlli + IIENII~, lI(mN, nN, EN) 11;2 = II mNlli + IlnNII~ + IIENII~· lI(mN,nN, EN) 11;1 =

Theorem 5.2 Suppose that (mo,no, Eo) E co and solution of (5.1)-(5.6) satisfies

1 E H-l(O),

g E H1(0), then the

(5.8)

where the positive constants ao and bo depend on the parameters of the system, IIINgll and liEN II· If II(mo,no,Eo)lleo ~ R, then there exists a positive constant B o such that II(mN,nN, EN)lIeo ~ ti; Theorem 5.3 Suppose that (mo,no,Eo) E solution of (5.1)-(5.6) satisfies

Cl

and

1 E L 2(0),

g E H1(0), then the

(5.9) where the positive constants al and b1 depend on the parameters of the system, IIINgII1, II IN 111 and lI(mN, nN, EN)lIeo· If II(mo,no,Eo)llel ~ R, there exists a positive constant B 1 such that lI(mN,nN,EN)IICl ~ B1· Theorem 5.4 Suppose that (mo,no,Eo) E C2 and 1 E H1(0), g E Hl(O), then the solution of (5.1)-(5.6) satisfies (5.10) where the positive constants a2 and bz depend on the parameters of the system, III Nglh, II IN llh and lI(mN,nN,EN)lIel. If II(mo,no,Eo)lIe2 ~ R, there exists a positive constant B 2 such that lI(mN,nN,EN)lIe2 ~ B2. In view of the uniform a priori estimates of Theorems 5.1-5.4, one can obtain the existence of the global attractors for (5.1)-(5.6) with respect to 1I·lle;, i = 0, 1, 2. Let conditions i, i = 0, 1, 2, stand for the conditions of Theorem 5.2-5.4 respectively. The following theorem holds. Theorem 5.5 Assume that a > 0, A > 0, '"Y > 0, then the solution of the discrete system (5.1)-(5.6) exists globally. Moreover if condition i is fulfilled and the initial value satisfies II(mo, no, Eo) lie; ~ u; R i > 0, there exist positive constants fi(R i) and Ti(R i) such that for f~ > I', the solution of the discrete system (5.1)-(5.6) satisfies

The semi-flow of (5.1)-(5.6) has a global attractor Ai under the norm II . lie;. The attractor lies in B(Ojfi ) C R N x R N X eN. where i = 0,1,2 and B(Ojf i ) is the ball centered at E R N X R N X eN with radius r..

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To majorize the dimensions of the global attractors we begin with the linearized system of (5.1)-(5.6)

:t(VN,UN, WN) = F'(mN,nN,EN)(VN,UN, WN).

(5.11)

In view of the fact that cPj, j = - -;, "', J¥- - 1, are N eigenvectors of the discrete Laplace operator and are corresponding N eigenvalues, by a process similar to [2] and [11], we have the following theorem. Theorem 5.6 There exists a sufficiently large positive integer No independent of Nand the initial value (mo, no, Eo) such that for any N > No the dimensions of the global attractor Al in El satisfy dH(A1 ) ~ No + 1,

41,/

(I + E~=1 I) I I

1\T) dF (A) 1 ~ (iVo + 1 . max

1

1~I~No+l

J.Lj J.Lj

N, +1

Ej~1

'

where J.Lj stands for the jth uniform Lyapunov exponent on AI.

6

A Linearized Analysis

In this section we show that there are exponential instabilities associated with some solutions of the dissipative Zakharov equations. Assume that f(x) = 0, g(x) = -noEo + i,Eo , where no is a real constant and Eo = E; + iEi is a complex constant. Then no and Eo is a steady state solution of the Zakharov equations. First let n = no + Sn, E = Eo + 8E, and the linearized equations of (1.1) and (1.2) around the steady state point no, Eo are given by

+ Q:A28nt - A2 A(8n + E'Q8E + E o8E*) = { i8Et + A8E - n o8E - E o8n + i,8E = 0. 8ntt

Now we introduce the same perturbation E(t)eiqx modulated respectively by

+ E*(t)e- iqx

(6.1)

to no and Eo, i.e., no, Eo are

n(x, t) = no + E(t)eiqx + E*(t)e- iqX , E(x, t) = Eo + E(t)eiqx

0,

+ E*(t)e- iqx ,

(6.2)

where q is a constant. Substituting (6.2) into (6.1) leads to

e« + Q:A2Et + q2 A2(2Er + l)E {

= 0,

+ i(q2 + no + Eo - h)E = 0, E t* + 1.( q 2 + no + E*' 0 - I, ) E * = Et

° .

(6.3) (6.4)

(6.5)

From (6.3)-(6.5), we have

q2 + no + s; = 0, { Et - (E i -,)E = 0. To make

E

(6.6)

grow exponentially, the relation (6.7)

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must be satisfied. So a.A2 + 2(Ei

-.,,)

q2 {

Vol.24 Ser.B

> O. Then we get from (6.6)

= -no - e;

- -E no r

+ (Ei-'Y)(Ei+ a,\2_'Y) ,\2(2Er + l }

(6.8) '

where n and E are L-periodic, So q should be an integer-multiple of L. Above discussion shows that under the conditions of (6.7) and (6.8) the steady state may be unstable and for very small perturbations there may arise exponential growth with the increasing of Ei. All our numerical simulation of the next section is based on these.

7

Algorithm and Numerical Analysis

In (2.4)-(2.5) we choose

(7.1) d dt mj

4

2'2

1r J - 2 2 = -a.A 2 mj - -V(nj + IENl j) -.A Ii, d 41r 2P - . -._ -E· = - - - E -1(nNEN)' - ",E· -lg'

dt

J

L2

J

I

J

J'

(7.2) (7.3)

iij(O) = (no,
(7.4)

= (nl'
(7.5)

Ej(O) = (Eo,
(7.6)

mj(O)

The fourth order Runge-Kutta method is used to discrete the time t. In all our numerical simulation the parameters of a, .A and." are fixed and take the same value one. The real part E; of Eo also take a fixed value -2.0. The numerical results for large values of N are similar to those where N is equal to 8. So almost all our experiments are simulated with N equal to 8. Only an experiment is simulated with N equal to 16 to show the similarity among different values of N. The relation among the no, Eo and q is determined by (4.8). For initial values we have chosen a slight departure from the steady state solution, namely,

= no + €cos(qXj), E(xj,O) = Eo + €cos(qXj),

n(Xj,O)

m(xj,O) = _€[q2 sin(qxj)

(7.7)

+ (-y - E r ) cos(qXj)],

where to is much smaller than one. The results show that there is a critical point Ep for E; (for N = 8 E? ~ 3.86). For negative E; - E? the asymptotic state is a steady state point. For small positve E, - E? a limit cycle arises. As E, continues to increase the limit cycle loses its stability and evolves into a trapping region which is nowadays called a strange attractor. These phenomena have been predicted by the linearized analysis of Section 6 as well as by the analysis of the largest Lyapunov exponent of Section 8. Now we illustrate the asymptotic behavior of the steady state points for different values of E i by phase portraits.

Ma & Chang: STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS

No.3

333

First we take a small E i , for example, E; = 2.0, the portrait of n(x, t) is shown in Fig.1. It seems that the solution tends to a steady state point. In fact when t = 25 the solution has attained its steady state point. It is very quick. The steady state point is a constant independent of x. The portrait (which is not shown here) for t ~ 25 is a single plane. The figure of E(x, t) is similar to that of n(x, t), so it isn't discussed here.

Fig.l

Figure of n(x, t) for E;

= 2.0.

Fig.2

(a) Limit cycle project onto the E 1 , E 6 plane for E i

= 4.05.

For small positive value of E; - E? a limit cycle emerges. Here we take a value of E; slightly larger than the critical point E?, for example, E, = 4.05. The result is interesting. We project the phase portraits onto the E I , E 2 plane. The figure is a closed cycle (Fig. 2 (a)). The figure in the phase space (In(x4' t)l, Int(X4' t)l) is of the same result (Fig. 2 (b)). From Fig. 2 (c) it is obvious that n(x, t) is periodic. With E, increasing continually two or three closed cycles are arisen. Fig. 3 (a) and (b) show these cases for E, = 4.1 and E, = 4.3 respectively.

0.3 . - - - - - - - - - - - 0.2 0.1

OH---------+-l IiJ

-0.1 -0.2 -0.3 Lc-L--'----'-----'----'----'_.L-.....J 2.5 2.6 2.7 2.8 2.9

Fig.2

(b) Orbit in the phase space

(In(x4, t)j, Int(X4, t)l) for E, = 4.05.

Fig.2

(c) Figure of n(x, t) for E, = 4.05 with t ~ 50.

Now a large value of E; is experimented. This time we take E; = 4.52. Fig. 4 (a)-(d) show this case. The portraits projected onto the n4, ns plane (Fig. 4 (a)) and the E I , E 6 plane (Fig. 4 (c)) are no more steady state points or limit cycles but trapping regions. These are the strange attractors. Figures in the phase space (In(x4' t)l, Int(X4, t)l) and (IE(X4, t)l, IE t(X4, t)l)

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ACTA MATHEMATICA SCIENTIA

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show the same conclusion (Fig. 4 (b) and (dj). -1.4

-0.8 , - - - - - - - - - - - - - - ,

~-----------,

-1.0

-1.6

-1.2

-1.8

-1.4

-2.0

-1.6

-2.2

-1.8 -2.0

-2.4

-2.2

-2.6

-2.4 -2.6

-2.8 L--L_-'-----'-_...L------'-_--'----' -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4

Fig.3

(a) Multi-closed-cycles project onto the El, E6 plane for E,

L----'------'-_-'-----'-_J...--'---'----"

-4

Fig.3

o

-1

-2

-3

(b) Multi-closed-cycles project

= 4.1.

onto the E l , E 6 plane for E,

60

200

40

150

= 4.3.

100

20

50

0

0 -20

-50

-40

-100

-60

-80 -30

Fig.4

-150 -20

-10

0

10

20

30

40

(a) strange attractor project onto the n4, ns plane for E, = 4.52.

-200

Fig.4

10

15

20

25

6

20

30

25

35

(b) Orbit in the phase space

(In(x4, t)l, Int(X4, t)l) for E,

8 ~-------,--------,

= 4.52.

.-.

15

4

0l-;-=-.

10

2

5 0

-2

-5

-4

-10

-6

-15

-8

-20

-10 L---'----L--l._L--...L--'----'-_..L-....J -10 -8 -6 -4 -2 0 2 4 6 8

Fig.4

5

0

(c) strange attractor project onto the E l , E 6 plane for Ei

= 4.52.

-25

'~~

0

Fig.4

2

3

.-....:>'"

4

5

6

(d) Orbit in the phase space

(IE(X4, t)l, IE t(X4, t)1) for Ei

= 4.52.

Ma & Chang: STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS

No.3

335

Finally there is an experiment for N = 16. Here the value of E; is 4.05. Phase portrait projected on to the E s, E g plane (Fig. 5) is similar to that of N = 8. It is a limit cycle too. So do the other phase portraits. Thus the case for N = 16 or larger will not be discussed here. 2.5

~r··-// -2.03 -2.Q4

/

~----~~~~~~~,

2.0 1.5 -

./

1.0

/

0.5 ol---~~~-~--+~~~----j I----~~~~~--'

-0.52.0

Fig.5

Fig.6

Limit cycle project onto the E a, E g plane for E,

= 4.05 with N = 16.

2.5

3.0

3.5

4.0

4.5

5.0

The largest Lyapunov exponent as a function of

s; with N = 8.

Characterization of the Attraetors by the Largest Lyapunov Exponent

8

In Section 7, we have presents the existence of a strange attractor by numerical methods. Now we apply the largest Lyapunov exponent to explain it. The value of the largest Lyapunov exponent /-Lmax is a quantitative measure of chaotic behavior. It measures the exponential divergence or convergence of nearby initial points in the phase space of a system. For /-Lmax > 0, one will observe bounded chaotic motions. In [5], A. Wolf has presented a popular algorithm for calculating Lyapunov exponents. Here we consider the equations of the motion (7)((7.1)-(7.6)). By single computation we have IENI; =

L

EJJ:t,

(nNEN)j =

&,1

IJ-l=j

L /J,'

TisE,.

iJ+l=j

So the Jacobian matrix J of (7) can be presented explicitly. We denote the linearization of (7) by d (8.1) dt6U = J6U. To obtain the largest Lyapunov exponent one can start with an arbitrary initial value, then integrate (7) and (8.1) simultaneously. We define the largest Lyapunov exponent as follows: 1

/-Lmax

K

= K-too lim k ut L A

log 116U(k~t)ll,

(8.2)

k=l

where II . II is the Euclidean norm. The limit of large K is necessary if we are to obtain a quantity that describes longterm behavior and is independent of initial conditions. In practice we normalize i5U(k~t) to unity after each time step. In Fig. 6 we have shown an example of

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ACTA MATHEMATIC A SCIENTIA

Vo1.24 SeLB

the largest Lyapunov exponents for N = 8 as a function of E i . Here we apply the classical fourth-order Runge-Kutta method to (7) and (8.1) simultaneously at a sampling of rate 0.001. The largest Lyapunov exponent is calculated via (8.2). the number of iterations J( used in the computation is 200 000. Fig. 6 shows the critical point E? ~ 3.86 for N = 8.

9

Conclusion

In this paper we use pseudospectral method to the dissipative Zakharov Equations. Its convergence is proved by a priori estimates. The existence of the global attractors and estimates of dimensions are presented. Furthermore, we have shown some interesting results of several of our numerical experiments. We have concentrated on the variable E i . Many phase portraits of different values of E, are given. Although in all numerical results the number modes, N, equals 8, we have observed the same behavior of N equal to 16 or larger. Only there is small difference among the critical point of E? the results provide to support for the existence of a strange attractor. Moreover the experiment data show that for IiI > 2 the Fourier coefficients nj and Ej are approximately zero. These demonstrate that the dimensions of the attractors are finite. Thus all the experiment results agree with the theoretical analysis. In this paper we not only present the existence of a global attractor of Zakharov equations but also show that the global attractor contains strange subattractors by numerical experiments. This is the first time for people to provide reference to the possibility of the existence of chaotic motions in numerical experiments on Zakharov equations. Therefore, it is of value and interest to introduce them to many investigators in computational mathematics. Acknowledgments The authors are pleased to acknowledge the very useful discussions and advices of Prof. B.L. Guo and Dr. B.L. Lu. References Flahaut 1. Attractors for the dissipative Zakharov system. Nonlinear Analysis TMA, 1991, 16: 599 2 Chang Q S, Guo B L. Attractors and dimensions for discretizations of dissipative Zakharov equations. Acta Mathematicae Applicatae Sinica, English Series, 2002, 18: 201 3 Sulem C, Sulem P L. Regularity properties for the equations of Langmuir turbulence. C R Acad Sci Paris Ser A Math, 1979, 289: 173 4 Temam R. Infinite dimensional dynamical systems in mechanics and physics. Berlin: Springer, 1988 5 Wolf A, Swift B. Determining Lyapunov exponents from a time series. Phys D, 1985, 16: 285 6 Butzer P L, Nessel R J. Fourier analysis and approximation. Academic Press, 1971 7 K\eiss H 0, Oliger J. Stability of the Fourier method. SIAM J Numer Anal, 1979, 16: 421 8 Zakharov V E. Collapse of Langmuir waves. SOy Phys JETP, 1972, 35: 908 9 Brefore B, Ghidaglia J, Temam R. Attractors for the penalized N-S equations. SIAM J Math Anal, 1988, 19: 1 10 Guo B L, Chang Q S. Attractors and dimensions for discretizations of generalized Landan-Ginzberg equation. J Partial Diff Eqs, 1996, 9: 365 11 Yin Yan. Attractors and error estimates for discretizations of impressible N-S equations. SIAM J Numer Anal, 1996, 33: 1451 12 Ablowitz M J, Herbst B M, Schober C M. On the numerical solution of the sine-Gordon equation. J computational Physics, 1997, 131: 354 13 Zhou Y 1. Applications of Discrete Functional Analysis to the Finite Difference Method. Beijing: International Academic Publishers, China, 1990