Eigenvalue problems and their application to the wavelet method of chaotic control

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Jonq Juang and Chin-Lung Li

Citation: Journal of Mathematical Physics 47, 072704 (2006); doi: 10.1063/1.2218674 View online: http://dx.doi.org/10.1063/1.2218674

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/7?ver=pdfcov Published by the AIP Publishing

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Eigenvalue problems and their application to the wavelet

method of chaotic control

Jonq Juanga兲 and Chin-Lung Li

Department of Applied Mathematics, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.

共Received 5 January 2006; accepted 5 June 2006; published online 31 July 2006兲

Controlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai关Phys. Rev. Lett. 89, 284103 共2002兲兴. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue ␭1共␣,␤兲 of the 共wavelet兲 transformed coupling matrix

C共␣,␤兲 for each␣ and␤. Here␤is a mixed boundary constant and ␣is a scalar factor. In particular,␤= 1共respectively, 0兲 gives the nearest neighbor coupling with periodic共respectively, Neumann兲 boundary conditions. The first, rigorous work to understand the eigenvalues of C共␣, 1兲 was provided by Shieh et al. 关J. Math. Phys. 共to be published兲兴. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of C共␣, 1兲 and C共␣, 0兲. This, in turn, yields some new information concerning␭1共␣, 1兲. Second, we shed

some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions. © 2006 American

Institute of Physics. 关DOI:10.1063/1.2218674兴

I. INTRODUCTION

Chaotic synchronization 共Refs. 1, 8, 12–14, and references cited therein兲 is a fundamental phenomenon in physical systems with dissipation. It was first observed in Ref. 8 for identical master-slave Lorenz equations. This phenomenon was later observed in many different fields— physics, electrical engineering, biology, laser systems, etc. Experimental observations show that chaotic subsystems in a lattice manifest synchronized chaotic behavior in time provided they are coupled with a dissipative coupling and its coupling strength is greater than some critical value. Specifically, let there be N nodes共oscillators兲. Assume uiis the m-dimensional vector of

dynami-cal variables of the ith node. Let the isolated共uncoupling兲 dynamics be u˙i= f共ui兲 for each node.

We assume that ui has a chaotic dynamics in the sense that its largest Lyapunov exponent is

positive. Let h:Rm_→Rm

be an arbitrary function describing the coupling within the components of each node. Thus, the dynamics of the ith node are

u˙i= f共ui兲 +⑀

兺

j=1 N

aijh共uj兲, i = 1,2, ... ,N, 共1.1a兲

where ⑀ is a coupling strength. Here sum ⌺N_j=1aij= 0. Let u =共u1, u2, . . . , uN兲T, F共u兲

=共f共u1兲, f共u2兲, ... , f共uN兲兲T, H共u兲=共h共u1兲,h共u2兲, ... ,h共uN兲兲T, and A =共aij兲. We may write 共1.1a兲 as

u˙ = F共u兲 +⑀A⫻ H共u兲. 共1.1b兲

a兲_{Electronic mail: jjuang@math.nctu.edu.tw}

47, 072704-1

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Here⫻ is the direct product of two matrices B and C defined as follows. Let B=共bij兲k₁_⫻k₂be

a k1⫻k2 matrix and C =共Cij兲k₂⫻k₃ be a k2⫻k3 block matrix, where each of Cij, 1艋i艋k2, 1艋 j

艋k3, is a k4⫻k5matrix. Then B⫻ C =

冉

兺

l=1 k₂ bilClj

冊

k₁⫻k₃ .

Many coupling schemes are covered by Eq.共1.1b兲. For example, if the Lorenz system is used and the coupling is through its three components x, y, and z, then the function h is just the matrix

I3=

冢

1 0 0 0 1 0

0 0 1

冣

. 共1.2兲

The choice of A will provide the connectivity of nodes. For instance, the nearest neighbor coupling with mixed boundary conditions is given as follows:

A = A共␤兲 =

冢

− 1 −␤ 1 0 ¯ ¯ ␤ 1 − 2 1 ¯ ¯ 0 0 1 − 2 1 ¯ 0 ] ] ] ] ␤ 0 ¯ ¯ 1 − 1 −␤

冣

N⫻N . 共1.3兲

Note that␤= 1 corresponds to periodic boundary conditions, while␤= 0 is associated with Neu-mann boundary conditions. The synchronous manifold of the chaotic system共1.1兲 can be studied by setting u1共t兲=u2共t兲= ¯ =uN共t兲=s共t兲. Here the chaotic solution s共t兲 satisfies the single oscillator

equation ds / dt = f关s共t兲兴. The stability property of the synchronous manifold can then be studied in the space of difference variables␦ui共t兲=ui共t兲−s共t兲, which are governed by7,10

d␦u

dt =共IN⫻ DF +⑀A⫻ DH兲␦u, 共1.4a兲

where DH = dH共u兲/du, and␦u =共␦u1,␦u2, . . . ,␦uN兲T. When H is just a matrix E, DH = E. The first

term in 共1.4a兲 is block diagonal. The second term can be treated by diagonalizing A. The trans-formation which does this does not affect the first term, since it acts only on the identity matrix IN.

This leaves us with a block diagonalized variational equation with each block having the form7

␦u˙i=关DF +⑀␭iDH兴␦ui, 共1.4b兲

where␭iis an eigenvalue of A, i = 0 , 1 ,¯ ,N−1. The Jacobian functions DF and DH are the same

for each block, since they are evaluated on the synchronized state. It then follows from共1.4b兲 that the largest eigenvalue␭0of A being equal to 0 governs the motion on the synchronized manifold,

and all of other eigenvalues␭i共i⫽0兲 control the transverse stability9of the chaotic synchronous

state. The stability condition is then given by Lmax+⑀␭1艋0, where Lmax⬎0 is the largest

Lyapunov exponent of a single chaotic oscillator. As a consequence, the second largest eigenvalue ␭1is dominant in controlling the stability of chaotic synchronization, and the critical strength⑀c

can be determined in term of␭1,

⑀c=

Imax

−␭1

. 共1.4c兲

Note that the eigenvalues of A = A共1兲 are given by ␭i= −4 sin2共␲i / N兲, i=0,1, ... ,N−1. In

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a larger ⑀c. As a consequence, controlling chaos is apparently of great interest and

importance.4–7,11,12 In Ref. 11, a new efficient strategy for controlling nonlinear dynamics was presented. To be self-contained, we briefly describe such procedures. Let

A =

冢

A11 ¯ A1n

] ]

A_n1 ¯ Ann

冣

n⫻n

, 共1.5a兲

be a matrix with the dimension of each block matrix Akl being 2i⫻2i. By an i-scale wavelet

operator W,2,11the matrix A is transformed into W共A兲 of the form

W共A兲 =

冢

A ˜ 11 ¯ A˜1n ] ] A ˜ n1 ¯ A˜nn

冣

n_⫻n , 共1.5b兲

where each entry of A˜klis the average of entries of Akl, 1艋k,l艋n.

For a given matrix, the above wavelet transform allows a perfect reconstruction 共inverse wavelet transform兲, by which there is nothing to gain: A=W−1_{共W共A兲兲. In Ref. 11, a simple}

opera-tor Okis introduced to attain a desirable coupling matrix. That is,

C = W−1共Ok共W共A兲兲兲 = A + 共k − 1兲W共A兲 ¬ A +␣W共A兲, 共1.5c兲

where Ok is the multiplication of a scalar factor K on each block matrix A˜kl. After such

recon-struction, the critical strength⑀cis again determined in terms of the second largest eigenvalue of

C. A numerical simulation of a coupled system of 512 Lorenz oscillators in Ref. 11 shows that

with h = I3 and A = A共1兲, the critical coupling strength ⑀c decreases linearly with respect to the

increase of␣up to a critical value␣c. The smallest⑀cis about 6, which is about 103times smaller

than the original critical coupling strength, indicating the efficiency of the proposed approach. To verify this phenomenon mathematically, we first consider the coupling matrix A = A共␤兲, as given in 共1.3兲. Let n=N/2i_{苸N, where i is a fixed positive integer. We then write A into an n}

⫻n block matrix of the form

₂i_⫻2i . 共1.6c兲

Then the newly transformed coupling matrix C = C共␣,␤兲 can be written as

C共␣,␤兲

=

冢

A₁共␤兲 + A˜₁共␤兲 A₂共1兲 + A˜₂共1兲 0 _¯ 0 A₂T共␤兲 + A˜₂T共␤兲 A₂T共1兲 + A˜₂T共1兲 A₁共1兲 + A˜₁共1兲 A₂共1兲 + A˜₂共1兲 0 _¯ 0

0 _]

] 0

0 _¯ 0 A₂T共1兲 + A˜₂T共1兲 A₁共1兲 + A˜₁共1兲 A₂共1兲 + A˜₂共1兲

A2共␤兲 + A˜2共␤兲 0 ¯ 0 A2 T_{共1兲 + A˜} 2 T 共1兲 A˜₁共␤兲 + A¯˜1共␤兲

冣

¬

冢

C1共␤兲 C2共1兲 0 ¯ 0 C2 T_共␤兲 C₂T共1兲 C1共1兲 C2共1兲 0 ¯ 0 0 _] ] 0 0 _¯ 0 C₂T共1兲 C₁共1兲 C₂共1兲 C₂共␤兲 0 _¯ 0 C₂T共1兲 C¯₁_共␤兲

冣

. 共1.7兲

Here for any matrix B of dimension 2i_⫻2i_{, the kl entry}_共B˜兲

klof B˜ is defined to be 共B˜兲kl= ␣ 22i

兺

l=1 2i

兺

k=1 2i 共B兲kl.

Here ␣ is a scalar factor. The matrix C共␣,␤兲 carries a new relationship among the coupled oscillators, which might not be as simple as the original matrix A. Nevertheless, the stability of the synchronous states can be determined by matrix C共␣,␤兲, whose eigenvalues ␭i共␣,␤兲共i

= 0 , 1 , 2 ,¯ ,N−1兲 determine the synchronous stability of the coupled chaotic system. The follow-ing theorem of Shieh et al.9 showed, indeed, the dramatic reduction in the critical coupling strength can be achieved with the periodic boundary conditions. We summarize their main results in the following.

Theorem 1.1: Let N⫻N, N=8k, k苸N, be the dimension of the matrix C共␣, 1兲. Let the

dimension of each block matrix in C共␣, 1兲 be 2i_⫻2i_{. Then the following assertions hold.}

(i) ␳iª2 cos共␲/ 2i兲−2 is an eigenvalue of C共␣, 1兲.

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when-ever␣艌−2i_␳

i/ 4 sin2共2i␲/ N兲.

Note that C共␣, 1兲 is a block circulant matrix 共see e.g., Ref. 3兲. A classical result of a block circulant matrix states that its eigenvalues exactly consist of those of certain linear combinations of its block matrices共see, e.g., Theorem 5.6.4 of Ref. 3兲. The proof of Theorem 1.1 was then reduced to working on the eigenvalues of those linear combinations of block matrices of C共␣, 1兲. Note that C共␣,␤兲,␤⫽1 are not block circulant matrix. The objective of the present work is to present another approach to study the eigenvalues of C共␣,␤兲. Specifically, we use this new method to study two coupling schemes, the nearest neighbor coupling with periodic boundary conditions and the nearest neighbor coupling with Neumann boundary conditions. To simplify our calculation, we consider only the case i = 1. In both coupling schemes, we are able to obtain, respectively, exact form of eigenvalues ␭m±共␣,␤兲 of its corresponding matrix C共␣,␤兲, see 共2.16兲

and 共3.9兲. Here␤= 0 or 1. For each ␣ and ␤, let ␭₁共␣,␤兲 be the second largest eigenvalue of

C共␣,␤兲. We prove that for N being a multiple of 4, then

␭1共␣,1兲 =

冦

␭1+共␣,1兲, 0艋␣艋 1 sin2 z n ␭n/2 + _共_␣_{,1兲 = − 2,} ␣艌 1 sin2␲ n .

冧

Let N = 2n be an even number which is not multiple of 4. We show that␭₁共␣, 1兲=␭_关n/2兴+ 共␣, 1兲 for

␣ sufficiently large, where 关n/2兴=the largest positive integer that is less than or equal to n/2. Moreover, we prove that for such N that ␭1共␣, 1兲⬍−2, whenever ␣⬎1/sin2共␲/ n兲. With those

results above, we get considerably more information than those obtained in Ref. 9. Among other, such result suggests that if the number N of oscillators is even but not a multiple of 4, then the wavelet method works even better. Specifically, it is better in the sense that the corresponding second largest eigenvalue␭1共␣, 1兲 is further away from 0, and, hence, gives even smaller critical

length. Our second main result is concerned with␭1共␣, 0兲 of C共␣, 0兲, which corresponds to the

nearest neighbor coupling with Neumann boundary conditions. We show that for all even number

N its second largest eigenvalue␭1共␣, 0兲 for each␣behaves like its periodic counterpart for which

its corresponding N is a multiple of 4.

II. PERIODIC BOUNDARY CONDITIONS

␣ 4 ␣ 4 1 4共4 +␣兲 ␣ 4

冣

. 共2.1c兲

We begin by identifying some trivial eigenvalues of C共␣, 1兲.

Proposition 2.1: For each␣, 0 and −4 are eigenvalues of C共␣, 1兲. If, in addition, n/2共⬎1兲 is a positive integer, then −2 is also an eigenvalue of C共␣, 1兲 for any ␣.

Proof: Let C共␣, 1兲+4I=共c1, c2, . . . , cN兲, where ci, 1艋i艋N, are column vectors. Then

⌺j=1 N _共−1兲j+1_c

j= 0. Thus −4 is an eigenvalue of C共␣, 1兲 for each ␣⬎0. Let C共␣, 1兲+2I

=共c1, c2, . . . , cN兲. If N=2n共⬎4兲 is a multiple of four, then ⌺j=1 N _␦_共j兲c

j= 0, where

␦共j兲 =

再

1 if j = 4k or 4k + 1 for some k − 1 if j = 4k + 2 or 4k + 3 for some k.

冎

Thus, −2 is an eigenvalue of C共␣, 1兲 for each␣with such N. 䊐

Writing the corresponding eigenvalue problem C共␣, 1兲b=␭b, where b=共b₁, b₂, . . . , bn兲Tand

bi苸C2, in block component form, we have

C₂T共1兲bi−1+ C1共1兲bi+ C2共1兲bi+1=␭bi, 1艋 i 艋 n. 共2.2a兲

Periodic boundary conditions would yield that

C₂T共1兲b₀+ C₁共1兲b₁+ C₂共1兲b₂=␭b₁= C₁共1兲b₁+ C₂共1兲b₂+ C₂T共1兲bn and C2 T_共1兲b n−1+ C1共1兲bn+ C2共1兲bn+1=␭bn= C2共1兲b1+ C2 T_共1兲b n−1+ C¯1共1兲bn, or, equivalently, b0= bn, 共2.2b兲 b1= bn+1. 共2.2c兲

To study the block difference equation共2.2兲, we first seek to find the solution biof the form

bi=␦i

冉

1

␯

冊

. 共2.3兲

Substituting共2.3兲 into 共2.2a兲, we get 关C2

T_{共1兲 +}_␦_共C

1共1兲 − ␭I兲 +␦2C2共1兲兴

冉

1

␯

冊

= 0. 共2.4兲

To have a nontrivial solution

共

_␯1

兲

to Eq.共2.4兲, we need to have det关C2 T_{共1兲 +}_␦_共C 1共1兲 − ␭I兲 +␦2C2共1兲兴 = 0, 共2.5a兲 or, equivalently, ␣␦4₊_共4_␣_{+ 4 + 2}_␣_␭兲_␦3₋_{共8 + 10}_␣_{+ 16␭ + 4}_␣_{␭ + 4␭}2_兲_␦2₊_共4_␣_{+ 4 + 2}_␣_␭兲_␦₊_␣_{= 0.} 共2.5b兲 Equation共2.5b兲 is to be called the characteristic equation of the block difference equation 共2.2a兲. To study the property of Eq.共2.5b兲, we need the following proposition.

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x2, x3, and x4 be roots of det关D1+ xD2+ x2D3兴=0, where x苸C. Then we may renumber the

sub-scripts if necessary so that

x₁x₂= 1 = x₃x₄. 共2.6a兲

If, in addition, diagonal elements of D1 and D2, respectively, are both equal, then

y1y2= 1 = y3y4. 共2.6b兲

Here

共

y1i

兲

, i = 1 , 2 , 3 , 4, are vectors satisfying

关D1+ xiD2+ xi 2 D3兴

冉

1 yi

冊

= 0. 共2.6c兲

Proof: If D₁, D₂, and D₃are as assumed, then det关D₁+ xD₂+ x2_D

3兴 = ax4+ bx3+ cx2+ bx + a 共2.7兲

for some constants a⫽0, b, and c. Letting y=x+1/x, then 共2.7兲 can be written as␣y2₊_␤_{y +}_␥_,

where␣,␤, and␥depend on the constants a, b, and c. Thus det关D1+ xD2+ x2D3兴=0 is equivalent

to x2₋_␭

±x + 1 = 0, where␭±are the roots a1y2+ b1y + c1= 0. Consequently, x1x2= 1 = x3x4.

Letting D1=

共

a1 c1 b1 a1

兲

= D3 T and D2=

共

a2 b2 b2

a2

兲

, we write共2.6c兲 in component form,

共a1+ yib1兲 + 共a2+ yib2兲xi+共a1+ yic1兲xi 2

= 0, i = 1,2,3,4, 共2.8a兲

共c1+ yia1兲 + 共b2+ yia2兲xi+共b1+ yia1兲xi2= 0, i = 1,2,3,4, 共2.8b兲

For i = 1,共2.8a兲 is equal to

共a1+ y1b1兲 + 共a2+ y1b2兲 1 x2 +共a₁+ y₁c₁兲1 x₂2= 0 or

共a1+ y1c1兲 + 共a2+ y1b2兲x2+共a1+ y1b1兲x22= 0

x22= 0. 共2.8c兲

Using Eqs.共2.8c兲 and 共2.8b兲 with i=2, and the uniqueness of yi, i = 1 , 2 , 3 , 4, we conclude that

y₁y₂= 1. Similarly, y₃y₄= 1. We just complete the proof of the proposition. 䊐 We are now in a position to further study Eq. 共2.5兲. We assume, momentarily, that Eq. 共2.5兲 has four distinct roots␦1,␦2,␦3, and␦4. The general solutions to 共2.2a兲 can then be written as

. 共2.9兲

c₁ c2 c₃ c4

冣

= 0. 共2.11兲

Now if, diag共␦1n

− 1 ,␦2n− 1 ,␦3n− 1 ,␦4n− 1兲 is singular, then Eq. 共2.9兲 has nontrivial solutions ci, i

= 1 , 2 , 3 , 4. Note that diag共␦1n− 1 ,␦2n− 1 ,␦3n− 1 ,␦4n− 1兲 is singular if and only if ␦i, i = 1 , 2 , 3 , 4,

satisfy

␦n

= 1 共2.12兲

and共2.5b兲. To solve the system of equations 共2.12兲 and 共2.5b兲, we first note that

␦m= ei2m␲/n, 0艋 m 艋 n − 1, 共2.13兲

= 0. 共2.14兲 Before we proceed to compute the real part of the resulting equation, we need the following lemma.

Lemma 2.1: Let a, b, and c be any complex number, then

cos 2␪共sin 4␪+ a sin 3␪+ b sin 2␪+ a sin␪兲 = sin 2␪共cos 4␪+ a cos 3␪+ b cos 2␪+ a cos␪+ 1兲. 共2.15兲 Since the proof of the lemma is straightforward, we will skip it.

Using共2.14兲 and 共2.15兲, we see immediately that the real part of 共2.5b兲 with ␦= ei2m␲/n is a constant multiple sin/ cos共4m␲/ n兲/共4m␲/ n兲 of its imaginary part. We next show that 共2.14兲 is indeed the characteristic equation of the matrix C共␣, 1兲.

Theorem 2.1: Let N⫻N, N=2k, k苸N, be the dimension of the matrix C共␣, 1兲. Let dimension

of each block matrix in C共␣, 1兲 be 2⫻2. Then the eigenvalues ␭m±共␣, 1兲 of C共␣, 1兲 are of the

following form: ␭m ±_共_␣_{,1兲 =}1 2

冉

␣cos 2m␲ n −␣− 4

冊

± 1 2

冋

冉

␣cos 2m␲ n −␣− 4

冊

2 + 4

冉

␣cos22m␲ n + 2共␣+ 1兲cos2m␲ n − 2 − 3␣

冊

册

1/2 ¬ ␭ˇm共␣,1兲 ± ␭ˆm共␣,1兲, m = 0,1, ... ,n − 1. 共2.16兲

Proof: Solving共2.14兲, we get 共2.16兲. Using Proposition 2.2, we see that if␦= 1 or −1 is a root of Eq. 共2.5b兲, then the multiplicity of ␦= 1 or −1 is both two. Thus, we have only proved the

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following. 共i兲 If n/2 is not a positive integer, then for each ␣, ␭m±共␣, 1兲, m=1,2, ... ,n−1, are

eigenvalues of C共␣, 1兲. 共ii兲 If n/2 is a positive integer, then for each ␣, ␭m±共␣, 1兲, m

= 1 , 2 , . . . , n / 2 − 1 , n / 2 + 1 , . . . , n − 1, are eigenvalues of C共␣, 1兲. To complete the proof of the theo-rem, it remains to show that for each␣,␭₀±共␣, 1兲共=0,−4兲 are eigenvalues of C共␣, 1兲 for each␣and that if, additionally, n / 2⬎1 is a positive integer, then for each␣,␭_n/2± 共␣, 1兲共=−2,−2−2␣兲 are also eigenvalues of C共␣, 1兲. Using Proposition 2.1, we only need to show that −2−2␣=共␭_n/2− 共␣, 1兲兲 is an eigenvalue of C共␣, 1兲 for fixed␣. To this end, we see that

trace of C共␣,1兲 = − n共␣+ 4兲. 共2.17兲

Let N = 2n⬎4 be a multiple of four, then ␭n/2+ 共␣,1兲 +

冉

兺

j=1,j⫽n/2 n

␭±j共␣,1兲

冊

+␭0±共␣,1兲 = − 2 − 共n − 2兲共␣+ 4兲 − 4. 共2.18兲

Using共2.17兲 and 共2.18兲, we have that the remaining eigenvalue of C共␣, 1兲 for each␣is −2 − 2␣, which is equal to␭_n/2− 共␣, 1兲. We thus complete the proof of the theorem. 䊐

Proposition 2.3: For all␣⬎0, we have that ␭ˆm共␣, 1兲⬎0, ␭ˇm共␣, 1兲⬍0 and ␭m±共␣, 1兲艋0.

Proof: Obviously,␭ˇm共␣, 1兲⬍0. Now, letting t=cos共2m␲/ n兲, we have that

4共␭ˆm共␣,1兲兲2=共t − 1兲2␣2+ 4共t2− 1兲␣+ 8共1 + t兲 = 共共t − 1兲␣+ 2共t + 1兲兲2+ 4共1 − t2兲 ⬎ 0

for any␣⬎0. Thus ␭ˆm共␣, 1兲⬎0. To prove the last assertion of the proposition, we note, via 共2.16兲,

that 0⬎ 4

冉

␣cos22m␲ n + 2共␣+ 1兲cos 2m␲ n − 2 − 3␣

冊

¬ l. Thus, 2␭m ±_共_␣_{,1兲 = 2␭ˇ} m共␣,1兲 ± 共4␭ˇm 2_共_␣_{,1兲 + l兲}1/2_{艋 0.}

We just complete the proof of the proposition. 䊐

Proposition 2.4: If n / 2 is not a positive integer, then the eigencurves ␭m

±_共_␣_{, 1兲, m}

= 1 , 2 , . . . , n − 1, are strictly decreasing in ␣苸共0,⬁兲. If n/2共⬎1兲 is a positive integer, then ␭m±共␣, 1兲, m=1,2, ... ,n/2−1,n/2+1, ... ,n−1, and ␭n/2− 共␣, 1兲 are strictly decreasing in ␣

苸共0,⬁兲.

Proof: Letting t = cos共2m␲/ n兲, we write 共2.16兲 as

␭m ±_共_␣_{,1兲 =}1 2兵␣共t − 1兲 − 4 ± 关共t − 1兲 2_␣2_{+ 4共t}2_{− 1兲}_␣_{+ 8共1 + t兲兴}1/2_其 ¬12兵␣共t − 1兲 − 4 ± 共t␣兲 1/2_{其 ¬ ␭} t ±_共_␣_兲. _共2.19兲 Then 2d␭m ±_共_␣_,1_兲 d␣ =共t − 1兲

冉

1 ± 共t − 1兲␣+ 2共t + 1兲

冑

t_␣

冊

.

A direct computation would yield that

t_␣艌共共t − 1兲␣+ 2共t + 1兲兲2.

Thus, d␭m±共␣, 1兲/d␣艋0. The equality holds only if t=1 or t=−1 for ␭m+. 䊐

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␣t,k=

2共1 + t兲 − k2

共1 − t兲共1 + t + k兲. 共2.20兲

(ii) For −1艋t⬍1, lim_␣→⬁␭t+共␣, 1兲=−共t+3兲.

Proof: Solving equation −2 + k =␭t+共␣, 1兲, we easily get that ␣t,k are as asserted. Rewriting

␭t+共␣, 1兲 as

␭t+共␣,1兲 =

− 2␣共t − 1兲共t + 3兲 + 4共1 − t兲

␣共t − 1兲 − 4 −

冑

t_␣ ,

we see that lim_␣→⬁␭t

+_共_␣_{, 1兲=−共t+3兲 for −1艋t⬍1.} _䊐

Theorem 2.2: Let N be any positive even integer. The dimension of each block matrix in

C共␣, 1兲 is 2⫻2. Then (i) suppose N is a multiple of four and N⬎4. For each␣⬎0, let ␭共␣, 1兲 be the second largest eigenvalue of C共␣, 1兲. Then ␭共␣, 1兲=␭₁+共␣, 1兲, for 0艋␣艋1/sin2共␲/ n兲ª␣1;

and␭共␣, 1兲=␭_n/2+ 共␣, 1兲=−2 for all␣苸关␣1,⬁兲. See Fig. 1.

(ii) Suppose N is not a multiple of four. Then there exists a␣˜csuch that␭共␣, 1兲=␭_关n/2兴+ 共␣兲 for

all␣艌␣˜c. Here 关n/2兴=the largest positive integer that is less than or equal to n/2. Moreover,

␭共␣, 1兲⬍−2 whenever␣⬎␣1. See Fig. 2.

Proof: For ␣_t,kto be positive, we must have

2共1 + t兲 ⬎ k2_. _共2.21兲 Now, 共1 − t兲2_{共1 + t + k兲}2d␣t,k dt = 2共t + 1兲 2_{− k}3_{+ 4k − 2tk}2_{⬎ 共1 + t兲k}2_{− k}3_{+ 4k − 2tk}2 = − k共k2₊_{共t − 1兲k − 4兲 = − k共k − t} +兲共k − t−兲,

where t±=共1−t±

冑

16+共1−t兲2兲/2. Note that we have used 共2.21兲 to justify the above inequality.

Moreover t−⬍0 and t+艌2. Thus, d␣t,k/ dt⬎0 whenever ␭=−2+k, 0艋k⬍2, and ␭=␭t+共␣, 1兲 have

the intersections intersect at the positive ␣_t,k. Upon using Proposition 2.4, we conclude that for 0艋m艋n−1, the portion of the graphs of ␭m+共␣, 1兲 lying above the line ␭=−2 do not intersect each

other. Thus,␭共␣, 1兲 is as asserted. By Proposition 2.5共ii兲, we have that FIG. 1. The curves␭m

±_共_␣_{, 1}_{兲 with N=2n=12 are provided. As predicted in Theorem 2.2–共i兲, ␭共}_␣_{, 1}_{兲 turns flat after}_␣ 1.

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lim ␣→⬁␭m +_共_␣_{,1兲 = −}

冉

cos2m␲ n + 3

冊

¬ ␭m ⬁₌_␭ t ⬁_.

Then␭m⬁, 0⬍m艋n−1, have a maximum at m=关n/2兴. Thus, there exists a ␣˜c such that␭共␣, 1兲

=␭_关n/2兴+ 共␣, 1兲 for all␣艌␣˜c. The last assertion of the theorem follows from Proposition 2.5-共i兲 and

Proposition 2.1. 䊐

Remark 2.1: 共i兲 Since ␭_t+共␣, 1兲 is increasing in t and ␭t⬁ is decreasing in t, the eigencurves

␭m

+_共_␣_{, 1}_{兲, 0⬍m艋关n/2兴 must be crossing each other.}

共ii兲 The first column in Table I contains the values of ␭m

±_{共1,1兲, m=0,1, ... ,5, while the second}

column contains the eigenvalues of C共1,1兲 obtained by usingMATHEMATICA. As indicated, the

C共1,1兲 and C共5,1兲 obtained by both methods are identical. The values ␭m

±_{共3,1兲, m=0,1, ... ,8, in}

the first and third columns of Table II are computed by MAPLE, while those in the second and fourth columns are computed byMATLAB. Some discrepancies between the values in the respec-tive columns occur due to the round-off errors.

共iii兲 Figure 1 illustrates the graph of ␭m

±_共_␣_{, 1兲, m=0,1, ... ,5, with n=6. The dotted part of the}

curve is␭共␣, 1兲. Figure 2 gives the same information with n=9.

共iv兲 We conclude, via the last assertion of Theorem 2.1, that the wavelet approach works even better when N is an even number but not a multiple of four. Indeed, in such case, it synchronizes faster when␣is chosen to be the critical value␣˜c.

III. NEUMANN BOUNDARY CONDITIONS

Here, we consider the nearest neighbor coupling with Neumann boundary conditions. The resulting coupling matrix A is then A共0兲, given as in 共1.6a兲. With i=1, we have

FIG. 2. The curves ␭m±共␣, 1兲 with N=2n=18 are provided. As predicted in Theorem 2.2–共ii兲, ␭共␣, 1兲 lies below −2 eventually.

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TABLE I. The first and third columns contain the values computed by using formulas␭m±共␣, 1兲 as given in 共2.16兲. The values in the second and fourth columns are eigenvalues of C共␣, 1兲 obtained by using MATHEMATICA.

C₂T共1兲bi−1+ C1共1兲bi+ C2共1兲bi+1=␭bi, 1艋 i 艋 n. 共3.3兲

With Neumann boundary conditions, b0, and bn+1must satisfy

C1共0兲b1+ C2共1兲b2=␭b1= C2 T_共1兲b 0+ C1共1兲b1+ C2共1兲b2 共3.4a兲 and C₂T共1兲b_n−1+ C¯₁共0兲bn=␭bn= C2 T_共1兲b n−1+ C1共1兲bn+ C2共1兲bn+1. 共3.4b兲

Solving共3.4a兲 and 共3.4b兲, respectively, we get b0=共C2 T_共1兲兲−1_共C 1共0兲 − C1共1兲兲b1=

冉

0 1 1 0

冊

b1 共3.5a兲 and

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bn+1= C2共1兲−1共C¯1共0兲 − C1共1兲兲bn=

冉

0 1

1 0

冊

bn. 共3.5b兲

We then see that the characteristic equation of the block difference equation共3.3兲 is

det关C2

T_{共1兲 +}_␦_共C

1共1兲 − ␭I兲 +␦2C2兴 = 0. 共3.6a兲

Here␦ is such that bi=␦i

共

1

␯

兲

, where␯is a constant depending on␦. Expanding the determinant in

共3.6a兲, we get

␣␦4_{+ 2共2}_␣_{+ 2 +}_␭_␣_兲_␦3_{− 2共4 + 5}_␣_{+ 2共}_␣_{+ 4兲␭ + 2␭}2_兲_␦2_{+ 2共2}_␣_{+ 2 +}_␭_␣_兲_␦₊_␣_{= 0.}

共3.6b兲 We assume, momentarily, that Eq. 共3.6b兲 has four distinct roots ␦1,␦2,␦3, and ␦4. The general solutions to共3.3兲 can then be written as

bi=

兺

j=1 4 cj␦j i

冉

1 ␯j

冊

. 共3.7兲

Substituting共3.7兲 into boundary conditions 共3.5兲, we get

冢

␦1␯1− 1 ␦2␯2− 1 ␦3␯3− 1 ␦4␯4− 1 ␦1−␯1 ␦2−␯2 ␦3−␯3 ␦4−␯4 ␦1n_共_␦1␯1 − 1兲 ␦2n共␦2␯2− 1兲 ␦3n共␦3␯3− 1兲 ␦4n共␦4␯4− 1兲 ␦1n_共_␦1 −␯1兲 ␦2n共␦2−␯2兲 ␦3n共␦3−␯3兲 ␦4n共␦4−␯4兲

冣

冢

c1 c2 c3 c4

冣

¬ Dc = 0, 共3.8兲

where c =共c₁, c₂, c₃, c₄兲T_{. We are now in a position to simplify det D,}

TABLE III. The first and third columns contain the values computed by using formulas␭m±共␣, 0兲 as given in 共3.9兲. The values in the second and fourth columns are eigenvalues of C共␣, 1兲 obtained by using MATH-EMATICA. n = 3 ␭m±共2,0兲 Eigenvalues of C共2,0兲 ␭m±共5,0兲 Eigenvalues of C共5,0兲 ␭0+共2,0兲=0 0 ␭0+共5,0兲=0 0 ␭1+共2,0兲=− 5 2+ 1 2冑7 − 5 2+ 1 2冑7 ␭1 +_{共5,0兲=−}13 4 + 1 4冑13 − 13 4+ 1 4冑13 ␭2 +_{共2,0兲=−}7 2+ 1 2冑7 − 7 2+ 1 2冑7 ␭2 +_{共5,0兲=−}23 4+ 1 4冑181 − 23 4+ 1 4冑181 ␭3 +_{共2,0兲=−2} ₋₂ _␭ 3 +_{共5,0兲=−2} ₋₂ ␭1−共2,0兲=− 5 2− 1 2冑7 − 5 2− 1 2冑7 ␭1−共5,0兲=− 13 4 − 1 4冑13 − 13 4− 1 4冑13 ␭2−共2,0兲=− 7 2− 1 2冑7 − 7 2− 1 2冑7 ␭2 −_{共5,0兲=−}23 4− 1 4冑181 − 23 4− 1 4冑181

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1 −␦1␯1 1 −␦3␯3 ␯1−␦1 ␯3−␦3

冏

冎

.

Therefore, det D being equal to zero amounts to␦i 2n

= 1 for i = 1 , 2 , 3 , 4.

To get the characteristic equation of C共␣, 0兲, we need to solve ␦2n_{= 1 and Eq.}_{共3.6b兲. This}

leads to the following theorem.

Theorem 3.1: Let N be any positive even integer. The dimension of each block matrix in

C共␣, 0兲 is 2⫻2. Let ␭m ±_共_␣_{, 0兲 be defined as follows:} ␭m±共␣,0兲 = 1 2

冉

␣cos m␲ n −␣− 4

冊

± 1 2

冋

冉

␣cos m␲ n −␣− 4

冊

2 + 4

冉

␣cos2 m␲ n + 2共␣+ 1兲 ⫻cosm␲ n − 2 − 3␣

冊

册

1/2 . 共3.9兲

Then␭m±共␣, 0兲, m=1,2, ... ,n−1, ␭0+共␣, 0兲=0 and ␭n+共␣, 0兲=−2 are eigenvalues of C共␣, 0兲 for each ␣⬎0.

Proof: Substituting␦= eim␲/n_{, 0}_{艋m艋n−1, into 共3.6b兲, we get 共3.9兲. Clearly, if}␦_{⫽1 or −1, or}

equivalently, cos共m␲/ n兲⫽1 or −1, then ␭m±共␣, 0兲, m=1,2, ... ,n−1, are eigencurves of C共␣, 0兲.

Since 0 =␭0+共␣, 0兲 is an eigenvalue of C共␣, 0兲 for all ␣, we only need to show that ␭n+共␣, 0兲 is,

indeed, the eigenvalue of C共␣, 0兲 for each ␣. To this end, we see that trace共C共␣, 0兲兲=−共n−2兲共␣

+ 4兲−6−␣. However, ␭₀+共␣, 0兲+兺n−1_j=1␭j±共␣, 0兲=−共n−1兲共␣+ 4兲= ¬k. Thus, trace共C共␣, 0兲兲−k=−2

=␭n+共␣, 0兲. We just complete the proof of the theorem. 䊐

Remark 3.1:共i兲 Letting t=cos共m␲/ n兲, ␭_m±共␣, 0兲=␭_t±共␣, 0兲 and treating t as a real parameter, we see that for fixed␣⬎0, the eigenvalues of C with periodic boundary conditions and Neumann

FIG. 3. The curves␭m

±_共_␣_{, 0}_{兲 with N=2n=6 are provided. As predicted in Theorem 3.2, ␭共}_␣_{, 0}_{兲 turns flat after}_␣ 1.

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boundary conditions, respectively, lie on the curve␭t±共␣, 0兲 in t−␭ plane.

共ii兲 Note that ␭m

±_共_␣_{, 0兲=␭} 2n−m ± _共_␣_{, 0兲.}

Theorem 3.2: For each ␣, let ␭共␣, 0兲 be the second largest eigenvalue of C共␣, 0兲. Then ␭共␣, 0兲=␭1

+_共_␣_{, 0兲, for 0艋}_␣_艋1/sin2_共_␲_{/ 2n}_兲¬_␣_¯

1; and␭共␣, 0兲=␭n

+_共_␣_{, 0兲=−2 for all}_␣_苸关_␣_¯ 1,⬁兲.

We skip the proof of theorem due to its similarity with that of Theorem 2.1 共ii兲.

Remark 3.2: Table III and Fig. 3 illustrate, again, the accuracy of our theorems.

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