Chaotic synchronization in lattices of two-variable maps coupled with one variable

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IMA Journal of Applied Mathematics (2009) 74, 827−850 doi:10.1093/imamat/hxp028

Advance Access publication on October 27, 2009

Chaotic synchronization in lattices of two-variable maps coupled with

one variable

WEN-WEILIN†

Department of Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan, Republic of China

CHEN-CHANGPENG‡

Department of Applied Mathematics, National Chiayi University, Chiayi City 60004, Taiwan, Republic of China

AND

YI-QIANWANG§

Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China [Received on 4 October 2007; revised on 8 September 2009; accepted on 30 September 2009] In this paper, we study chaotic synchronization in 1D lattices of two-variable maps coupled with one variable. We give a rigourous proof for the occurrence of chaotic synchronization of spatially homo-geneous solutions in such coupled map lattices (CMLs) of lattice size n = 4 with suitable coupling coefficients. For the case of lattice size n> 4, we demonstrate numerical results of synchronized chaotic behaviour of the CMLs. Moreover, we show numerically that the difference between two variables man-ifests chaotic behaviour. This behaviour combined with the special coupling method in the CMLs guar-antees high security in applications using our new model.

Keywords: chaotic synchronization; coupled map lattices; Lyapunov method; hyperchaos. 1. Introduction

Secure communication faces more and more serious challenges. In recent years, decryption techniques have been developed very rapidly. For example, as an Internet standard, MD5 (message-digest algorithm 5) has been employed in a wide variety of security applications and is also commonly used to check the integrity of files.Wang & Yu(2005) demonstrated collision attacks against MD5, SHA-0 (SHA stands for secure hash algorithm) and other related hash functions. Later,Wang et al.(2005) found a method to find collisions in the SHA-1 hash function, which is used in many of today’s mainstream security products. Their attack is estimated to require far fewer operations than previously thought needed to find a collision in SHA-1. Although no attacks have yet been reported on the SHA-2 variants, which are algorithmically similar to SHA-1, a new hash function, to be known as SHA-3, is currently under development. It shows the necessity of developing alternative methods in secure communication.

With the combination of synchronization and unpredictability, chaotic synchronization has attracted a lot of attention since 1990 for its promising potential in secure communication. A secret message can

†_{Email: [email protected]} ‡_{Email: [email protected]}

§_{Corresponding author. Email: [email protected]}

c

at National Chiao Tung University Library on April 24, 2014

http://imamat.oxfordjournals.org/

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be modulated on the chaotic signal of a sender, and a receiver with an identical system which is driven by the modulated signal can decrypt this message. Many encryption models based on chaotic synchro-nization have been proposed (seePecora & Carroll,1990;Vohra et al.,1992;Cuomo & Oppenheim,

1992,1993;Wu & Chua,1994;Heagy et al.,1995;Pecora et al.,1997).

On the other hand, it has been pointed out that the proposed chaos-based communication systems have many flaws and need to be improved (seeP´erez & Cerdeira,1995; Yang et al.,1998; Short & Parker,1998;Zhou & Lai,1999;Li et al.,2005;Hu & Guo,2008). Prompted by these decryption meth-ods, many countermeasures have been developed to improve the security of communication systems based on chaotic synchronization. Although some of them have been shown to be insecure still, more and more complicated and effective countermeasures have been proposed. For example,Kanter et al.

(2008a) showed that for non-identical partners which use private commutative filters and can synchro-nize, it may be difficult for the attacker to synchronize and to reveal the time-dependent output signal of the parties. Another work ofKanter et al.(2008b) even maps the task of the attacker onto the nonde-terministic polynomial time-complete problems, for which all known denonde-terministic algorithms require running time that is exponential with some tunable parameters of the problem. Thus, it is computation-ally infeasible for an attacker to extract the message from the transmitted signal.

These works stimulated intensive research on communication with synchronized chaos which is still ongoing. For example, communication with chaos synchronization has recently been demonstrated with semiconductor lasers which were synchronized over a distance of 120 km in a public fiber network in Greece (seeArgyris et al.,2005).

In this paper, we consider chaotic synchronization 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 intermediate 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 difficulty with complex behaviour. On the other hand, the computer simulation is utilized as an effective and powerful tool to study dynamical systems with complex behaviour. In such a study, the dynamical system shall be discretized in space as well as time. This is one of the motivations to introduce new models of CMLs (seeAfraimovich & Bunimovich,1993;Bunimovich,1997;Bunimovich & Carlen,

1995;Giberti & Vernia,1994;Kaneko,1993).

The simplest type of chaotic synchronization of CMLs occurs in stable spatially homogeneous regimes corresponding to the existence of attractive spatially homogeneous solutions. In other words, in such cases, there is a large (open) set of initial conditions such that a solution starting from an initial condition in the set becomes spatially homogeneous as the discrete time k becomes very large, i.e. the coordinates of the individual maps become almost equal to each other (and the differences approach to 0 as k → ∞). In established regimes, individual maps become indistinguishable and we observe exact perfect synchronization. Recently, synchronization in a lattice of one-variable maps has been studied in

Lin et al.(1999),Lin & Wang(2002) andJost & Joy(2002). The model in these 1D CMLs is given by xi(k+ 1) = f (xi(k))+ c( f (xi₋₁(k))+ f (xi₊₁(k))− 2 f (xi(k))) (1.1)

for 1 6 i 6 n, with periodic boundary conditions f (x0(k)) = f (xn(k)) and f (xn₊₁(k))= f (x1(k)).

Here, f : [0, 1]→ [0, 1] is a 1D map. For instance, f is usually chosen to be the well-known logistic

map:

x(k+ 1) = f (x(k)) = γ x(k)(1 − x(k)), 0 < γ 6 4. (1.2)

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It is known that (Gleick,1987) the logistic map (1.2) has a chaotic attractor forγ ∈ (γ∞≈ 3.57, 4].Lin

et al.(1999) gave a rigourous proof for chaotic synchronization of (1.1) and (1.2) with n = 2, 3, 4 and

γ ∈ (γ∞, 3.82]∈ (γ∞, 4], provided the coupling coefficient c is sufficiently close to 1/3. The result

is generalized byLin & Wang (2002) for γ ∈ (γ∞, 4] by the Lyapunov function method. Lin et al.

(1999) also provided a complete numerical experiment for chaotic synchronization of 1D and 2D CMLs of (1.1) and (1.2) with various lattice sizes.Jost & Joy(2002) gave a necessary and sufficient condition for the occurrence of local synchronization as well as a sufficient condition for the occurrence of global synchronization of (1.1) with more general one-variable maps.

In the following, we propose a model on synchronization of discrete hyperchaotic systems:

         xi(k+ 1) = g(xi(k), yi(k))+ c(g(xi₋₁(k), yi₋₁(k)) + g(xi+1(k), yi+1(k))− 2g(xi(k), yi(k))), yi(k+ 1) = h(xi(k), yi(k)), for 16 i 6 n, (1.3)

with periodic boundary conditions (x0(k), y0(k)) = (xn(k), yn(k)) and (x1(k), y1(k)) = (xn₊₁(k),

yn₊₁(k)), where

(_g_{(x, y)}_{= f}

γ(x)+ θ( fδ(y)− fγ(x)),

h(x, y)= fδ(y)+ θ( fγ(x)− fδ(y))

(1.4)

defined on [0, 1]2 _{with 0} _{< θ < 1, 1 < δ, γ < 4 and δ} _{6= γ , in which f}

γ(x) = γ x(1 − x) and

fδ(y)= δy(1 − y) are the logistic maps.

In the CMLs of (1.3), we put two-variable maps of (1.4) on the i th node of a circle lattice, i = 1, . . . , n, and only couple the xi-variable with xi₋₁- and xi₊₁-variables of its two neighbours. In other

words, in the CMLs of (1.3), the yi-variable connects only with the xi-variable in the i th node, and the

coupling occurs only through the xi-variable with the nearest nodes. The topological structure of the

CMLs of (1.3) with lattice size n= 4 is shown in Fig.1.

In (1.4), we construct a two-variable map by connecting two logistic maps with the parameter

θ ∈ (0, 1). We shall prove that the two-variable system (1.4) is chaotic in the type of snap-back re-peller (Marotto,1978) for some suitable θ and show the fast fourier transformation (FFT) values of

the difference of x(k) and y(k) as k → ∞ which forms a chaotic behaviour. We shall also prove the

occurrence of chaotic synchronization of (1.3), i.e. lim

k_→∞(|xi(k)− xj(k)| + |yi(k)− yj(k)|) = 0, (1.5)

for i, j = 1, 2, . . . , n, with some suitable coupling strengths c and the lattice size n = 4.

It is worth pointing out that usually it is a difficult task to find an analytic proof for globally chaotic synchronization in CMLs. In fact, the study of an uncoupled discrete chaotic dynamical system itself is still a challenge to mathematicians. For example, one of the most important works of the Wolf prize winner Carleson is Benedicks & Carleson(1985), a partial result on the logistic map. Moreover, in CMLs, one cannot obtain synchronization by increasing the coupling strength, which is often the reason for the occurrence of synchronization in coupled continuous systems. Thus, the proof for the occurrence of global synchronization in CMLs seems more difficult. We note that by now most of the mathematical results in this area focus on the local stability of the synchronous manifold. Thus, from the point of view of mathematics, these results cannot predict whether or when synchronization will occur.

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FIG. 1. Topological structure of CMLs with the lattice size n= 4.

Here are some motivations for the study of chaotic synchronization of the CMLs (1.3) and (1.4). (a) In many applications, such as in secure communication, in contrast to the CMLs of (1.1) and

(1.2), the duplexing coupling of xi-variables in (1.3) induces the chaotic synchronization of

yi-variables which can be used to make a chaotic mask of message and send it out to the

neighbours via yi-variables. Then, the secret message can be decoded by the synchronization of

yi-variables. For example, when synchronization is obtained, Node 1 encodes the secret message

mkby y-variable to obtain the signal ˜mk = mk+ y1(k) and sends it to Node 2. Then, Node 2 can

recover the message easily by ˜mk− y2(k) since y1(k) and y2(k) are synchronized with each other.

On the other hand, since only x-variables of all nodes are transmitted to induce synchronization, an eavesdropper knows nothing about y-variables. Thus, he cannot recover the message. (b) The logistic map used in (1.4) is a well-studied simple model which has chaotic behaviour over

a wide range of parameters in(γ_∞, 4].

(c) In contrast to the other two-variable maps, such as theH´enon(1976) map, the differences of xi(k) and yi(k) in (1.3) form a chaotic behaviour. Thus, one channel (duplexingly coupled with

xi-variables) makes the CMLs of (1.3) synchronized and the other channel (simplexingly

con-nected with yi-variables) is used to realize secure communication. On the contrary, the H´enon

map has the relation xi(k+ 1) = yi(k) which cannot be used in secure communication because

the values of yi(k) can be encoded by the duplexing coupling of xi-variables.

In practice, we have further measures against general attacks. For example, a variational logistic map (VLM) has been proposed (Chen et al.,2008) with a large parameter space without windows. The VLM with a disturbing method can pass the most stringent statistical testing suite in TestU01. With up to 3200 Mbps throughput and complex output properties, VLM is suitable for security applications. A chaotic cryptographical scheme (Schneier,1996), constructed by coupling four VLMs, generates the output sequence with a minimal length equal to 2128 by a 128-bit external key.

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Besides secure communication, chaotic synchronization has the same importance in biology and life science, which is another reason why we focus on this area. For example, people found that fireflies are able to synchronize the timing of their light emission within a flashing population by adjusting the frequency and phase of their own flashing (see Mirollo & Strogatz,1990). For fireflies, this kind of capability plays a critical role in the processing of mating. People believe that modelling networks after such biological systems may potentially be more efficient than current networking schemes allow. In the last decade, many people have pointed out that synchronization among large groups of neurons is a fundamental mechanism that allows us to understand how the brain solves the binding problem. For instance, Parkinsonian tremor and epileptic seizures are believed to be caused by such a mechanism (seeGray,1999;Haken,2002;Singer,1999a,b;Tass et al.,1998). Recently, Kaneko and his coauthors obtained a lot of results in studying a series of biology-related problems with chaotic synchronization theory, such as the origin of heredity, cell differentiation, universal features of a cell with recursive growth, stability and irreversibility in the development of cell societies, pattern formation and the origins of positional information and multicellular organisms, etc. (seeFurusawa & Kaneko,2000,2001,2003;

Kaneko & Yomo,1997,2002).

This paper is organized as follows.Marotto(1978) introduced the ‘snap-back repeller’ of a differen-tiable map and proved that the existence of a snap-back repeller is sufficient to imply chaotic behaviour of the map. In Section2, based on the theorem ofMarotto(1978) and a generalized version (Shiraiwa & Kurata,1979), we give a rigourous proof for the chaotic behaviour of the CMLs (1.3) and (1.4) for 3.678 < γ ≈ δ < 4. In Section3, we prove that the system (1.3) and (1.4) is synchronized, i.e. the conditions in (1.5) hold or a spatially homogenous solution of (1.3) exists for n = 4, c ∈ (0.41, 0.43),

θ ∈ (0.62, 0.64) and γ ≈ δ ∈ (3.7 − , 3.7 + ) with 0 < << 1. In Section4, we show numerical results for the chaotic synchronized behaviour of (1.3) and (1.4).

2. Chaos for the two-variable map

In this section, we shall prove the chaotic behaviour for a two-variable map of (1.4). The proof is based on a theorem ofMarotto(1978) and a generalized version inShiraiwa & Kurata(1979).

DEFINITION2.1 (Marotto) Let F :RN → RNbe aC1-map. Let z∗be a fixed point of F such that all the eigenvalues of_DF(z∗) have absolute values larger than 1. Then, z∗is called a snap-back repeller if there exists a point z0in W_locu (z∗), the local unstable set of z∗, and some integer m such that Fm(z0)= z∗and

det_DFm(z0)6= 0.

THEOREM2.1 (Marotto) Let F :RN → RNbe aC1-map. Let z∗be a snap-back repeller of F . Then, the following holds:

(i) There is a positive integer p0such that for each p> p0, F has a point of period p.

(ii) There is an uncountable set S⊂ RN containing no periodic points of F such that (iia) F(S)⊂ S;

(iib) for everyξ, η∈ S with ξ 6= η,

lim sup

k→∞ |F

k_{(ξ )}_{− F}k_(η)_{| > 0;}

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(iic) for everyξ ∈ S and any periodic point η of F,

lim sup

k_→∞ |F

k_{(ξ )}_{− F}k_(η)_{| > 0;}

(iii) There is an uncountable subset S0⊂ S such that for every ξ, η ∈ S0,

lim inf

k→∞ |F

k_{(ξ )}_{− F}k_(η)_{| = 0.}

Conditions (i)–(iii) were first defined as ‘chaos’ of a one-variable map and proved as necessary conditions of a ‘period-3’ map byLi & Yorke(1975).

Note that the original proof ofMarotto (1978) has some logical error which has been corrected recently byChen et al.(1998).

REMARK Shiraiwa & Kurata (1979) proved that conditions (i)–(iii) in Theorem2.1hold by modifying

the assumption as follows:

‘Let z∗∈ RNbe a hyperbolic fixed point of F such that

(1) there exists a point z1 ∈ W_locu (z∗) (z1 6= z∗) and a positive integer m such that Fm(z1) ∈

W_locs (z∗);

(2) there exists a u-dimensional disk Buembedded in W_locu (z∗) such that Bu_{is a neighbourhood of}

z1in Wlocu (z∗), Fm|Bu: Bu→ RNis an embedding and Fm(Bu) intersects Wlocs (z∗) transversely

at Fm(z1), where u= dim Wu

loc(z∗) > 0’.

In case u = dim RN and fm(z1)= z∗, the above assumptions reduce to the snap-back repeller of the original Marotto’s theorem.

In the following, we use the generalized version ofShiraiwa & Kurata(1979) to prove the existence of chaotic behaviour of (1.4).

2.1 Two-variable map connected with logistic maps

Consider a special case of a two-variable map connected with logistic maps as in (1.4):

F(x, y)= (1− θ) fγ(x)+ θ fγ(y) (1− θ) fγ(y)+ θ fγ(x)

!

(2.1)

withγ = δ.

We give an elementary stability analysis of fixed points of (2.1), which is useful in Section2.2to determine if a snap-back repeller exists for a two-variable map (1.4).

LEMMA2.1 In the invariant region [0, 1]× [0, 1], the fixed point γ_γ−1,γ_γ−1of (2.1) exists for all 16 γ 6 4, 0 6 θ 6 1.

Proof. Obvious.

LEMMA2.2

(i) Ifγ < 3, then the fixed point γ−1_γ ,γ−1_γ is a stable point.

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(ii) Ifγ > 3 and 0 < θ < ₂_(γγ−3₋₂₎ or 1> θ >₂_(γγ−1₋₂₎, then the fixed point γ−1_γ ,γ_γ−1is a repelling fixed point.

(iii) Ifγ > 3 and ₂_(γγ−1₋₂₎ > θ > ₂_(γγ−3₋₂₎, then the fixed point γ−1_γ ,γ−1_γ is a saddle fixed point of the map F as in (2.1).

Proof. The Jacobian matrix at γ−1_γ ,γ−1_γ of (2.1) is

J= DF = " (θ− 1)(−2 + γ ) −θ(−2 + γ ) −θ(−2 + γ ) (θ− 1)(−2 + γ ) # .

The eigenvaluesλ1andλ2of J can be computed by

λ1= 2 − γ, λ2= (−1 + 2θ)(−2 + γ ).

Therefore, we have the following:

(i) |λ1| < 1, |λ2| < 1 for γ < 3, i.e. the fixed point γ−1_γ ,γ−1_γ is a stable point.

(ii) |λ1| > 1, |λ2| > 1 for γ > 3 and θ > ₂_(γγ−1₋₂₎ orθ < ₂_(γγ−3₋₂₎, i.e. the fixed point γ−1_γ ,γ−1_γ is a repelling fixed point.

(iii) |λ1| > 1, |λ2| < 1 for γ > 3 and ₂_(γγ−1₋₂₎ > θ > ₂_(γγ−3₋₂₎, i.e. the fixed point γ−1_γ ,γ−1_γ is a saddle fixed point.

2.2 Snap-back repeller of two-variable maps

In this section, we shall prove the existence of a snap-back repeller of (1.4).

We first prove that the fixed point x∗ = γ−1_γ of the logistic map is a snap-back repeller forγ > γ∗≈

3.678. Letξ = fγ(x)= γ x(1 − x). Then, x=γ ± p γ2_{− 4γ ξ} 2γ .

We choose pre-images of the fixed point x∗from backward orbits (if they exist) by the following ‘best’ way: x₋₁=γ − p γ2_{− 4γ x}_∗ 2γ ∈ f −1_(x∗_), x_{−( j+1)}= γ + q γ2_{− 4γ x} − j 2γ ∈ f−1(x− j), for j= 1, 2, . . . . (2.2)

REMARK The above way for choosing pre-images of x∗ is the best in the sense that if we choose

˜x−1= γ− √ γ2_{−4γ x}∗ 2γ , ˜x−2= γ−√γ2_{−4γ ˜x−1} 2γ ,˜x−( j+1)= γ₊√γ2_{−4γ ˜x− j}

2γ , for j = 2, 3, . . . , then it is easy

to show that ˜x₋₃ > x₋₂. Since a point x ∈ (0, 1) has no pre-image if and only if x ∈ (γ /4, 1), it is easily seen that if x_−k, chosen by the best way, does not exist for some k, then ˜x_−(k+1)does not exist either.

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From the best way for the choice of{x_{− j}}, we have x_{− j} > 1₂ as j > 2 (if they exist). Moreover, x₋₁ = _γ1 and x₋₂ = γ+

√

γ2₋₄

2γ always exist. However, x− j may not exist as j > 3. The following

lemma is a criteria for the existence of x₋₃. LEMMA2.3 Ifγ > 3.678, then x₋₂=γ+ √ γ2_{−4γ x−1} 2γ < γ 4, where x−1= 1 γ.

Proof. It is easily seen that x₋₂ < γ₄ if and only if f(x₋₂) > f ₄γ, i.e. x₋₁ = f (x₋₂) > f γ₄, which is equivalent to γ−1_γ > f2 γ 4 = f3 1 2 . By direct computation, γ_γ−1 > f3 1 2 is equivalent to γ − 1 − γ f3 1 2 > 0. Denote Γ (γ )= γ − 1 − γ3_f 1 2 . Then, Γ (γ )= γ − 1 − γ3_f ₁ 2 = γ − 1 −γ 4 4 1−γ 4 " 1−γ 2 4 1−γ 4 # = γ − 1 −1 4γ 4₊ 1 16γ 6₋ 1 32γ 5₊ 1 16γ 5₊ 1 256γ 8 = 1 256(γ 8_{− 8γ}7_{+ 16γ}6_{+ 16γ}5_{− 64γ}4_{+ 256γ − 256)} = 1 256(γ + 2)(γ 3_{− 2γ}2_{− 4γ − 8)(γ − 2)}4_.

Since γ ∈ [0, 4], we have Γ (γ ) > 0 if and only if γ3_{− 2γ}2_{− 4γ − 8 > 0 and γ 6= 2. Denote} Γ1(γ ) = γ3_{− 2γ}2_{− 4γ − 8. Then, Γ}0 1(γ ) = 3γ2− 4γ − 4 = 0 implies that γ = 2 or γ = − 2 3. Obviously,Γ₁00(2) > 0 and Γ₁00 − 2 3

< 0. Since Γ1(0)= −8, Γ1(2) = −16 and Γ1(4) > 0, by the

intermediate value theorem, there exists aγ∗∈ (2, 4) such that Γ1(γ∗₎_{= 0. By numerical computation,}

we haveγ∗≈ 3.678. So Γ1(γ ) > 0, for γ ∈ (γ∗_{, 4] and Γ1(γ )}_{6 0, for γ ∈ [0, γ}∗_]_.

THEOREM2.2 Ifγ > 3.678, then x∗=γ−1

γ is a snap-back repeller of the logistic map fγ(x).

Proof. We prove that x∗satisfies the conditions as in Definition2.1. (i) x∗is a fixed point of fγ, i.e.| fγ0(x∗)| > 1.

(ii) For all > 0, there exists a ξ ∈ B(x∗, ε) such that fm

γ (ξ )= x∗for some m.

(iii) |( f_γm(x∗))0| 6= 0.

Condition (i) is easy to check. To prove (ii) and (iii), we perform the following six steps. Step 1: Since f_γ−1(x∗)=1

γ, x∗

, from the best way we choose, we choose x₋₁= 1

γ < γ−1

γ = x∗.

Step 2: Since f_γ−1(x₋₁)= {x−2, 1− x−2}, where x−2 > 12 and fγ is strictly increasing on [0, 1/2]

with fγ 0,1₂=0,γ₄, there exists a ξ∗∈0,_γ1such that fγ(ξ∗)= _γ1 = x−1. It is easily seen that

ξ∗= 1 − x₋₂and 0< 1− x₋₂<_γ1. This implies that x∗< x₋₂< 1. Step 3: Since x₋₃=γ+

√

γ2_{−4γ x−2}

2γ and fγ0(x) < 0 for x ∈ [x∗, 1], by Lemma2.2, we have 12< x−3<

x∗.

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Step 4: Since f_γ−1(x₋₃)= {x−4, 1− x−4}, fγ−1 12 ∩0,1 2 < 1− x−4< 1r and fγ−1 γ1 ∩0,1 2 < f_γ−1 1₂∩0,1 2 , we have x₋₂> x₋₄> x∗.

Step 5: Suppose x₋₃ > x₋₅ > x∗. Then, x₋₂ = fγ(x−3) 6 fγ(x−5) 6 fγ(x∗) = x∗, which

contradicts that x₋₂> x∗. So 1₂ < x₋₃< x₋₅.

Step 6: Since f_γ−1(x₋₅)= {x₋₆, 1− x₋₆} and fγ−1(x−5)∩

0,1₂< 1− x₋₆ < 1_r, we have x₋₄ >

x₋₆> x∗.

According to the above steps, we have

x₋₂> x₋₄> x₋₆>∙ ∙ ∙ > x∗,

x₋₁> x₋₃< x₋₅<∙ ∙ ∙ < x∗.

It is easily shown that limn→∞x−2n = x∗and limn→∞x−(2n−1) = x∗. For any > 0, there exists a

ξ ∈ B(x∗, ε) such that fm

γ (ξ )= x∗for some m, thus (ii) holds. Since fγ0(x)= 0 if and only if x = 12,

condition (iii) is satisfied.

In the following, we shall prove that the two-variable map (2.1) has a snap-back repeller. THEOREM2.3 Ifγ > 3.678, then γ−1_γ ,γ−1_γ is a snap-back repeller of the two-variable map (2.1). Proof. We prove that γ−1_γ ,γ_γ−1satisfies (i)–(iii) as in Theorem 1. From Lemma2.2, we know that forγ > 3.678, u= dim Wu

loc

γ₋₁

γ ,

γ₋₁

γ > 1, so condition (i) is satisfied. Obviously, F γ₋₁ γ , γ₋₁ γ = _f γ γ−1_γ fγ γ−1_γ

. In Theorem 2.5, we proved that the fixed point x∗=γ−1_γ of fγ(x) is a snap-back repeller.

So for any > 0, there exists a ξ ∈ B(x∗_{, /2) such that f}m

γ (ξ )= γ−1γ = x∗for some m. Therefore,

(ξ, ξ ) ∈ B γ−1 γ , γ−1 γ , such that_Fm(ξ, ξ ) = γ−1 γ γ−1 γ

. Hence, condition (ii) is satisfied. From Lemma2.3, it follows that W_locu (x∗, x∗) and Ws

loc(x∗, x∗) are small deformations of the manifold of {(x, y)| x = y} and the manifold of {(x, y)|x + y = 1}, respectively, on a neighbourhood of (x∗_{, x}∗_{). It}

is easily seen that condition (iii) is satisfied. Hence, we complete the proof. Now, we consider (1.4) withγ 6= δ. Rewrite (1.4) as

˜F((x, y), γ, δ) = (1− θ) fγ(x)+ θ fδ(y)

(1− θ) fδ(y)+ θ fγ(x)

!

, (2.3)

where fγ(x)= γ x(1 − x) and fδ(y)= δy(1 − y).

THEOREM2.4 Ifγ ≈ δ > 3.678, then the fixed point ‘near’ γ−1γ , γ₋₁

γ

is a snap-back repeller of (2.3).

Proof. We shall check that the three conditions (i)–(iii) as in Theorem2.2hold. Ifγ = δ > 3.678,

then from Theorem 2.6, we proved that the fixed point γ−1_γ ,γ−1_γ = (x∗_{, x}∗_{) is a snap-back repeller}

of (2.3). Therefore, whenγ = δ, we have the following:

(a) _{D ˜F(x}∗, x∗, γ , γ ) has an eigenvalue with absolute value larger than 1, i.e. dim Wu

loc(x∗, x∗)> 1.

(b) For any > 0, there exists a point (ξ₁∗, ξ₂∗)∈ Bu

loc(x∗, x∗, ) such that ˜F

m_(ξ_∗

1, ξ2∗)= (x∗, x∗)

for some m.

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(c) det_{D ˜F}m((ξ₁∗, ξ₂∗), γ , γ )6= 0.

For a fixedγ , let δ= γ + η and we define Z((x, y), η) by

Z((x, y), η)= ˜F((x, y), η) − (x, y) = ˜F((x, y), γ, γ + η) − (x, y). (2.4) It is easy to check that

Z ∈ C1, Z((x∗, x∗), 0)= ˜F((x∗, x∗), 0)− (x∗, x∗)= (0, 0) (2.5) and the matrix

D(x,y)Z((x, y), η)= D(x,y) ˜F((x, y), η) − I2

is invertible at(x, y)= (x∗, x∗) and η= 0.

By the implicit function theorem and (a), for sufficiently small η, there exists a q1 > 0 and a

function ζ∗ on(−q1, q1) such that Z (ζ∗(η), η) = 0 for η ∈ (−q1, q1), i.e. ˜F(ζ∗(η), η) = ζ∗(η)

withζ∗(0)= (x∗, x∗) andD(x,y)Z(ζ∗(η), η) has a positive eigenvalue, i.e.D(x,y) ˜F(ζ∗(η), η) has an

eigenvalue with absolute value larger than 1 for η ∈ (−q1, q1). Thus, condition (i) holds. Next, we

define a function W((x, y), η) by

W((x, y), η)= ˜Fm_{((x, y), η)}_{− ζ}_∗_(η). _(2.6)

From (b), we have

W ∈ C1, W((x∗, x∗), 0)= ˜Fm_((x∗_{, x}∗_{), 0)}_{− ξ}∗₍₀₎_{= (0, 0)}

and

D(x,y)W((x∗, x∗), 0)= D(x,y)˜Fm((x∗, x∗), 0) invertible.

By the implicit function theorem, there exists a q2with 0 < q2 < q1and a functionω defined

on (−q2, q2) such that ω ∈ C1 _with_ω(0) _{= (x}_∗_{, x}_∗_{) and W (ω(η), η)} _{= 0 for η ∈ (−q2, q2), i.e.} ˜Fm_{(ω(η), η)} _{= ζ}∗_{(η). Thus, condition (ii) is satisfied. Since ˜}_Fm_and_ω _{∈ C}1_{, from (c), there is a q}

3

with 0 < q3 < q2such that det_D(x,y)˜Fm(ζ∗(η)) 6= 0 and ω(η) ∈ (−q1, q1), for all η ∈ (−q3, q3).

If we chooseη < q3, then condition (iii) is satisfied. We complete the proof. 3. Synchronization for 1D CMLs of two-variable maps coupled with one variable

In this section, we shall prove that the chaotic synchronization occurs for 1D CMLs of two-variable maps coupled with one variable as in (1.3) and (1.4) with n= 4.

First, we state a proposition and a lemma.

PROPOSITION 3.1 For c ∈ [0, 1], d ∈ [0, 1/2] and every (x1(0), y1(0), x2(0), y2(0), x3(0), y3(0), x4(0), y4(0))∈ (0, 1)8_{, there exists a k}

0∈ N such that for all k> k0,

(x1(k), y1(k), x2(k), y2(k), x3(k), y3(k), x4(k), y4(k))

generated by (1.3) and (1.4) lie in

D0= [(4 − max(γ, δ))/4, max(γ, δ)/4]8.

Proof. The proof is similar to Theorem2.3inLin et al.(1999).

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From this proposition, we w.l.o.g. assume that (1.3) and (1.4) are defined on D0.

LEMMA3.1 Consider the map

Xi(k+ 1) = fi(X (k), Ci) (3.1)

with i = 1, 2, . . . , N, where Ci are parameters, X(k) = (X1(k), . . . , XN(k)) with Xi(k)∈ RM and

the differential function vector f = ( f1, . . . , fN) possess the proposition: if X1 = ∙ ∙ ∙ = XN and

C1= ∙ ∙ ∙ = CN, then fi = fj, i, j = 1, 2, . . . , N. Then, for any l ∈ N, X(k) = (X1(k), . . . , XN(k))

and C = (C1, . . . , CN), there exists a cl0> 0 such that it holds that

|Xi(k+ l) − Xj(k+ l)| 6 cl0(|Xi(k)− Xj(k)| + |Ci− Cj|).

Proof. Since fi(X1, . . . , X1, Ci)= fi₊₁(X1, . . . , X1, Ci), we have

|Xi(k+ 1) − Xj(k+ 1)|

= | fi(X (k), Ci)− fi(X1, . . . , X1, Ci)+ fj(X1, . . . , X1, Ci)− fj(X (k), Cj)|

6 | fi(X (k), Ci)− fi(X1, . . . , X1, Ci)| + | fj(X1, . . . , X1, Ci)− fj(X (k), Cj)|,

which implies this lemma for l= 1. The case for l > 2 is similar. Define the set Nη,η0 to be the subset of D0which satisfies the following:

(i) |xi − xj| + |yi − yj| < η2for|i − j| is even,

(ii) |xi − xj| + |yi − yj| < η for |i − j| is odd and

(iii) |xi − yj| < η0for 16 i, j 6 4,

where i, j = 1, 2, 3, 4.

THEOREM3.1 Assumeγ , δ∈ (3.7 − , 3.7 + ) with > 0, the connected parameter θ ∈ (0.62, 0.64)

and the coupling coefficient c ∈ (0.41, 0.43), respectively, where > 0. Then, there exist an 0 > 0

such that for any 0< < 0, the spatially homogeneous chaotic solutions for (1.3) and (1.4) with n= 4 are stable, i.e. there exist anη0> 0 and an η0₀> 0 such that for any initial points in Nη0,η00, it holds that

lim

k_→∞|xi(k)− xj(k)| = 0, klim_→∞|yi(k)− yj(k)| = 0, i, j = 1, 2, 3, 4. (3.2)

REMARK

(i) Here, the spatially homogeneous solution is of the form

{(x1, y1, x2, y2, x3, y3, x4, y4)∈ D0| xi = xj, yi = yj, i, j = 1, 2, 3, 4}.

(ii) From0 1, we have δ ≈ γ . For this case, almost synchronization can occur between x-variables

and y-variables, i.e.|xi(k)− yj(k)| 1 for k large enough. However, perfect synchronization

never occurs between them ifδ6= γ . In fact, numerical results in Section4show that|xi(k)−yj(k)|

is chaotic, which shows that almost synchronization between x-variables and y-variables does not influence the security of the CMLs in (1.3) and (1.4).

The proof of Theorem3.1can be reduced to the following theorem.

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THEOREM3.2 There exist K0 ∈ N, 1 > 0, η1 > 0 and η0₁ > 0 such that for every 0 6 < 1,

06 η < η1and 06 η0< η0₁, we haveΦK0_(N

η,η0)∈ Nη/2,η0, whereΦ is the map defined by (1.3) and

(1.4).

Proof of Theorem 3.1. Define K0,η1andη₁0 as in Theorem3.2. From (1.3) and (1.4), we have

|xi(k+ 1) − xj(k+ 1)| + |yi(k+ 1) − yj(k+ 1)| 6 ci, j(|xi(k)− xj(k)| + |yi(k)− yj(k)|)

for|i − j| is even and

|xi(k+ 1) − xj(k+ 1)| + |yi(k+ 1) − yj(k+ 1)| 6 ci, j

4 X

i, j₌₁

(|xi(k)− xj(k)| + |yi(k)− yj(k)|),

where ci, jare constants independent of k.

From Lemma3.1, for the mapΦ, we obtain that there exist an η2> 0 and an η0₂> 0 such that for

any 0< η < η2, 0< η0 < η0₂and k6 K0, there exists a c8> 0 independent of η and η0such that for

any initial point(x1(0), y1(0), . . . , x4(0), y4(0)) in Nη,η0, it holds that

|xi(k)− xj(k)| 6 c112 η2, for|i − j| even,

|xi(k)− xj(k)| 6 c8η, for|i − j| odd. (3.3)

Setη0= max(η1, η2) and η0₀= max(η0₁, η0₂). From Theorem3.2, we haveΦ(K0)_(N

η,η0)⊂ Nη/2,η0

for 06 η < η0and 06 η0 < η0₀. By iteration, we haveΦ(l K0)(Nη,η0)⊂ Nη 2l,η0

. Hence, for any given  > 0, there exists an l0 ∈ N such that if l > l0, thenΦ(l K0)_(N_η,η0₎⊂ N_/c

8,η0. Combining this with

(3.3), we have

|xi(k)− xj(k)| 6 2, for|i − j| even,

|xi(k)− xj(k)| 6 , for|i − j| odd, (3.4)

for every i> l0K0. This completes the proof of Theorem 3.1.

The remaining part of this section is devoted to the proof of Theorem 3.2. Following the idea inLin & Wang(2002), we shall use the Lyapunov method to show the synchronization for 1D CMLs. Due to the complicated topological structure of (1.3) and (1.4), the construction of the appropriate Lyapunov function is much more complicated than that ofLin & Wang(2002), and thus, we can only obtain the local synchronization of (1.3) and (1.4).

By direct computation, we have

x1(k+ 1) − x3(k+ 1) = (1 − 2c)(g(x1(k), y1(k))− g(x3(k), y3(k)))

= (1 − 2c)(1 − θ)γ (1 − (x1(k)+ x3(k)))(x1(k)− x3(k)) +(1 − 2c)θδ(1 − (y1(k)+ y3(k)))(y1(k)− y3(k))

and

y1(k+ 1) − y3(k+ 1) = h(x1(k), y1(k))− h(x3(k), y3(k))

= (1 − θ)δ(1 − (y1(k)+ y3(k)))(y1(k)− y3(k)) +θγ (1 − (x1(k)+ x3(k)))(x1(k)− x3(k)).

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From|γ − δ| < 2, we obtain that x1(k+ 2) − x3(k+ 2)

= (1 − 2c)2₍₁_{− θ)}2_γ2₍₁_{− (x1(k}_{+ 1) + x3(k}_{+ 1)))(1 − (x1(k)}_{+ x3(k)))(x1(k)}_{− x3(k))} +(1−2c)2_{θ (1}_{− θ)γ}2₍₁_{− (x1(k}_{+ 1) + x3(k}_{+ 1)))(1−(y1(k}_{+ 1)−y3(k}_{+ 1)))(y1(k)}_{− y3(k))} +(1 − 2c)θ(1 − θ)γ2₍₁_{− (y1(k}_{+ 1) − y3(k}_{+ 1)))(1 − (y1(k)}_{+ y3(k)))(y1(k)}_{− y3(k))} +(1 − 2c)θ2_γ2₍₁_−(y1(k_{+ 1) −y3(k}_{+ 1)))(1−(x1(k)}_{+ x3(k)))(x1(k)}_{− x3(k))}_+c1_dist_13(k),

where dist13(k)= |(x1(k)− x3(k))| + |(y1(k)− y3(k))| and c1is a constant independent of. In the

last inequality, we use the fact that|γ − δ| < 2. Similarly, we have y1(k+ 2) − y3(k+ 2)

= (1 − θ)2_δ2₍₁_{− (y1(k}_{+ 1) + y3(k}_{+ 1)))(1 − (y1(k)}_{+ y1(k)))(y1(k)}_{− y3(k))} +(1 − θ)θγ2₍₁_{− (y1(k}_{+ 1) + y3(k}_{+ 1)))(1 − (x1(k)}_{+ x3(k)))(x1(k)}_{− x3(k))}

+(1 − 2c)(1 − θ)θγ2₍₁_{− (x1(k}_{+ 1) + x3(k}_{+ 1)))(1 − (x1(k)}_{+ x3(k)))(x1(k)}_{− x3(k))} +(1 − 2c)θ2_δ2₍₁_{− (x1(k}_{+ 1) + x3(k}_{+ 1)))(1 − (y1(k)}_{+ y3(k)))(y1}_{− y3)}_{+ c2}_dist_13(k),

where c2is a constant independent of.

After direct computation by using the definition of N_η,η0, we have

x1(k+ 3) − x3(k+ 3) = A(x1(k)− x3(k))+ B(y1(k)− y3(k))+ c3(+ η + η0)dist13(k),

y1(k+ 3) − y3(k+ 3) = C(x1(k)− x3(k))+ D(y1(k)− y3(k))+ c4(+ η + η0)dist13(k) (3.5) with dist13(k) = |x1(k)− x3(k)| + |y1(k)− y3(k)|, where c3and c4are constants independent of, η and η0and A, B, C and D are of the form:

A= mAγ3(1− (x1(k+ 2) + x3(k+ 2)))(1 − (x1(k+ 1) + x3(k+ 1)))(1 − (x1(k)+ x3(k))),

B= mBγ3(1− (y1(k+ 2) + y3(k+ 2)))(1 − (y1(k+ 1) + y3(k+ 1)))(1 − (y1(k)+ y3(k))),

C= mCγ3(1− (x1(k+ 2) + x3(k+ 2)))(1 − (x1(k+ 1) + x3(k+ 1)))(1 − (x1(k)+ x3(k))),

D= mDγ3(1− (y1(k+ 2) + y3(k+ 2)))(1 − (y1(k+ 1) + y3(k+ 1)))(1 − (y1(k)+ y3(k))), (3.6)

in which

mA= (1 − 2c)(1 − θ)[(1 − 2c)2(1− θ)2+ 2(1 − 2c)θ2+ θ2],

mB= (1 − 2c)3(1− θ)2θ+ (1 − 2c)2(1− θ)2θ+ (1 − 2c)θ(1 − θ)2+ (1 − 2c)2θ3,

mC= θ(1 − θ)2+ θ(1 − θ)2(1− 2c) + θ(1 − θ)2(1− 2c)2+ θ3(1− 2c),

mD= (1 − θ)3+ 2(1 − θ)θ2(1− 2c) + (1 − θ)θ2(1− 2c)2.

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For any pointζ (k) = (x1(k), y1(k), x2(k), y2(k), x3(k), y3(k), x4(k), y4(k)) ∈ Nη,η0, we define a

Lyapunov function L13as follows:

L13(ζ (k))= √ |x1(k)− x3(k)|

(x1(k)+ x3(k))(2− (x1(k)+ x3(k)))+

|y1(k)− y3(k)| √

(y1(k)+ y3(k))(2− (y1(k)+ y3(k))).

(3.7) Substituting (3.5) into (3.7), we obtain that

L13(ζ (k+ 3)) = √ |x1(k+ 3) − x3(k+ 3)|

(x1(k+ 3) + x3(k+ 3))(2 − (x1(k+ 3) + x3(k+ 3))) +√ |y1(k+ 3) − y3(k+ 3)|

(y1(k+ 3) + y3(k+ 3))(2 − (y1(k+ 3) + y3(k+ 3)))

6 |(mA+ mC)γ3(1− (x1(k+ 2) + x3(k+ 2)))(1 − (x1(k+ 1) + x3(k+ 1)))(1 − (x1(k)+ x3(k)))|

√

(x1(k+ 3) + x3(k+ 3))(2 − (x1(k+ 3) + x3(k+ 3))) ×|x1(k)− x3(k)|

+|(mB+ mD)γ3(1−(y1(k+ 2) + y3(k+ 2)))(1 − (y1(k+ 1) + y3(k+ 1)))(1−(y1(k)+ y3(k)))|

√

(y1(k+ 3) + y3(k+ 3))(2 − (y1(k+ 3) + y3(k+ 3))) ×|y1(k)− y3(k)|

6 (mA+ mC)γ3(1− 2x1(k+ 2))(1 − 2x1(k+ 1))(1 − 2x1(k))

√

2x1(k+ 3)(2 − 2x1(k+ 3)) |x1(k)− x3(k)|

+(mB+ mD)γ3(1− 2y1(k+ 2))(1 − 2y1(k+ 1))(1 − 2y1(k))

√

2y1(k+ 3)(2 − 2y1(k+ 3)) |y1(k)− y3(k)| +c5(+ η + η0)dist13(k) 6 (mA+ mC)γ3/2(1− 2x1(k))(1− 2 fγ(x1(k)))(1− 2 fγ( fγ(x1(k)))) p (1− fγ(x1(k)))(1− fγ( fγ(x1(k))))(1− fγ( fγ( fγ(x1(k))))) ×√ |x1(k)− x3(k)| (x1(k)+ x3(k))(2− x1(k)− x3(k))

+(mB+ mD)γ3/2(1− 2y1(k))(1− 2 fγ(y1(k)))(1− 2 fγ( fγ(y1(k))))

p

(1− fγ(y1(k)))(1− fγ( fγ(y1(k))))(1− fγ( fγ( fγ(y1(k)))))

×√ |y1(k)− y3(k)|

(y1(k)+ y3(k))(2− y1(k)− y3(k))+ c5(+ η + η

0_)dist13(k),

where c5is a constant independent of, η and η0. In the last inequality, we use the definition of Nη,η0.

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LEMMA3.2 There exist an2> 0, an η2> 0 and an η₂0 > 0 such that for any 0 < < 2, 06 η < η2,

06 η0< η₂0 and anyζ (k)∈ Nη,η0, there exists aλ∈ (0, 1) such that

L13(ζ (k+ 3)) 6 λ ∙ L13(ζ (k)).

Proof. We first estimate the following two numbers:

λ1= (mA+ mC)γ3/2 max 1−γ /46x6γ/4 (1− 2x)(1 − 2 fγ(x))(1− 2 fγ( fγ(x))) p (1− fγ(x))(1− fγ( fγ(x)))(1− fγ( fγ( fγ(x)))) , λ2= (mB+ mD)γ3/2 max 1−γ /46y6γ/4

(1− 2y)(1 − 2 fγ(y))(1− 2 fγ( fγ(y)))

p

(1− fγ(y))(1− fγ( fγ(y)))(1− fγ( fγ( fγ(y))))

. Since max 1−γ /46x6γ/4 (1− 2x)(1 − 2 fγ(x))(1− 2 fγ( fγ(x))) p (1− fγ(x))(1− fγ( fγ(x)))(1− fγ( fγ( fγ(x)))) 6 0.6 and mA+ mC≈ 0.169,

we haveλ1≈ 0.9 < 1. Similarly, we can prove that λ2≈ 0.9 < 1. Let λ= max(λ1, λ2)+ 2c5(+ η + η0)/μ, where μ= min 1−γ /46x6γ/4 q (1− fγ(x1))(1− fγ( fγ(x1)))(1− fγ( fγ( fγ(x1)))) > 0. (3.8)

It is easily seen that L13(ζ (k+ 3)) 6 λL13(ζ (k)).

Obviously, if

2+ η2+ η02< (1− max(λ1, λ2))μ/(2c5), (3.9)

then 0< λ < 1. This completes the proof of the lemma.

By direct computation, we have the following equalities:

x1(k+ 2) − x2(k+ 2) = E(x1(k)− x2(k))+ F(y1(k)− y2(k))+ c6(+ η + η2)dist12(k),

y1(k+ 2) − y2(k+ 2) = G(x1(k)− x2(k))+ H(y1(k)− y2(k))+ c7(+ η + η2)dist12(k). (3.10)

Here, we use the fact that|xi(k)− xj(k)| + |yi(k)− yj(k)| < η2if|i − j| is even and c6and c7are

constants independent of and η and

E= mEγ2(1− (x1(k+ 1) + x2(k+ 1)))(1 − (x1(k)+ x2(k))),

F= mFγ2(1− (y1(k+ 1) + y2(k+ 1)))(1 − (y1(k)+ y2(k))),

G= mGγ2(1− (x1(k+ 1) + x2(k+ 1)))(1 − (x1(k)+ x2(k))),

H= mHγ2(1− (y1(k+ 1) + y2(k+ 1)))(1 − (y1(k)+ y2(k))),

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in which

mE= (1 − 4c)2(1− θ)2+ (1 − 4c)θ2,

mF= 2(1 − 4c)(1 − θ)θ(1 − 2c),

mG= 2(1 − θ)θ(1 − 2c),

mH= (1 − θ)2+ 2cθ(1 − 4c).

We define a Lyapunov function L12in Nηfor the first and the second nodes:

L12(ζ (k))=√ |x1(k)− x2(k)|

(x1(k)+ x2(k))(2− (x1(k)+ x2(k))) +√ |y1(k)− y2(k)|

(y1(k)+ y2(k))(2− (y1(k)+ y2(k))). (3.11)

In a similar way to that of the above discussion, we can also prove the following lemma.

LEMMA3.3 There exist 0< 3 < 2, 0< η3 < η2, 0< η0₃ < η0₂and 0< ˜λ < 1 such that for any ζ (k)∈ Nη3,η03, it holds that

L12(ζ (k+ 2)) 6 ˜λL12(ζ (k)).

LEMMA3.4 Consider the map

x(k+ 1) = (1 − θ) fγ(x(k))+ θ fδ(y(k)), y(k+ 1) = (1 − θ) fδ(y(k))+ θ fγ(x(k)), (3.12)

whereθ ∈ (0.62, 0.64), γ, δ ∈ (3.7 − 0_{, 3.7}₊0_{) and f}_γ _{and f}_δ_{are logistic maps. Then, for every}

> 0, there exists an ₀0 > 0 such that for every 0 < 0 < ₀0 it holds that if|x(0) − y(0)| > , then

|x(k) − y(k)| decreases exponentially as k increases until it becomes less than .

Proof. Define the Lyapunov function for (3.12):

L(x, y)= (x− y)2 (x+ y)(2 − x − y).

Then, there exists a k0such that L(x(k0), y(k0)) < λ0L(x(0), y(0)) with 0 < λ0< 1 and (x(0), y(0))∈ (1− γ /4, γ /4)2_{. Hence, if}₀ _{= 0, then L(x(k), y(k)) decreases exponentially to zero, which implies}

the exponential decrease of|x(k) − y(k)| to zero. For 0> 0, we have

L(x(k0), y(k0)) < λ0L(x(0), y(0))+ c130.

For any > 0, let ₀0 < (1−λ_c 0)ν

13 . Then, L(x(k0), y(k0)) will decrease exponentially until it becomes

less thanν, which implies the exponential decrease of|x(k) − y(k)| until it becomes less than .

The following lemma is useful later.

LEMMA 3.5 For any 4 6 k ∈ N, there exist an ηk and an η0_k such that Φi Nηk,η0k

⊂ Nη3,η03 for

16 i 6 k.

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Proof. It is easy to see that for any initial point in N_η

4,η04, whereη4 < 1 and η 0

4< 1 are small positive

numbers determined later, we have

|x1(k+ 1) − x3(k+ 1)| + |y1(k+ 1) − y3(k+ 1)| 6 c8(|x1(k)− x3(k)| + |y1(k)− y3(k)|) 6 c8η2

4,

and c9is independent of k and

|x1(k+ 1) − x2(k+ 1)| + |y1(k+ 1) − y2(k+ 1)|

6 c10(|x1(k)− x2(k)| + |y1(k)−y2(k)|) + c11(|x1(k)− x3(k)| + |y1(k)− y3(k)| + |x2(k)− x4(k)| +|y2(k)− y4(k)|)

6 c10η4+ 3c11η2 4,

where we use the definition of Nη,η0 and c9, c10 and c11 are independent of k. Hence, if η4 <

It is easy to obtain the similar estimates for

|xi(k+ 1) − xj(k+ 1)| + |yi(k+ 1) − yj(k+ 1)| and |xi(k+ 1) − yj(k+ 1)| for other i, j= 1, 2, 3, 4. Letη4 = η3 C4 andη 0 4 = η0₃

C₄0. The above inequalities imply thatΦ Nη1,η10

∈ Nη0,η00. By induction,

assume that for k ∈ N, there exist a Ck and a Ck0 such thatΦi Nηk,ηk0

∈ Nη3,η03 withηk = η0

Ck and

η0_k = η00

C_k0, for 16 i 6 k. Then, similarly we can find Ck+1and Ck0₊₁such thatΦ Nηk+1,η0_k₊₁

∈ Nηk,η0k withηk₊₁ = _Cη0 k+1 andη0k₊₁ = η0₀ C_k0₊₁. Thus, we haveΦ k+1 _N ηk+1,η0_k₊₁

∈ Nη3,η03. This completes the

proof.

From Lemmas3.2–3.5, we have the following lemma.

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LEMMA3.6 There exist c13 > 0, K1∈ N, 0 < ˉη < η3and 0< ˉη0< η₃0 such that for any 06 η < ˉη,

06 η0< ˉη0_{and initial points in N}_η,η0_{, it holds that}

(i) |xi(K1)− xj(K1)| + |yi(K1)− yj(K1)| < ν

4_η2

4 for|i − j| even,

(ii) |xi(K1)− xj(K1)| + |yi(K1)− yj(K1)| < ν

2_η

2 for|i − j| odd and

(iii) |xi(K1)− yj(K1)| < c₁₃K1η0.

Proof of Theorem 3.2. Define K1, ˉη and ˉη0as in Lemma3.5. Obviously, for k > K1, if only|xi(k)−

yj(k)| < η₃0, i, j = 1, 2, 3, 4, then

|xi(k)− xj(k)| + |yi(k)− yj(k)| 6 η23/4 for|i − j| even

and

|xi(k)− xj(k)| + |yi(k)− yj(k)| 6 η3/2 for|i − j| odd.

Let K2be a small positive integer such thatλ0K2c₁₃K1 < 1/2. Define η₀0 = 1₂c−(K₁₃ 1+k0)η₃0. Then, we

Thus, defining K0= K1+ k0K2, η0= ˉη and η0₀as above, we finish the proof of Theorem 3.2.

4. Numerical results

In this section, we present some numerical results to illustrate the chaotic synchronization behaviour of (1.3) and (1.4) in established regimes (see the last pages). In Figs 2–4, we show the regions of parametersθ and the coupling coefficients c for which synchronization occurs in 1D CMLs (1.3) and (1.4) with fixedγ , δ and n= 4. Figure5shows the region ofθ and c for which synchronization of (1.3) and (1.4) occurs is very small with the lattice n= 8. In Figs6–7, we show the regions of parametersγ

andδ for some fixed θ and c. In Fig.8, we present the difference of x(k)− y(k) for CMLs of (1.3) and (1.4) with n= 4 and plot the FFT of |x(k) − y(k)|. The numerical behaviour shows that the difference of x(k) and y(k) forms a chaotic behaviour.

5. Conclusion

In this paper, we have designed CMLs of two-variable maps (connected with two logistic maps) cou-pled with one variable. We have proved that our 1D CMLs with the lattice size n = 4 have chaotic synchronized behaviour for some suitable coupling coefficients. We also present the numerical results

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FIG. 2. Range of c andθ for 1D CMLs with γ= 3.685, δ = 3.68 and n = 4.

FIG. 3. Range of c andθ for 1D CMLs with γ= 3.9, vδ = 3.75 and n = 4.

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FIG. 6. Range ofγ and δ for 1D CMLs with c= 0.8, θ = 0.42 and n = 4.

FIG. 7. Range ofγ and δ for 1D CMLs with c= 0.95, θ = 0.25 and n = 4.

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FIG. 8. The difference of x(k) and y(k) in CMLs (1.4) with γ= 3.68, δ = 3.681, c = 1/3 and θ = 2/3. The below picture is the

FFT of|x(k) − y(k)|.

of synchronization of 1D cases with various coupling coefficients, connected parameters and lattice sizes. The two-variable map as in (1.4) connected with logistic maps produces chaotic behaviour over a certain wide connecting range. Due to the special topological structure of security in private commu-nication, the new designed topological structure of (1.3) and (1.4) appears to be attractive from both theoretical and practical points of view.

Acknowledgements

We are very grateful to the referees for valuable comments and pointing out a number of inaccuracies and misprints in the first version of the manuscript. Y-QW thanks the hospitality of Prof. Song-Sun Lin at the Center for Theoretical Sciences, Taiwan, and many helpful suggestions of Mr John S. Brock.

Funding

Natural Science Fund of China (10871090 to Y.-Q.W).

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