A secure communication scheme basedgeneralized function projectivesynchronization of a new 5D hyperchaoticsystem

Xiangjun Wu1,2,3, Zhengye Fu1 and Jürgen Kurths2,3

1 College of Software, Henan University, Kaifeng 475004, People’s Republic of China2 Potsdam Institute for Climate Impact Research (PIK), D-14473 Potsdam, Germany3Department of Physics, Humboldt University, D-12489 Berlin, Germany

E-mail: xiangjung.wu@gmail.com

Received 19 October 2014, revised 3 January 2015Accepted for publication 30 January 2015Published 27 March 2015

AbstractIn this paper, a new five-dimensional hyperchaotic system is proposed based on the Lühyperchaotic system. Some of its basic dynamical properties, such as equilibria, Lyapunovexponents, bifurcations and various attractors are investigated. Furthermore, a new securecommunication scheme based on generalized function projective synchronization (GFPS) of thishyperchaotic system with an uncertain parameter is presented. The communication scheme iscomposed of the modulation, the chaotic receiver, the chaotic transmitter and the demodulation.The modulation mechanism is to modulate the message signal into the system parameter. Thenthe chaotic signals are sent to the receiver via a public channel. In the receiver end, by designingthe controllers and the parameter update rule, GFPS between the transmitter and receiver systemsis achieved and the unknown parameter is estimated simultaneously. The message signal can befinally recovered by the identified parameter and the corresponding demodulation method. Thereis no any limitation on the message size. Numerical simulations are performed to show thevalidity and feasibility of the presented secure communication scheme.

Keywords: five-dimensional (5D) hyperchaotic system, Lü hyperchaotic system, generalizedfunction projective synchronization (GFPS), secure communication, parameter estimation,message size

(Some figures may appear in colour only in the online journal)

1. Introduction

It is well known that there is at least one positive Lyapunovexponent in chaotic systems. However, in the case wherethere is just one positive Lyapunov exponent, the system isnot safe enough to mask messages [1]. Higher-dimensionalhyperchaotic systems are often recommended for addressingthis issue. Hyperchaos is characterized as a chaotic systemwith more than one positive Lyapunov exponent, indicatingthat its dynamics are expanded in more than one directionsimultaneously, which can increase randomness and unpre-dictability. Because of its higher unpredictability than simple

chaotic systems, hyperchaotic systems may be more useful insome fields such as secure communication, encryption, etc. Inrecent years, many hyperchaotic systems have been devel-oped numerically and experimentally by adding a simple statefeedback controller or a sinusoidal parameter perturbationcontroller in the generalized Lorenz system, Chen system, Lüsystem and a unified chaotic system [2, 3]. It was also noticedthat hyperchaos can be generated based on lower-dimensionalchaotic systems by employing an additional state input [4, 5].Up to now, almost all hyperchaotic systems are four-dimen-sional systems, which have double-wing hyperchaoticattractors with three or five equilibrium points [6, 7].

| Royal Swedish Academy of Sciences Physica Scripta

Phys. Scr. 90 (2015) 045210 (12pp) doi:10.1088/0031-8949/90/4/045210

0031-8949/15/045210+12$33.00 © 2015 The Royal Swedish Academy of Sciences Printed in the UK1

Generating a hyperchaotic attractor from a smooth dynamicalsystem with only one equilibrium point is a very rare phe-nomenon [8]. Motivated by the above discussions, this paperpresents a new five-dimensional (5D) hyperchaotic systemwith only one equilibrium point, which is generated from the4D Lü hyperchaotic system. The hyperchaotic dynamicbehavior of the new system is demonstrated by computersimulations.

Since the seminal work of Pecora and Carrol [9], syn-chronization of chaotic systems for secure communication hasreceived much attention [10–15]. So far, various types ofsynchronization phenomena in the chaotic systems have beenreported, such as complete synchronization [9], generalizedsynchronization [16], phase synchronization [17], lag syn-chronization [18], projective synchronization [10] etc.Recently, the concept of function projective synchronization(FPS) has been introduced [19–21], where the drive andresponse systems could be synchronized up to a scalingfunction. Du et al [22] presented a new type of synchroni-zation, the modified function projective synchronization,where the drive and response systems could be synchronizedup to a desired scaling function matrix. Yu and Li [23] studiedadaptive generalized function projective synchronizationbetween two different uncertain chaotic systems. In theapplication to secure communication, the scaling functionmatrix may also be a useful utility to improve the security ofthe secure communication scheme. Therefore, it is essential tostudy FPS of hyperchaotic systems and its application tosecure communication.

The general idea for transmitting a message via chaoticsystems is that a message signal is embedded in the trans-mitter system which produces a chaotic signal. The chaoticsignal is emitted to the receiver through a public channel.Finally, the message signal is recovered by the receiver. Inrecent years, many types of secure communication schemeshave been presented [24, 25]. The techniques of chaoticcommunication can be divided into three categories: chaosmasking [11, 12], chaos modulation [26–28] and chaos shiftkeying [29, 30]. In chaotic masking, the message signal isadded to a chaotic signal and the combined signal is thentransmitted to the receiver. The message can be extractedunder certain conditions in the receiver terminal. In chaoticmodulation, the message is injected into the states or theparameters of the chaotic system, or is modulated by usingan invertible transformation. If the transmitter and thereceiver are synchronized, the message signal can berecovered by a receiver. In chaos shift keying, we assumethat the message is binary, and it is mapped into thetransmitter and the receiver. The message signal can berecovered by a receiver as synchronization between thetransmitter and the receiver occurs. To our knowledge, inmost of secure communication schemes [10–13, 24, 27], themessage size is required to be sufficiently small, otherwiseit may induce a chaotic system to be asymptotically stableor emanative, which may render the failure of recoveringthe emitted signal. In addition, another method consideredin [26, 28] is that the upper and lower bounds of the mes-sage signal must be known in advance. However, in real

situations, some messages to be transmitted may be verylarge or unbounded. For example, the messages are t ,2 e ,t

t tsin ( ) etc, < ∞t . The existing secure communicationmethods are clearly invalid for unbounded messages.Recently, chaotic complex systems are attempted to applyfor secure communications because doubling the number ofvariables increases the content and security of the trans-mitted information [31, 32]. Mahmoud et al [31] trans-mitted more than one large or bounded message by thepassive projective synchronization of uncertain hyperch-aotic complex nonlinear system. Liu and zhang [32] pro-posed a secure communication scheme based on complexfunction projective synchronization of complex chaoticsystems and chaotic masking. However, there are defects intheir method such as complex controllers, high control costand only applying to bounded or very small signals. So it isan important issue to investigate how to transmit unboundedmessage signals.

In this work, a new secure communication scheme isintroduced based on generalized function projective syn-chronization (GFPS) of the novel 5D hyperchaotic system andparameter modulation. The message signal to be emitted maybe bounded or unbounded. In the transmitter side, the mes-sage signal is firstly modulated by an invertible function.Then this modulated signal is taken as the parameter of the5D hyperchaotic system to guarantee communication secur-ity. We only transmit the unpredictable chaotic states througha public channel to the receiver. Suppose that the parameter ofthe receiver system is unknown. In the receiver end, thecontrollers and corresponding parameter update rule aredesigned to achieve GFPS between the transmitter andreceiver systems and estimate the unknown parametersimultaneously. Finally, the original message signal trans-mitted from the transmitter can be successfully recovered bythe estimated parameter and the presented invertible function.Simulation results show the effectiveness and feasibility ofthe proposed communication scheme.

The rest of this paper is organized as follows. Insection 2, a new 5D hyperchaotic system is generated fromthe Lü hyperchaotic system. The properties and dynamics ofthe hyperchaotic system are investigated numerically. Insection 3, a secure communication scheme via GFPS of the5D hyperchaotic system with uncertain parameter is pro-posed. The controllers and the parameter update rule aredevised for obtaining the desired synchronization and identifythe unknown parameter simultaneously. Numerical simula-tions are given to illustrate and validate the proposed com-munication scheme in section 4. Our conclusions are finallydrawn in section 5.

2. Generation and dynamical analysis of a new 5Dhyperchaotic system

Based on the original Lü chaotic system, the Lü hyperchaoticsystem was constructed by employing a feedback controller

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Phys. Scr. 90 (2015) 045210 X Wu et al

[33], which is described by

= − + = − + = − = +

x a y x wy xz cyz xy bzw xz dw

( ) ,,

,,

(1)

⎧⎨⎪⎪

⎩⎪⎪

where x, y, z and w are state variables and a, b, c, d are realconstant parameters. When =a 36, =b 3, =c 20 and− ⩽ ⩽d0.35 1.3, the system (1) exhibits a hyperchaoticbehavior.

By eliminating the linear feedback controller from thefirst equation and introducing a linear feedback controller tothe second one of system (1), a new 5D hyperchaotic systemcan be obtained as follows:

= − = − + + = − = + = − −

x a y xy xz cy vz xy bzw xz dwv x y

( ),,

,,

,

(2)

⎧

⎨⎪⎪

⎩⎪⎪

where a, b, c and d are again system parameters and ⩽d 0.

2.1. Symmetry

System (2) is symmetric respect to the z-axis, i.e., it isinvariant under the following coordinate transformations

→ − − − −x y z w v x y z w v( , , , , ) ( , , , , ).

2.2. Dissipation

The divergence of the system (2) is

= ∂ ∂

+ ∂ ∂

+ ∂ ∂

+ ∂ ∂

+ ∂ ∂

= − + − + = − +

Vx

x

y

y

z

z

w

w

v

va c b d d19 .

Obviously, for the parameter values considered here, weget − + <d19 0, i.e., system (2) is dissipative. Hence, thenew dynamical system (2) converges to a set of measure zeroexponentially, i.e., = − +V t( ) e .d19 It indicates that

= − +V t V t( ) ( )e d t0

( 19 ) for any initial cubage V t( ),0 and theorbit flows into a certain bounded region as → ∞t .

2.3. Equilibria and stability

The equilibria of system (2) can be obtained by solving thefollowing equations:

− = − + + = − = += − − =

a y x xz cy v xy bz xz dwx y

( ) 0, 0, 0,0, 0. (3)

It is easy to find that system (2) has only one trivialequilibrium point E (0,0,0,0,0).0 By linearizing system (2)

around E ,0 we yield the following Jacobian matrix:

=

−

−

− −

J

a ac

bd

0 0 00 0 0 10 0 0 00 0 0 01 1 0 0 0

.

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

The corresponding characteristic equation is:

λ λ λ λ λ+ − + − − + =b d a c ac( )( ) ( ) 1 0. (4)2⎡⎣ ⎤⎦

For =a 36, =b 3, =c 20 and ⩽d 0, the eigen values ofthe Jacobian matrix J are:

λ λ λ λ λ= = = = = −d20, 19.98, 0, and 35.98.1 2 3 4 5

Here λ1 and λ2 are two positive real numbers. If <d 0, λ4 andλ5 are two negative real numbers; otherwise, λ5 is a negativereal number. Therefore, the equilibrium E (0,0,0,0,0)0 is anunstable saddle point.

2.4. Dynamics in the new 5D system

In the following, the basic dynamics of the new 5D system (2)is studied by means of the Lyapunov exponents spectrum andcorresponding bifurcation diagrams. We compute the Lya-punov exponents Li =i( 1,2,3,4,5) by the Wolf algorithm[34]. Some simulations were performed with varying para-meters to analyze the dynamics of system (2), and the findingsare summarized as follows.

= = =a b c dCase I. Fix 36, 3, 20 and vary .

Figure 1(a) displays the Lyapunov exponents spectrum ofthe 5D system (2) with respect to parameter d. Figure 1(b)shows the bifurcation diagram of state y versus parameter d.Obviously, when ∈ −d [ 50,0], there are >L 0,1 >L 0,2

=L 0,3 <L 04 and <L 0,5 which implies that the newsystem (2) is always hyperchaotic. In order to further observethe new hyperchaotic attractors, some phase portraits of the5D system (2) with different d are plotted in figure 2.

= = = −b c d aCase II. Fix 3, 20, 1 and vary .

Figure 3 plots the spectrum of Lyapunov exponents andcorresponding bifurcation diagram of system (2) with respectto parameter a. When ∈a [21,26.4), =L 01 and <τL 0τ =( 2,3,4,5), implying that the 5D system (2) is periodicshown in figures 4(a) and (b). As ∈a [26.4,60.2], >L 0,1

>L 0,2 =L 0,3 <L 04 and <L 0,5 which means that sys-tem (2) is hyperchaotic. When ∈a (60.2,61], >L 01 and

<τL 0 τ =( 2,3,4,5), so system (2) is chaotic. As∈a (61,120], =L 01 and <τL 0 τ =( 2,3,4,5), indicating

that system (2) eventually evolves to a periodic orbit. Fromthe bifurcation diagram shown in figure 3(b), one can clearlysee that, starting from the periodic region (for instance, thephase diagrams are shown in figures 4(a) and (b)), with aincreasing, the state y goes through a process of hyperchaos(figure 4(c)), chaos and finally to the periodic orbit

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Phys. Scr. 90 (2015) 045210 X Wu et al

(figure 4(d)).

= = = −a c d bCase III. Fix 36, 20, 1 and vary .

When the parameters =a 36, =c 20 and = −d 1 arefixed while parameter b is varied, the spectrum of Lyapunovexponents and the corresponding bifurcation diagram of statey versus b are shown in figure 5. As ∈b [0,0.24),

∈b (13.3,15.7], ∈b [16.03,16.15], ∈b [16.35,16.42),∈b (16.65,17.56), ∈b [17.84,17.92], ∈b [18.2,18.44],∈b [19,19.1], ∈b (19.48,19.52] and ∈b [19.68,19.76], the

maximum Lyapunov exponent L1 equals zeros and other fourLyapunov exponents are negative, representing that system(2) has a periodic orbit. When ∈b [0.24,0.58],

∈b [0.61,0.66], ∈b (6.2,13.3], ∈b (15.7,16.03),∈b (16.15,16.35), ∈b [16.42,16.65], ∈b [17.56,17.84),∈b (17.92,18.2), ∈b (18.44,19), ∈b (19.1,19.48],∈b (19.52,19.68) and ∈b (19.76,20], only >L 0,1 implying

that system (2) is chaotic. When ∈b (0.58,0.61) and∈b (0.66,6.2], the Lyapunov exponents L1 and L2 are

positive, which means that system (2) is hyperchaotic.Figure 5 reveals that periodic orbit, chaos and hyperchaosappear alternately with b increasing gradually from 0 to 20.

= = = −a b d cCase IV. Fix 36, 3, 1 and vary .

In this case, we fix =a 36, =b 3, = −d 1 and onlychange c. Figure 6 shows the Lyapunov exponents spectrumand the bifurcation diagram with respect to parameter c.When ∈c [0,11.8) and ∈c (29.24,30], system (2) is peri-odic, in which the maximum Lyapunov exponent L1 equalszero; while ∈c [11.8,24.78], ∈c (24.88,26.5],

∈c (27.35,28.08), ∈c (28.5,28.8] and ∈c (29,29.1], system(2) has two positive Lyapunov exponents, implying that it is ahyperchaotic system. When ∈c (24.78,24.88],

∈c (26.5,27.35], ∈c [28.08,28.5], ∈c (28.8,29] and∈c (29.1,29.24], system (2) has a single positive Lyapunov

exponent, which means that this system has a chaotic orbit.Figure 6(b) displays clearly the whole evolution process of

Figure 1. Lyapunov exponents spectrum and bifurcation diagram of system (2) with =a 36, =b 3, =c 20, ∈ −d [ 50, 0].

Figure 2. Phase portraits of the hyperchaotic system (2) with different d .

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Phys. Scr. 90 (2015) 045210 X Wu et al

Figure 3. Lyapunov exponents spectrum and bifurcation diagram of system (2) with =b 3, =c 20, = −d 1, ∈a [21, 120].

Figure 4. Phase diagrams of the 5D system (2) with different a.

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Phys. Scr. 90 (2015) 045210 X Wu et al

system (2), i.e., from period to hyperchaos, chaos and finallyto period again.

3. A new secure communication scheme via GFPS ofthe uncertain 5D hyperchaotic system

In this section, we will introduce a new secure communica-tion method based on GFPS of the new 5D hyperchaoticsystem (2) with unknown parameter d. Figure 7 describes theproposed communication system consisting of a modulation,a transmitter (drive), a receiver (response) at the receiving endof the communication, and a demodulation. The messagesignal to be transmitted is modulated into the parameter of thechaotic transmitter system by employing an invertible func-tion, and the resulting system is still hyperchaotic. Theresulting chaotic signals are sent to the receiver end through apublic channel. In the receiver side, by designing suitablecontrollers, the desired synchronization between the trans-mitter and receiver systems can be obtained, and the unknownparameter can also be identified at the same time. Then themessage signal can be recovered from the estimated para-meter by performing the proposed demodulation method. Inthe following, we will illustrate the secure communicationsystem via GFPS of the new 5D hyperchaotic system withunknown parameter in detail.

3.1. Modulation

For transmitting an arbitrary continuous-time message signalregardless of its size, we consider modulating it into theparameter d of system (2). Let s t( ) represent the messagesignal. Suppose ⩽ ⩽d d d ,1 2 where = −d 11 and =d 0.2

Obviously, system (2) is always hyperchaotic in this range. Itis known that the arc tangent function satisfies:

π π⋅ ∈ −arctan () ( 0. 5 , 0. 5 ). Let us define a new parameterρ t( ). In order to obtain ρ ∈ −t( ) ( 1, 0), we present the fol-lowing modulation technique:

ρπ

π

=−

++

= −

( ) ( )t

d ds t

d d

s t

( ) arctan ( ( ))2

1arctan ( ( )) 0.5. (5)

2 1 1 2

In the following, we will use ρ t( ) as the parameter ofsystem (2), which will be explained in detail later.

3.2. Transmitter

Replacing the parameter d in system (2) with the new para-meter ρ t( ) obtained in section 3.1, we get

ρ

= − = − + + = − = + = − −

x a y x

y x z cy v

z x y bz

w x z t wv x y

( ),

,

,

( ) ,.

(6)

1 1 1

1 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

Since ρ ∈ −t( ) ( 1, 0), the resulting system (6) is stillhyperchaotic. We take system (6) as the transmitter system.x ,1 y ,1 z ,1 w1 and v1 are the chaotic signals and need to betransmitted to the receiver via a public channel. Since system(6) is hyperchaotic, it is hard to extract a message from thesignals transmitted in the channel.

3.3. Receiver

Consider the receiver system as follows:

ρ

= − + = − + + + = − + = + + = − − +

x a y x u

y x z cy v u

z x y bz u

w x z t w uv x y u

( ) ,

,

,

ˆ ( ) ,,

(7)

2 2 2 1

2 2 2 2 2 2

2 2 2 2 3

2 2 2 2 4

2 2 2 5

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

where ρ tˆ ( ) is the unknown parameter to be estimated, u t( )i

( =i 1,2,3,4, 5) are the controllers to be designed. Thus, thecontrol objective is to find u t( )i and ρ tˆ ( ) such that both thetransmitter system and the receiver system achieve GFPS, andρ tˆ ( ) converges to the actual value of ρ t( ).

Let us introduce the following state errors:

ααααα

= −= −= −= −= −

e x t x

e y t y

e z t z

e w t w

e v t v

( ) ,( ) ,

( ) ,( ) ,

( ) ,

(8)

1 2 1 1

2 2 2 1

3 2 3 1

4 2 4 1

5 2 5 1

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

and the parameter estimate error as

ρ ρ= −ρe t tˆ ( ) ( ). (9)

The time derivative of the error signals (8) is

α αα αα αα αα α

= − − = − − = − − = − − = − −

e x t x t x

e y t y t y

e z t z t z

e w t w t w

e v t v t v

( ) ( ) ,( ) ( ) ,

( ) ( ) ,( ) ( ) ,

( ) ( ) .

(10)

1 2 1 1 1 1

2 2 2 1 2 1

3 2 3 1 3 1

4 2 4 1 4 1

5 2 5 1 5 1

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

Substituting equations (6) and (7) into equation (10), wehave

α αα α

αα α

ρ α αα

α α α

= − + − − + = − + + −

− + = − + − − + = + + −

− + = − − + + − +

ρ

e ae ay a t y t x u

e ce x z t x z v t v

t y u

e be x y t x y t z u

e e t e w x z t x z

t w u

e x y t x t y t v u

( ) ( ) ,

( ) ( )( ) ,

( ) ( ) ,

ˆ ( ) ( )

( ) ,( ) ( ) ( ) .

(11)

1 1 2 1 1 1 1 1

2 2 2 2 2 1 1 2 2 1

2 1 2

3 3 2 2 3 1 1 3 1 3

4 4 4 1 2 2 4 1 1

4 1 4

5 2 2 5 1 5 1 5 1 5

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

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Phys. Scr. 90 (2015) 045210 X Wu et al

Taking the time derivative on both sides of equation (9)yields

ρπ

= − +

ρ ( )e t

s t

s tˆ ( )

( )

1 ( ). (12)

2

Hence the synchronization problem becomes the stabilityproblem of the error dynamics (11).

We get the following main theorem.

Theorem 1. For given nonzero scaling functions α t( )i

=i( 1,2,3,4, 5), GFPS between the transmitter (6) and thereceiver system (7) can be achieved, and the uncertainparameter ρ tˆ ( ) can be estimated, if the controllers and theparameter update rule are constructed as follows:

Figure 5. Lyapunov exponents spectrum and bifurcation diagram of system (2) with =a 36, =c 20, = −d 1, ∈b [0, 20].

Figure 6. Lyapunov exponents spectrum and bifurcation diagram of system (2) with =a 36, =b 3, = −d 1, ∈c [0, 30].

Figure 7. Block diagram of the proposed secure communicationsystem.

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Phys. Scr. 90 (2015) 045210 X Wu et al

αα

αα

αα

αα

α ρα

α α

= − ++ −

= −− ++ −

= − ++ −

= − ++ − +

= + −− + × −

( )

u ay a t y

t x k e

u x z t x z

v t v

t y k e

u x y t x y

t z k e

u x z t x z

t w k e

u x y t x

t y

t v k e

( )

( ) ,( )( )

( ) ,

( )

( ) ,( )

( ) ˆ ,

( )

( )

( ) ,

(13)

1 2 1 1

1 1 1 1

2 2 2 2 1 1

2 2 1

2 1 2 2

3 2 2 3 1 1

3 1 3 3

4 2 2 4 1 1

4 1 4 4

5 2 2 5 1

5 1 5

1 5 5

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

and

ρ απ

= − + +( )

t t w es t

s tˆ ( ) ( )

( )

1 ( ), (14)4 1 4

2

respectively, where ki =i( 1,2,3,4, 5) are positive controlgains and >k c2 .

Proof. Choose the following Lyapunov function candidate:

= + + + + + ρ( )V t e e e e e e( )1

2. (15)1

222

32

42

52 2

By calculating the derivative of V t( ) along thetrajectories of the error system (11), and using equations (13)and (14), we have

α α

α α

α

α α

ρ α α

α

α α

α

ρ π

= + + + + +

= − + − − +

+ − + + −

− +

+ − + − − +

+ + + −

− +

+ − − + +

− +

+ − +

= − + − − − +

× − − <

ρ ρ

ρ

ρ ( )

[

]

]

( ) ( ) ( )

V t e e e e e e e e e e e e

e ae ay a t y t x u

e ce x z t x z v t v

t y u

e be x y t x y t z u

e e t e w x z t x z

t w u

e x y t x t y

t v u

e t s t s t

a k e k c e b k

e k e k e

( )

( ) ( )

( ) ( )

( )

( ) ( )

ˆ ( ) ( )

( )

( ) ( )

( )

ˆ ( ) ( ) 1 ( )

0.

1 1 2 2 3 3 4 4 5 5

1 1 2 1 1 1 1 1

2 2 2 2 2 1 1 2 2 1

2 1 2

3 3 2 2 3 1 1 3 1 3

4 4 4 1 2 2 4 1 1

4 1 4

5 2 2 5 1 5 1

5 1 5

2

1 12

2 22

3

32

4 42

5 52

⎡⎣ ⎤⎦

⎤⎦⎡⎣ ⎤⎦⎡⎣

⎡⎣

⎡⎣ ⎤⎦

Obviously, V t( ) is positive definite and V t( ) is negativedefinite. By the Lyapunov stability theorem, the

synchronization errors ei =i( 1,2,3,4, 5) asymptoticallyconverge to zero, i.e., GFPS between the transmitter system(6) and the receiver system (7) is obtained, and the zero pointof the parameter error ρe is globally and asymptotically stable.It implies that the uncertain parameter ρ t( ) is also estimatedin the receiver simultaneously. This completes the proof. □

3.4. Demodulation

As GFPS between the transmitter and receiver systemsappears, one can identify the parameter ρ tˆ ( ) based on theo-rem 2. According to the invertible transformation function(5), the original message signal can be recovered as

π ρ= +( )( )s t tˆ ( ) tan ˆ ( ) 0.5 . (16)

Here s tˆ ( ) represents the recovered signal. When thedesired synchronization takes place, we have ρ ρ→t tˆ ( ) ( ) as

→ ∞t . One further gets

π ρ π ρ= + → = +( )( )s t t s t tˆ ( ) tan ˆ ( ) 0.5 ( ) tan( ( ( ) 0.5))

as → ∞t . Therefore, the receiver can extract the messagesignal successfully from ρ tˆ ( ) by the above modulationmethod.

Remark 1. The cordinate rotation digital computer (CORDIC)algorithm is traditionally used for the implementation oftrigonometric functions. Besides general scientific andtechnical computation, the CORDIC algorithm has beenutilized for various applications such as signal and imageprocessing, communication systems and robotics [35–38]. Inpractical engineering, the arctan and tangent functions can beimplemented based on the CORDIC algorithm. The hardwarearchitectures of implementing arctan function and tangentfunction are shown in figures 8 and 9, respectively.

Remark 2. In figure 8, the inputs χ χ ∈ −∞ +∞, ( , )1 2 areIEEE 754 standard single precision floating point numbers.The floating point numbers χ1 and χ2 are firstly preprocessedin the input data format transform unit to obtain two fixedpoint integer numbers, i.e., χ1 and χ ,2 whereχ χ ∈ − +¯ , ¯ [ 1, 1].1 2 χ1 and χ2 are considered as the inputs ofCORDIC unit. Next, the CORDIC algorithm is employed tocompute the arctan function by using χ1 and χ ,2 and one canget the result μ. The CORDIC algorithm has beenimplemented by many ways with software and hardware[36, 38]. Finally, in the output data format transform unit, weconvert the fixed point integer number μ into the IEEE 754standard single precision floating point number μ. Corre-sponding software simulation and hardware experiment canbe conducted on FPGAs [39].

Figure 8. Hardware architecture of implementing arctan function.

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Remark 3. In figure 9, the circular CORDIC algorithm isfirstly applied to get two results, i.e., ϖsin and ϖcos , withthe input ϖ. Then we use ϖsin and ϖcos as the inputs of thelinear CORDIC-based computation unit, and the value of

ϖtan can be obtained by means of the linear CORDIC

algorithm. The implementation of the circular CORDICalgorithm has been reported in the literature [37, 39–41]. Thelinear CORDIC algorithm only involves with addition andshift operations, which makes it very convenient implementon the hardware.

Figure 9. Hardware architecture of implementing tangent function.

Figure 10. The hyperchaotic attractors of the resulting system (6).

Figure 11. Simulation results of secure communication based on GFPS of the 5D hyperchaotic system when the information signal is abounded signal s(t) = 2 sin(0.5t) + 6 cos(5t).

Figure 12. The hyperchaotic attractors of the resulting system (6).

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4. Simulation results

In this section, computer simulations are performed to showthe efficiency of the proposed communication system. TheODE45 algorithm is adopted to solve the differentialequations. The system parameters are set as =a 36, =b 3and =c 20. In the following simulations, for saving space,we consider two cases: (i) choose a bounded message signalfor secure communication; (ii) choose an unbounded messagesignal for secure communication.

4.1. A bounded information signal for secure communication

Here the message signal hidden in the transmitter system is= +s t t t( ) 2 sin(0.5 ) 6 cos(5 ). Obviously, ⩽s t( ) 8. By

equation (5), ρ t( ) can be obtained as follows:

ρπ

= + −t t t( )1

arctan (2 sin (0.5 ) 6 cos (5 )) 0. 5. (17)

The initial conditions for the transmitter system (6) andthe receiver system (7) are arbitrarily chosen as = −x (0) 2,1

=y (0) 1,1 =z (0) 3,1 =w (0) 01 and =v (0) 4;1 = −x (0) 5,2

=y (0) 0,2 =z (0) 2,2 = −w (0) 12 and =v (0) 3,2 respec-tively. The initial value of the unknown parameter ρ tˆ ( ) istaken as ρ = −ˆ (0) 0. 1. Let the control gains be

= = = =k k k k 101 3 4 5 and =k 30.2 The scaling functionsare selected randomly as α =t v t( ) ( ),1 1 α = −t t( ) cos( ) 3,2

α = +t z t( ) ( ) 1,3 1 α = − +t tsin(10 ) 3 cos(2 ) 54 andα = − t8 5 sin(0. 2 )5 .

Figure 10 displays the hyperchaotic behavior of theresulting system (6). The simulation results of the proposedsecure communication system are given in figure 11. Infigure 11(a), we see that the synchronization errors ei

=i( 1,2,3,4,5) asymptotically converge to zero quickly. Thatis, GFPS between the transmitter system and the uncertainreceiver system is achieved under the controllers (13) and theparameter update rule (14). Figure 11(b) depicts the originalmessage signal s t( ) (blue and solid line) and the recoveredsignal s tˆ ( ) (red and dotted line) via the demodulator (16). It iseasily seen that the reconstructed signal s tˆ ( ) coincides withthe message signal s t( ) with good accuracy. This can be

Figure 13. Simulation results of secure communication based on GFPS of the 5D hyperchaotic system when the information signal is anunbounded signal s(t) = 10 + t.

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Phys. Scr. 90 (2015) 045210 X Wu et al

further validated by figure 11(c) which shows the messagesignal recovery error −s t s tˆ ( ) ( ). As expected, the signalrecovery error quickly converges to zero and the commu-nication goal is attained.

4.2. An unbounded information signal for securecommunication

In this section, we simulate the presented secure commu-nication system with an unbounded message signal. Forexample, the message signal = +s t t( ) 10 , where∣ ∣ < ∞s t( ) . From equation (5), we get

ρπ

= + −t t( )1

arctan (10 ) 0.5. (18)

Corresponding initial values are arbitrarily set as follows:=x (0) 3,1 =y (0) 2,1 = −z (0) 1,1 =w (0) 2,1 =v (0) 5,1

=x (0) 2,2 = −y (0) 4,2 = −z (0) 3,2 =w (0) 5,2 =v (0) 02

and ρ = −ˆ (0) 0. 1. The control gains are given as= = = =k k k k 401 3 4 5 and =k 60.2 We randomly choose

the scaling functions as α = +t t( ) 6 2 sin(0.5 ),1

α = +t t( ) 2 3 cos( ),2 α = −t t( ) 2 sin( ) 1,3 α = −24 andα = − t2 sin( )5 .

The hyperchaotic attractors of the resulting system (6) aredepicted in figure 12. Numerical results for GFPS between thetransmitter and receiver systems via the controllers (13) andthe parameter update rule (14) and its application to securecommunication are illustrated in figure 13. Figure 13(a)describes the time evolution of the synchronization errors ei

=i( 1,2,3,4,5), which shows that the time response of theerrors approach the origin very fast. So the desired synchro-nization is obtained. The original message signal s t( ) and therecovered one s tˆ ( ) are plotted in figures 13(b) and (c),respectively. Figure 13(d) displays the error signal

−s t s tˆ ( ) ( ) between the message signal s t( ) and the recov-ered one s tˆ ( ). As seen, the signal error tends to zero in a veryshort time. Thus, the message signal is recovered accurately.

5. Conclusion

This paper presents a new 5D hyperchaotic system generatedfrom the Lü hyperchaotic system. Some basic dynamicalbehaviors of the system are explored by investigating itsLyapunov exponents spectrum and bifurcation diagrams. Andvarious phase portraits of the system has been demonstratedby computer simulations. Furthermore, combining GFPS ofthe new 5D hyperchaotic system with the parameter mod-ulation technique, we have introduced a new chaotic securecommunication method. In contrast to existing chaotic securecommunication approaches, there are no limitations on themessage size in our scheme. Under this structure, the messagesignal can successfully and secretly be transmitted throughfour main functions, i.e., modulation, chaotic transmitter,chaotic receiver and demodulation. Finally, numerical simu-lations have been provided to verify the effectiveness and thefeasibility of the presented secure communication scheme.

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (Grant Nos. 61004006 and61203094), China Postdoctoral Science Foundation (GrantNo. 2013M530181), the Natural Science Foundation ofHenan Province, China (Grant No. 13230010254), Programfor Science & Technology Innovation Talents in Universitiesof Henan Province, China (Grant No. 14HASTIT042), theFoundation for University Young Key Teacher Program ofHenan Province, China (Grant No. 2011GGJS-025), Shang-hai Postdoctoral Scientific Program (Grant No.13R21410600), the Science & Technology Project Plan ofArchives Bureau of Henan Province (Grant No. 2012-X-62)and the Natural Science Foundation of Educational Com-mittee of Henan Province, China (Grant No. 13A520082).The authors would like to thank the anonymous reviewers andthe editor for their helpful comments.

References

[1] Perez G and Cerdeira H A 1995 Extracting messages maskedby chaos Phys. Rev. Lett. 74 1970–3

[2] Li Y X, Chen G and Tang W K S 2005 Controlling a unifiedchaotic system to hyperchaotic IEEE Trans. Circuit Syst. II52 204–7

[3] Li Y X, Tang W K S and Chen G R 2005 Generatinghyperchaos via state feedback control Int. J. BifurcationChaos 15 3367–75

[4] Jia Q 2007 Generation and suppression of a new hyperchaoticsystem with double hyperchaotic attractors Phys. Lett. A 371410–5

[5] Niu Y, Wang X, Wang M and Zhang H 2010 A newhyperchaotic system and its circuit implementationCommun. Nonlinear Sci. Numer. Simul. 15 3518–24

[6] Chen C H, Sheu L J, Chen H K, Chen J H, Wang H C,Chao Y C and Lin Y K 2009 A new hyper-chaoticsystem and its synchronization Nonlinear Anal. RWA 102088–96

[7] Zheng S, Dong G and Bi Q 2010 A new hyperchaoticsystem and its synchronization Appl. Math. Comput. 2153192–200

[8] Chen Z, Yang Y, Qi G and Yuan Z 2007 A novel hyperchaossystem only with one equilibrium Phys. Lett. A 360696–701

[9] Pecora L M and Carroll T L 1990 Synchronization in chaoticsystems Phys. Rev. Lett. 64 821–4

[10] Li Z G and Xu D L 2004 A secure communication schemeusing projective chaos synchronization Chaos SolitonsFractals 22 477–81

[11] Uyaroğlu Y and Pehlivan Ī 2010 Nonlinear sprott94 case achaotic equation: synchronization and maskingcommunication applications Comput. Electr. Eng. 361093–100

[12] Wu X, Wang H and Lu H 2012 Modified generalizedprojective synchronization of a new fractional-orderhyperchaotic system and its application in securecommunication Nonlinear Anal. RWA 13 1441–50

[13] Cheng C J 2012 Robust synchronization of uncertainunified chaotic systems subject to noise and its applicationto secure communication Appl. Math. Comput. 2192698–712

11

Phys. Scr. 90 (2015) 045210 X Wu et al

[14] Boccaletti S, Bianconi G, Criado R, del Genio C I,Gómez-Gardeñes J, Romance M, Sendiña-Nadal I,Wang Z and Zanin M 2014 The structure and dynamics ofmultilayer networks Phys. Rep. 544 1–122

[15] Wang Z, Wang L and Matjaž P 2014 Degree mixing inmultilayer networks impedes the evolution of cooperationPhys. Rev. E 89 052813

[16] Zheng Z G and Hu G 2000 Generalized synchronization versusphase synchronization Phys. Rev. E 62 7882–5

[17] Rosenblum M G, Pikovsky A S and Kurths J 1996 Phasesynchronization of chaotic oscillators Phys. Rev. Lett. 761804–7

[18] Pikovsky A S, Rosenblum M G and Kurths J 1997 From phaseto lag synchronization in coupled chaotic oscillators Phys.Rev. Lett. 78 4193–7

[19] Chen Y and Li X 2007 Function projective synchronizationbetween two identical chaotic systems Int. J. Mod. Phys. C18 883–8

[20] Luo R Z 2008 Adaptive function project synchronization ofRössler hyperchaotic system with uncertain parametersPhys. Lett. A 372 3667–71

[21] Du H Y, Zeng Q S and Wang C H 2008 Function projectivesynchronization of different chaotic systems with uncertainparameters Phys. Lett. A 372 5402–10

[22] Du H, Zeng Q and Wang C 2009 Modified function projectivesynchronization of chaotic system Chaos Solitons Fractals42 2399–404

[23] Yu Y and Li H 2010 Adaptive generalized function projectivesynchronization of uncertain chaotic systems NonlinearAnal. RWA 11 2456–64

[24] Fallahi K and Leung H 2010 A chaos secure communicationscheme based on multiplication modulation Commun.Nonlinear Sci. Numer. Simul. 15 368–83

[25] Bocaletti S, Kurths J, Osipov G, Valladares D and Zhou C2002 The synchronization of chaotic systems Phys. Rep. 3661–101

[26] Megam Ngouonkadi E B, Fotsin H B and Louodop Fotso P2014 Implementing a memristive Van der Pol oscillatorcoupled to a linear oscillator: synchronization andapplication to secure communication Phys. Scr. 89035201

[27] Lin J S, Huang C F, Liao T L and Yan J J 2010 Design andimplementation of digital secure communication based onsynchronized chaotic systems Digit. Signal Process. 20229–37

[28] Wu X J, Wang H and Lu H T 2011 Hyperchaotic securecommunication via generalized function projectivesynchronization Nonlinear Anal. RWA 12 1288–99

[29] Dedieu H, Kennedy M P and Hasler M 1993 Chaos shiftkeying: modulation and demodulation of a chaotic carrierusing self-synchronizing Chua’s circuits IEEE Trans.Circuit Syst. II 40 634–42

[30] Hou Y Y, Chen H C, Chang J F, Yan J J and Liao T L 2012Design and implementation of the Sprott chaotic securedigital communication systems Appl. Math. Comput. 21811799–805

[31] Mahmoud G M, Mahmoud E E and Arafa A A 2013 Onprojective synchronization of hyperchaotic complexnonlinear systems based on passive theory for securecommunications Phys. Scr. 87 055002

[32] Liu S and Zhang F 2014 Complex function projectivesynchronization of complex chaotic system and itsapplications in secure communication Nonlinear Dyn. 761087–97

[33] Chen A M, Lu J A, Lü J H and Yu X M 2006 Generatinghyperchaotic Lü attractor via state feedback control PhysicaA 364 103–10

[34] Wolf A, Swift J B, Swinney H L and John A W 1985Determining Lyapunov exponents from a time seriesPhysica D 16 285–317

[35] Volder J E 1959 The CORDIC trigonometric computingtechnique IRE Trans. Electron. Comput. EC-8 330–4

[36] Meher P K, Valls J, Juang T B, Sridharan K and Maharatna K2009 50 Years of CORDIC: algorithms, architectures,and applications IEEE Trans. Circuit Syst. I 56 1893–907

[37] Hu X, Harber R G and Bass S C 1991 Expanding the range ofconvergence of the CORDIC algorithm IEEE Trans.Comput. 40 13–21

[38] Andraka R 1998 A survey of CORDIC algorithms forFPGA based computers Proc. 6th ACM/SIGDA Int. Symp.on Field Programmable Gate Arrays (FPGA’98)pp 191–200

[39] Parhi K K 1999 VLSI Digital Signal Processing Systems:Design and Implementation (New York: Wiley)

[40] Antelo E, Lang T and Bruguera J D 2000 Very-high radixcircular CORDIC: vectoring and unified rotation/vectoringIEEE Trans. Comput. 49 727–39

[41] Antelo E and Villalba J 2005 Low latency pipelined circularCORDIC Proc. 17th IEEE Symp. on Comput. Arithm.pp 280–7

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Phys. Scr. 90 (2015) 045210 X Wu et al