PROOF OF SYNCHRONIZED CHAOTIC BEHAVIORS IN COUPLED MAP LATTICES

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c

World Scientific Publishing Company

DOI:10.1142/S0218127411029069

PROOF OF SYNCHRONIZED CHAOTIC BEHAVIORS

IN COUPLED MAP LATTICES*

WEN-WEI LIN†and YI-QIAN WANG‡

†_{Department of Mathematics, National Chiao-Tung University,}

Hsinchu 30010, Taiwan [email protected]

‡_{Department of Mathematics, Nanjing University,}

Nanjing 210093, P. R. China [email protected] Received February 3, 2010

In this paper, we consider chaotic synchronization in coupled map lattices (CMLs) with periodic boundary conditions. We give a rigorous proof of the occurrence of synchronization for 1D such CMLs with lattice sizen = 5 for suitable parameters in the chaotic regime by Lyapunov method. Keywords: Synchronization; coupled map lattices; chaos; Lyapunov method.

1. Introduction

Chaotic synchronization is a fundamental phe-nomenon in physical systems with dissipation. Experimental simulations show that chaotic sub-systems in a lattice manifest synchronized chaotic behavior in time provided they are coupled with a dissipative coupling and a coefficient of this cou-pling is greater than some critical value. This phe-nomenon has been observed and well-studied in many different fields — synchronization of cou-pled chaotic circuits [Carroll & Pecora, 1991; Chua et al., 1993; Goldsztein & Streogatz, 1995], cou-pled chaotic oscillators [Afraimovich et al., 1997; Afraimovich & Lin, 1998; Chiu et al., 1998; Chan & Chao, 1998; Ermentrout, 1985; Fujisaka & Yamada, 1983; Mirollo & Strogatz, 1990] and master-slave chaotic Lorenz equations [Pecora & Carroll, 1990; He & Vaidya, 1992], etc.

In practice, with the combination of syn-chronization and unpredictability, chaotic synchro-nization has attracted a lot of attention for its promising potential in secure communication.

A secret message can be modulated on the chaotic signal of a sender, and a receiver with an identi-cal system which is driven by the modulated signal can decrypt this message. Many encryption mod-els based on chaotic synchronization have been pro-posed, see [Pecora & Carroll, 1990; Vohra et al., 1992; Cuomo & Oppenheim, 1992, 1993; Wu & Chua, 1994; Heagy et al., 1995; Pecora et al., 1997]. More recently, communication with chaos synchro-nization has been demonstrated with semiconduc-tor lasers which were synchronized over a distance of 120 km in a public ﬁber network in Greece, see [Argyris et al., 2005].

A mathematical foundation of synchronization of these coupled systems was heavily dependent on the bounded dissipativeness, the coupling rule and the type of chaotic subsystems. Thus, the problem of coming up with a rigorous mathematical proof of chaotic synchronization for speciﬁed coupled systems appears to be attractive and important from both theoretical and practical points of view.

∗_{This work was supported by the NNSF of China 10871090 and National Basic Research Program of China 2007CB814804.}

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Int. J. Bifurcation Chaos 2011.21:1493-1500. Downloaded from www.worldscientific.com

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In this paper, we consider chaotic synchroniza-tion in coupled map lattices (CMLs) which can be considered as systems of interacting maps, where the individual map is characterized not only by its internal state but also by the position in the physical space. CMLs are, in general, the interme-diate between partial differential equations (PDEs) and cellular automata which form a wide class of extended dynamical systems. PDEs are usually used to describe the physical phenomenon of spatial-temporal dynamical systems. However, the analytic study of solutions of PDEs suffers from extreme dif-ficulty with complex behavior. On the other hand, the computer simulation is utilized as an effective and powerful tool to study dynamical systems with complex behavior. In such a study the dynamical system shall be discretized in space as well as in time. This is one of the motivations to introduce new models of CMLs (see [Afraimovich & Buni-movich, 1993; BuniBuni-movich, 1997; Bunimovich & Carlen, 1995; Giberti & Vernia, 1994; Kaneko, 1993]).

A popular model in CMLs is deﬁned as follows:

(1) 1D lattice, for 1≤ i ≤ n,

xi(k + 1) = f(x_i(k)) + c(f(x_i−1(k))

+f(x_i+1(k)) − 2f(x_i(k))) (1) with periodic boundary conditions f(x₀(k)) = f(xn(k)) and f(x_n+1(k)) = f(x₁(k)), and (2) 2D lattice, fori = (i₁, i₂) with 1≤ i₁, i₂≤ n,

xi(k + 1) = f(xi(k)) + c(f(xi1+1,i2(k)) +f(x_i₁_−1,i₂(k)) + f(x_i₁_,i₂₊₁(k)) +f(x_i₁_,i₂₋₁(k)) − 4f(x_i(k))) (2) with f(x_0,i₂(k)) = f(x_n,i₂(k)), f(x_n+1,i₂(k)) = f(x1,i2(k)), f(xi1,0(k)) = f(xi1,n(k)) and

f(xi1,n+1(k)) = f(xi1,1(k)), where f is a one-dimensional logistic mapx(k + 1) = f(x(k)) = γx(k)(1 − x(k)) with f : (0, 1) → (0, 1) and γ ∈ [γ∞, 4] with γ∞≈ 3.57. It is well known (see

[Campbell, 1989; Gleick, 1987], etc.) that the map f becomes chaotic whenever γ increases from 3.57 to 4, except that γ is at a very nar-row interval of periodic windows near 3.63, 3, 73 or 3.83.

The simplest type of synchronization of CMLs in Eq. (1) or Eq. (2) occurs in stable spatially

homogeneous regimes corresponding to the exis-tence of attractive spatially homogeneous solutions. In other words, in such cases there is a large (open) set of initial conditions such that a solution start-ing from an initial condition in the set becomes spa-tially homogeneous as discrete timek becomes very large, i.e. the coordinates of the individual maps become almost equal to each other (and are equal ask → ∞). In established regimes, individual maps become indistinguishable and we observe exact per-fect synchronization. Thus, it may occur that suit-able coupling strength permits the existence of a spatially homogeneous solution provided all individ-ual maps are identical.

In 1999, [Lin et al., 1999] considered the syn-chronized chaotic behavior of Eqs. (1) and (2). They provided a complete numerical analysis for the range of parameters γ, the lattices size n and the coupling strengthsc such that chaotic synchroniza-tion occurs in the corresponding 1D and 2D CMLs. Moreover, they gave a rigorous proof for chaotic synchronization in the case of 1D CMLs with lattice size n = 2, 3 for γ ∈ [γ_∞, 4] and with lattice size n = 4 for γ ∈ [γ∞, 3.82] ⊂ [γ∞, 4] of the chaotic

regime. As the authors know, it is the ﬁrst rigorous proof on chaotic synchronization of CMLs.

In 2001, [Lin & Wang, 2002] generalized the mathematical result of [Lin et al., 1999] on the case of n = 4 to the full chaotic interval [γ_∞, 4] by Lya-punov method.

On the other hand, the numerical simulations in [Lin et al., 1999] also showed that to ensure the occurrence of chaotic synchronization, the larger is the lattice size n, the less is the measure of the parameter set (γ, c). Moreover, when n ≥ 12, no chaotic synchronization behavior can be observed, which is due to the fact that the spatially homoge-neous regime becomes unstable for large n.

In this paper, we will prove the occurrence of chaotic synchronization for the case n = 5. For this purpose, however, we cannot follow the approach in [Lin et al., 1999] or [Lin & Wang, 2002] for the case n = 4. The reason is that CMLs with size n = 4 have some special property of “decoupling”, which is critical in the proof of [Lin et al., 1999; Lin & Wang, 2002]; in comparison, unfortunately, CMLs with sizen ≥ 5 have no such property.

In the following, by modifying the method in [Lin & Wang, 2002], we will construct a more eﬃ-cient Lyapunov function for 1D CMLs (1) with size n = 5 to prove the occurrence of the synchronized chaotic behavior. More precisely, we will prove the

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following theorem:

Theorem 1. For every parameter γ ∈ [γ_∞, 4) for

the logistic map in the CMLs (1) with lattices size n = 5, there exists δ = δ(γ) > 0 such that for any initial values x_i(0) ∈ (0, 1), i = 1, . . . , 5 and any c ∈ (2/5 − δ(γ), 2/5 + δ(γ)), it holds that

lim

k→∞|xi(k) − xj(k)| = 0

for i, j = 1, 2, 3, 4, 5.

Remark 1.1. The Lyapunov method used here can also be used to generalize the result of [Lin & Wang, 2002], i.e. we can obtain the proof of chaotic synchronization for more parameters (c, γ) than in [Lin & Wang, 2002]. Hence this modiﬁed Lyapunov method is more eﬃcient.

Remark 1.2. The authors believe that by careful computation, it is possible to use our idea of Lyapunov method to prove the synchronized chaotic behavior for CMLs (1) with more lattice sizes.

2. Lyapunov Function and Proof of Theorem 1

In this section, we will construct a Lyapunov func-tion to prove the synchronized chaotic behavior for 1D coupled map lattices with sizen = 5.

First, we introduce the following proposition:

Proposition 2.1. For c ∈ [0, 1/2], 3 < γ < 4

and (x₁(0), x₂(0), x₃(0), x₄(0), x₅(0)) in (0, 1)5, there exists K ∈ N such that for all k ≥ K, (x₁(k), x₂(k), x₃(k), x₄(k), x₅(k)) generated by (1) lie in D0= 4− γ 4 , γ 4 ₅ .

Proof. The proof is the same as Theorem 2.3 in [Lin et al., 1999] and we omit it here.

By Proposition 2.1, without loss of general-ity, we assume (1) is deﬁned on D₀, i.e. x_i(0) ∈ [(4− γ)/4, γ/4], i = 1, 2, 3, 4, 5.

We rewrite the iteration in lattice of (1) for n = 5 by replacing xi(k + 1), xi(k) and f(xi) by

xi, x_i and f_i, i = 1, 2, 3, 4, 5 respectively as follows: x1 =f(x1) +c(f(x2) +f(x5)− 2f(x1)), x2=f(x2) +c(f(x1) +f(x3)− 2f(x2)), x3=f(x3) +c(f(x2) +f(x4)− 2f(x3)), x4=f(x4) +c(f(x3) +f(x5)− 2f(x4)), x5=f(x5) +c(f(x4) +f(x1)− 2f(x5)), (3) wheref(x) = γx(1 − x).

By direct computation, we have

(x₁− x₂)2= [(1− 3c)(f₁− f₂) +c(f₅− f₃)]2 = (1− 3c)2(f₁− f₂)2+c2(f₅− f₃)2 + 2(1− 3c)c(f₁− f₂)(f₅− f₃), (x₁− x₃)2= [(1− 2c)(f₁− f₃) +c(f₅− f₄)]2 = (1− 2c)2(f₁− f₃)2+c2(f₅− f₄)2 + 2(1− 2c)c(f₁− f₃)(f₅− f₄), (x₁− x₄)2= [(1− 2c)(f₁− f₄) +c(f₂− f₃)]2 = (1− 2c)2(f₁− f₄)2+c2(f₂− f₃)2 + 2(1− 2c)c(f₁− f₄)(f₂− f₃), (x₁− x₅)2= [(1− 3c)(f₁− f₅) +c(f₂− f₄)]2 = (1− 3c)2(f₁− f₅)2+c2(f₂− f₄)2 + 2(1− 3c)c(f₁− f₅)(f₂− f₄), (x₂− x₃)2= [(1− 3c)(f₂− f₃) +c(f₁− f₄)]2 = (1− 3c)2(f₂− f₃)2+c2(f₁− f₄)2 + 2(1− 3c)c(f₂− f₃)(f₁− f₄), (x₂− x₄)2= [(1− 2c)(f₂− f₄) +c(f₁− f₅)]2 = (1− 2c)2(f₂− f₄)2+c2(f₁− f₅)2 + 2(1− 2c)c(f₂− f₄)(f₁− f₅), (x₂− x₅)2= [(1− 2c)(f₂− f₅) +c(f₃− f₄)]2 = (1− 2c)2(f₂− f₅)2+c2(f₃− f₄)2 + 2(1− 2c)c(f₂− f₅)(f₃− f₄), (x₃− x₄)2= [(1− 3c)(f₃− f₄) +c(f₂− f₅)]2 = (1− 3c)2(f₃− f₄)2+c2(f₂− f₅)2 + 2(1− 3c)c(f₃− f₄)(f₂− f₅), (x₃− x₅)2= [(1− 2c)(f₃− f₅) +c(f₂− f₁)]2 = (1− 2c)2(f₃− f₅)2+c2(f₂− f₁)2 + 2(1− 2c)c(f₃− f₅)(f₂− f₁),

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(x₄− x₅)2 = [(1− 3c)(f₄− f₅) +c(f₃− f₁)]2 = (1− 3c)2(f₄− f₅)2+c2(f₃− f₁)2

+ 2(1− 3c)c(f₄− f₅)(f₃− f₁). Especially, whenc = 2/5, it follows that

5 i,j=1 (x_i− x_j)2 = 1 5 5 i,j=1 (f_i− f_j)2. (4) Deﬁne a Lyapunov function for (3) as follows

L(x1, . . . , x5) = 5 i,j=1 (x_i− x_j)2 5 i=1 xi 5− 5 i=1 xi , (x₁, . . . , x₅)∈ D₀. (5) Note that from the condition (x₁, . . . , x₅)∈ D₀, the functionL(x₁, . . . , x₅) is well deﬁned. Obviously, L is positive and continuous on D₀ and satisﬁes that L(x1, . . . , x5) = 0⇔ x_i =x_j,i, j = 1, 2, 3, 4, 5.

The following result is critical for this paper:

Theorem 2. Assume γ ∈ [γ_∞, 4). Then there exist

λ(γ) ∈ (0, 1) and small δ = δ(γ) > 0 indepen-dent of (x1, . . . , x5) ∈ D₀ such that for each c ∈ (2/5 − δ(γ), 2/5 + δ(γ)), it holds that

L(x1, . . . , x5)≤ λ(γ)L(x1, . . . , x5). (6)

Proof of Theorem 1. Theorem 2 shows that forγ ∈ [γ_∞, 4], there exist 0 < λ(γ) < 1 and δ = δ(γ) > 0 such that

L(x1(k), . . . , x₅(k)) ≤ λ(γ)k· L(x₁(0), . . . , x₅(0)) for c ∈ (2/5 − δ(γ), 2/5 + δ(γ)), (x₁(0), . . . , x5(0)) ∈ D₀ and k ∈ N. It implies L(x₁(k), . . . , x5(k)) → 0 as k → ∞. Thus |xi(k) − xj(k)| → 0

as k → ∞, i, j = 1, 2, 3, 4, 5. This completes the proof.

3. Proof of Theorem 2

In this section, we will prove Theorem 2.

In fact, we only need to prove Theorem 2 for the case c = 2/5. Then by continuation and the compactness of D₀, we can prove the existence of a small interval (2/5 − δ(γ), 2/5 + δ(γ)) such that the same conclusion holds true. Thus, without loss of generality, we assumec = 2/5.

Substituting (4) into (5), we have that for c = 2/5, L(x1, . . . , x5) = 5 i,j=1 (f_i− f_j)2 5 5 i=1 fi 5− 5 i=1 fi . (7)

In the following, to illustrate the idea of the proof, we ﬁrst prove the assertion for a special case:

max

1≤i,j≤5|xi− xj| < 0, (8)

where ₀ is a small positive number determined later. Then we will deal with the general case. Proof of the special case. From Eq. (8), we have

min

1≤i,j≤5|1 − xi− xj| ≤ max1≤i,j≤5|1 − xi− xj|

≤ min

1≤i≤5|1 − 2xi| + 20. (9)

Consequently, by the deﬁnition of f_i and f_j, it is easily seen that

(f_i− f_j)2=γ2(1− x_i− x_j)2(x_i− x_j)2 ≤ γ2· max 1≤i,j≤5|1 − xi− xj| ₂ (x_i− x_j)2 ≤ γ2· min 1≤i≤5|1 − 2xi| ₂ +c₁₀ × (xi− xj)2,

wherec₁ ≤ 4 is a constant independent of ₀. Similarly, we have 5 i=1 fi 5 ·        1− 5 i=1 fi 5        ≥ f(x)(1 − f(x)) + γ max1≤i≤5(xi− x)2 ≥ f(x)(1 − f(x)) − c320, where x = (x₁+x₂+x₃+x₄+x₅)/5 and c₃ ≤ γ are two constants independent of₀.

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Hence, from Eq. (7), we have L(x1, . . . , x5)≤ γ2_min 1≤i≤5|1 − 2xi| 2₊_c 10 5 i,j=1 (x_i− x_j)2 5 5 i=1 fi 5− 5 i=1 fi ≤ γ2_min 1≤i≤5|1 − 2xi| 2₊_c 10 5 i,j=1 (x_i− x_j)2 125(f(x)(1 − f(x)) − c₃2₀) = γ2_min 1≤i≤5|1 − 2xi| 2₊_c 10 5 i=1 xi· 5− 5 i=1 xi 125(f(x)(1 − f(x)) − c₃2₀) · L(x1, . . . , x5) ≤ γ2_min 1≤i≤5|1 − 2xi| 2₊_c 10 · max 1≤i,j≤5xi(1− xj) 5(f(x)(1 − f(x)) − c₃2₀) · L(x1, . . . , x5) ≤ max 1−γ/4≤x≤γ/4 γ2_{|1 − 2x|}2_{x(1 − x)} 5f(x)(1 − f(x)) · L(x1, . . . , x5) +c4(0)L(x1, . . . , x5), where c₄(₀) is a rational function of ₀ satisfying

c4(0)→ 0 as 0→ 0. Since max 1−γ/4≤x≤γ/4 γ2_{|1 − 2x|}2_{x(1 − x)} 5f(x)(1 − f(x)) = max 1−γ/4≤x≤γ/4 γ(1 − 2x)2 5(1− γx(1 − x)) ≤ γ 5, we obtain L(x1, . . . , x5)≤γ 5 +c4(0) L(x1, . . . , x5). Obviously, if ₀ is small enough, it holds that γ/5 + c4(₀)< 1. Hence, we complete the proof for the special case.

Proof of the general case. It is easily seen that L(x1, . . . , x5) L(x1, . . . , x5) = 5 i=1 xi 5− 5 i=1 xi · 5 i,j=1 (f_i− f_j)2 5 5 i=1 fi 5− 5 i=1 fi · 5 i,j=1 (x_i− x_j)2 . (10) Let F1 = 5 i=1 xi· 5− 5 i=1 xi , F2= 5 i,j=1 (f_i− f_j)2 5 , G1 = 5 i=1 fi· 5− 5 i=1 fi , G2 = 5 i,j=1 (x_i− x_j)2, and H(x1, . . . , x5) = _GF1· F2 1· G2 = F G.

In the following, we will analyze the maximum value of the function H on the domain D₀.

Obviously, we have ∂H

∂xi =

FxiG − F Gxi

G2 , i = 1, 2, 3, 4, 5.

Here F_x_i and G_x_i denote the partial derivatives of F and G with respect to x_i, respectively, i = 1, 2, 3, 4, 5.

Let∂H/∂x_i = 0, i = 1, 2, 3, 4, 5, then we have FxiG − F Gxi = 0, i = 1, . . . , 5. (11)

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By direct computation, we obtain F x1 = 1 5 5− 2 5 i=1 xi F2+ 2F1 5f₁− 5 i=1 fi f₁, G x1 = 5− 2 5 i=1 fi G2f1 + 2G1 5x₁− 5 i=1 xi , (12) F x2 = 1 5 5− 2 5 i=1 xi F2+ 2F₁ 5f₂− 5 i=1 fi f₂, G x2 = 5− 2 5 i=1 fi G2f2 + 2G1 5x₂− 5 i=1 xi , (13) F x3 = 1 5 5− 2 5 i=1 xi F2+ 2F1 5f₃− 5 i=1 fi f₃, G x3 = 5− 2 5 i=1 fi G2f3 + 2G1 5x₃− 5 i=1 xi , (14) F x4 = 1 5 5− 2 5 i=1 xi F2+ 2F₁ 5f₄− 5 i=1 fi f₄, G x4 = 5− 2 5 i=1 fi G2f4 + 2G1 5x₄− 5 i=1 xi , (15) F x5 = 1 5 5− 2 5 i=1 xi F2+ 2F₁ 5f₅− 5 i=1 fi f 5, G x5 = 5− 2 5 i=1 fi G2f5 + 2G1 5x₅− 5 i=1 xi . (16)

Lemma 3.1. For Eqs. (11), one of the following

equations must hold true: (i) x_i₁ =x_i₂,

(ii) x_i₁ =x_i₃, (iii) x_i₂ =x_i₃,

(iv) x_i₁ +x_i₂+x_i₃ = 3/2,

where i₁, i₂, i₃ = 1, 2, 3, 4, 5, i₁ = i₂, i₁ = i₃, i₂ = i₃.

Proof. From Eqs. (11)–(13), we have 2F₁G (5f₁f₁ − 5f₂f₂)− 5 i=1 fi· (f1 − f2) =F 5− 2 5 i=1 fi G2(f1− f2) + 10G1(x1− x2) , which is equivalent to 2F₁G 5γ2k(x₁, x₂) + 2γ 5 i=1 fi (x₁− x₂) =F −2γ 5− 2 5 i=1 fi G2+ 10G₁ (x₁− x₂), where k(x, y) = 1 − 3(x + y) + 2(x2 +xy + y2). Similarly, from (11)–(16), we have

2F₁G 5γ2k(x_j, x_k) + 2γ 5 i=1 fi (x_j− x_k) =F −2γ 5− 2 5 i=1 fi G2+ 10G₁ × (xj− xk), j, k = 1, 2, 3, 4, 5. (17)

Obviously, Eq. (17) implies that either (x_i₁− x_i₂)(x_i₁− x_i₃) = 0 or

k(xi1, xi2) =k(xi1, xi3).

If the former holds true, then we have proved this lemma. Otherwise, then the latter holds true, which is equivalent to 3 2 − xi1− xi2 − xi3 (x_i₂− x_i₃) = 0. It completes the proof.

Claim. If (11) holds true, then x₁, x₂, x₃, x₄, x₅ satisﬁes one of the following statements:

(i) x₁ =x₃ =x₄=x₅. (ii) x₁ =x₃ =x₄, x₂=x₅.

(iii) x₁ =x₄ =x₅, x₁+x₂+x₃= 3/2. (iv) x₁ =x₄, x₂ =x₅, x₁+x₂+x₃= 3/2.

Proof. Otherwise, without loss of generality, we assume that x₁, x₂, x₃, x₄ are diﬀerent from each

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other. Then from Lemma 3.1, we have x₁ +x₂ + x3 = 3/2 and x₁ +x₂ +x₄ = 3/2, which implies x3=x4. It contradicts with the assumption.

For case (i), we rewrite the function H as follows: H = F_G = γ 2₍₄_x 1+x2)(5− 4x1− x2)(1− x1− x2)2 (4f₁+f₂)(5− 4f₁− f₂) . It is easy to verify that

F_x₁ = 4 5(5− 2(4x1+x2))(1− x1− x2) 2 −2 5(4x1+x2)(5− 4x1− x2)(1− x1− x2), G x1 = 4(5− 2(4f1+f2))f1, F x2 = 1 5(5− 2(4x1+x2))(1− x1− x2) 2 −2 5(4x1+x2)(5− 4x1− x2)(1− x1− x2), G_x₂ = (5− 2(4f₁+f₂))f₂. Let F_x₁G = F G_x₁, F_x₂G = F G_G₂. Which implies (F_x₁ − 4F_x₂)G = F (G_x₁ − 4G_x₂). Since F_x₁− 4F_x₂ = 6 5(4x1+x2)(5− 4x1− x2)(1− x1− x2) and G x1 − 4Gx2 = 4(5− 2(4f1+f2))(f1 − f2), we obtain 3(4f₁+f₂)(5− 4f₁− f₂) =−4(5 − 2(4f₁+f₂))(f₁− f₂), which is equivalent to 16(f₁− f₁2) =f₂− f2₂. (18) However, since we assume that f is deﬁned on the interval [1− γ/4, γ/4], it is easily seen that

min 1−γ/4≤x≤γ/4(f(x) − f(x) 2₎_{≥ 0.925 × 0.075, and} max 1−γ/4≤x≤γ/4(f(x) − f(x) 2_{) =} 1 4.

It shows that Eq. (18) is impossible. Thus case (i) is excluded.

We can exclude cases (ii)–(iv) in the same way. Thus maximum values of H can be attained only on the boundary point set ofD₀, i.e.

xi= 1−γ₄ or γ₄, i ∈ {1, 2, 3, 4, 5}.

Now the situation for the boundary point set is sim-pler than the inner point set since there are less variables and we can deal with it in a similar way as above. Thus we omit the details. This completes the proof of Theorem 2.

Acknowledgment

The second author thanks the hospitality of the Center for Theoretical Science, Taiwan and Prof. Song-Sun Lin of National Chiao Tung University, Taiwan.

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