Dualcombinationcombinationmultiswitching ...arXiv:1702.06398v1 [cs.SY] 17 Feb 2017 Dualcombinationcombinationmultiswitching synchronizationofeightchaoticsystems Ayub Khana, Dinesh

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Dual combination combination multi switching

synchronization of eight chaotic systems

Ayub Khana, Dinesh Khattarb, Nitish Prajapatic,∗

aDepartment of Mathematics, Jamia Millia Islamia, Delhi, India.bDepartment of Mathematics, Kirorimal College, University of Delhi, Delhi, India.

cDepartment of Mathematics, University of Delhi, Delhi, India.

Abstract

In this paper, a novel scheme for synchronizing four drive and four responsesystems is proposed by the authors. The idea of multi switching and dualcombination synchronization is extended to dual combination-combinationmulti switching synchronization involving eight chaotic systems and is a firstof its kind. Due to the multiple combination of chaotic systems and multiswitching the resultant dynamic behaviour is so complex that, in commu-nication theory, transmission and security of the resultant signal is moreeffective. Using Lyapunov stability theory, sufficient conditions are achievedand suitable controllers are designed to realise the desired synchronization.Corresponding theoretical analysis is presented and numerical simulationsperformed to demonstrate the effectiveness of the proposed scheme.

Keywords: Chaos synchronization, multi switching synchronization,combination combination synchronization, dual synchronization, nonlinearcontrol

1. Introduction

Much has been written and said about the concept of synchronization ofchaotic systems since it was first introduced by Pecora and Caroll [1]. Be-cause of its interdisciplinary nature the chaos synchronization problem hasreceived interest from researchers across the academic fields such as physics,

∗Corresponding authorEmail address: [email protected] (Nitish Prajapati)

Preprint submitted to Elsevier September 13, 2018

mathematics, engineering, biology, chemistry, etc. The potential applicationsof chaos synchronization to engineering systems, information processing, se-cure communications, and biomedical science amongst many others has ledto a vast variety of research studies in this topic of nonlinear science [2–5]. Various kinds of synchronization such as complete synchronization, antisynchronization, projective synchronization, reduced order synchronization,etc. have been reported and presented in a variety of chaotic systems usingmany effective methods such as active control, adaptive control, backsteppingcontrol, sliding mode control and so on [6–13].

Amongst many synchronization schemes, dual synchronization of chaoticsystems is one which has successfully piqued the scientific curiosity of re-searchers because of its challenging and non traditional nature. Deviatingfrom the traditional approach of synchronizing one drive and one responsesystem, in dual synchronization two drive systems are synchronized with tworesponse systems. Since the first inception of the idea by Liu and Davids[14], it has been extensively investigated in various synchronization studies[15–17]. In all prior work the common theme is to consider one pair of drivesystem with one pair of response system. Only recently the idea of dualsynchronization was extended to two pair of drive systems and one pair ofresponse system [18, 19].

Synchronization studies involving multiple drive and response systems is arelatively unexplored area of research. New ideas have recently been initiatedin the study of chaos synchronization where multiple chaotic systems are in-volved. In the literature of chaos synchronization the addition of combinationsynchronization [20, 21], combination combination synchronization [22, 23],compound synchronization [24, 25], double compound synchronization [26],compound combination synchronization [27, 28] etc. has opened new researchdirections to be explored. These significant ideas have strengthened the secu-rity of information transmission because of the complexity which they bringin transmitted signals.

Multi switching synchronization of chaotic systems is yet another rela-tively unexplored area of research [29]. In this non conventional scheme,different states of the drive system are synchronized with different state ofthe response system. Due to this, a wide range of synchronization directionexists for multi switching synchronization schemes. The importance of suchkind of studies to information security cannot be emphasised enough andthus makes them a very relevant topic to be investigated. A few reportedwork in this direction can be studied in [30–33]. To the knowledge of authors

2

the diverse possibilities of multi switching synchronization have not yet beenexplored with the dual synchronization schemes.

In this paper, motivated by the above discussion, the authors have com-bined the idea of multi switching with dual synchronization and extendedit to combination combination synchronization of four chaotic systems. Thenovel scheme, dual combination combination multi switching synchronizationinvolves eight chaotic systems of which four are drive systems and four areresponse systems. In contrast to double compound synchronization involvingfour drive and two response systems, this work is a significant improvementand extension. Using Lyapunov stability theory, sufficient conditions havebeen achieved to realise the desired synchronization. To demonstrate theeffectiveness of the proposed method numerical simulations have been per-formed. The main contribution and advantages of this study are: a) Thedual synchronization study has been extended to pair of two drive and tworesponse systems to achieve dual combination combination synchronization.no such work involving two pair of response systems has earlier been re-ported. b) The multi switching synchronization scheme is combined withdual combination combination synchronization to achieve the novel schemewhich is a first of its kind. No previous work on dual multi switching studiesexist. c) The proposed scheme successfully synchronizes eight chaotic sys-tems of which four are drive and four are response systems. The complexityof signals due to multiple combination and the number of synchronizationdirections due to multi switching vastly enhances the anti attack ability ofany signal that will be transmitted using combination of two pair of drivesystems. d) Several existing synchronization schemes are obtained as specialcases of dual combination combination multi switching synchronization.

2. Formulation of dual combination combination multi switching

synchronization

In this section, we formulate the scheme of DCCMS of chaotic systems.We require two pair of four chaotic drive systems and two pair of four chaoticresponse systems. Let the first two drive systems be described as

x1 = f1(x1) (1)

x2 = f2(x2) (2)

where x1 = (x11, x12, ..., x1n)T , x2 = (x21, x22, ..., x2n)

T , f1, f2: Rn → Rn are

known continuous vector functions. Linear combination of the states of two

3

drive systems (1) and (2) gives a resultant signal of the form

S1 =[a11x11, a12x12, ..., a1nx1n, a21x21, a22x22, ..., a2nx2n]T

=

[

A1 00 A2

] [

x1

x2

]

= Ax(3)

where A1 = diag(a11, a12, ..., a1n), and A2 = diag(a21, a22, ..., a2n) are twoknown matrices and a1i, a2j are not all zero at the same time (i, j = 1, 2, ..., n).

Next two drive systems are written as

y1 = g1(y1) (4)

y2 = g2(y2) (5)

where y1 = (y11, y12, ..., y1n)T , y2 = (y21, y22, ..., y2n)

T , g1, g2: Rn → Rn areknown continuous vector functions. Hence, the linear combination of thestates of two drive systems (4) and (5) gives a resultant signal of the form

S2 =[b11y11, b12y12, ..., b1ny1n, b21y21, b22y22, ..., b2ny2n]T

=

[

B1 00 B2

] [

y1y2

]

= By(6)

where B1 = diag(b11, b12, ..., b1n), and B2 = diag(b21, b22, ..., b2n) are twoknown matrices and b1i, b2j are not all zero at the same time (i, j = 1, 2, ..., n).

Let the first two response systems be given by

z1 = h1(z1) + u1 (7)

z2 = h2(z2) + u2 (8)

where z1 = (z11, z12, ..., z1n)T , z2 = (z21, z22, ..., z2n)

T , h1, h2: Rn → Rn

are known continuous vector functions, and u1 = (u11, u12, ..., u1n), u2 =(u21, u22, ..., u2n) are the controllers to be designed. By linear combination ofthe states of two response systems (7) and (8) a resultant signal is obtainedof the form

S3 =[c11z11, c12z12, ..., c1nz1n, c21z21, c22z22, ..., c2nz2n]T

=

[

C1 00 C2

] [

z1z2

]

= Cz(9)

where C1 = diag(c11, c12, ..., c1n), and C2 = diag(c21, c22, ..., c2n) are twoknown matrices and c1i, c2j are not all zero simultaneously (i, j = 1, 2, ..., n).

4

Let the next two response systems be described as

w1 = k1(w1) + u3 (10)

w2 = k2(w2) + u4 (11)

where w1 = (w11, w12, ..., w1n)T , w2 = (w21, w22, ..., w2n)

T , k1, k2: Rn → Rn

are known continuous vector functions, and u3 = (u31, u32, ..., u3n), u4 =(u41, u42, ..., u4n) are the controllers to be designed. Linear combination ofthe states of two response systems (10) and (11) gives a resultant signal ofthe form

S4 =[d11z11, d12z12, ..., d1nz1n, d21z21, d22z22, ..., d2nz2n]T

=

[

D1 00 D2

] [

z1z2

]

= Dz(12)

where D1 = diag(d11, d12, ..., d1n), and D2 = diag(d21, d22, ..., d2n) are twoknown matrices and d1i, d2j are not all zero at the same time (i, j = 1, 2, ..., n).

The error signal for dual combination combination synchronization is

e =S1 + S2 − S3 − S4

=Ax+By − Cz −Dw

=

[

A1 00 A2

] [

x1

x2

]

+

[

B1 00 B2

] [

y1y2

]

−

[

C1 00 C2

] [

z1z2

]

−

[

D1 00 D2

] [

w1

w2

]

=

[

A1x1 +B1y1 − C1z1 −D1w1

A2x2 +B2y2 − C2z2 −D2w2

]

(13)

Definition 1. If there exist four constant diagonal matrices A,B,C,D ∈R2n×2n and C 6= 0 or D 6= 0 such that

limt→∞

‖e‖ = limt→∞

‖Ax+By − Cz −Dw‖ = 0, (14)

where ‖.‖ is the vector norm, then the drive systems (1), (2), (4), and (5)realise dual combination combination synchronization with the response sys-tems (7), (8), (10), and (11).

Remark 1. The diagonal matrices A, B, C, and D are called the scalingmatrices and can be extended to functional matrices of state variables x, y,z, and w.

5

Comment 1. From (14) we get that dual combination combination syn-chronization is achieved when

limt→∞

‖e‖ = limt→∞

‖Ax+By − Cz −Dw‖ = 0,

which is equivalent to say that

limt→∞

‖e1‖ = limt→∞

‖A1x1 +B1y1 − C1z1 −D1w1‖ = 0

limt→∞

‖e2‖ = limt→∞

‖A2x2 +B2y2 − C2z2 −D2w2‖ = 0

where e = (e1, e2)T . This can be further written as

limt→∞

e1m = limt→∞

a1mx1m + b1my1m − c1mz1m − d1mw1m = 0

limt→∞

e2m = limt→∞

a2mx2m + b2my2m − c2mz2m − d2mw2m = 0

where e1 = (e11, e12, ..., e1n), e2 = (e21, e22, ..., e2n), and m = 1, 2, ..., n.

Comment 2. Let us rewrite the components of e1, and e2 as{

e1m(ijlm)= a1ix1i + b1jy1j − c1lz1l − d1mw1m

e2m(ijlm)= a2ix2i + b2jy2j − c2lz2l − d2mw2m

(15)

where i, j, l,m = 1, 2, ..., n and the subscript (ijlm) denotes ith component ofx1 and x2, j

th component of y1 and y2, lth component of z1 and z2, and mth

component of w1 and w2. In relation to Definition 1, the indices (ijlm) of theerror states e1m(ijlm)

, and e2m(ijlm)are strictly chosen to satisfy i = j = l = m

(i, j, l,m = 1, 2, ..., n).

Definition 2. If the indices of the error states e1m(ijlm), and e2m(ijlm)

areredefined such that i = j = l 6= m or i = j = m 6= l or i = l = m 6= j orj = l = m 6= i; or i = j 6= l = m or i = l 6= j = m or i = m 6= j = l; ori = j 6= l 6= m or i = l 6= j 6= m or i = m 6= l 6= j or i 6= j = l 6= m ori 6= j 6= l = m or i 6= l 6= j = m; or i 6= j 6= l 6= m and

limt→∞

‖e‖ = limt→∞

‖Ax+By − Cz −Dw‖ = 0, (16)

where i, j, l,m = 1, 2, ..., n and ‖.‖ is the vector norm, then the drive systems(1), (2), (4), and (5) are said to be in dual combination combination multiswitching synchronization with response systems (7), (8), (10), and (11).

6

Remark 2. If A1 = B1 = C1 = D1 = 0, or A2 = B2 = C2 = D2 = 0, thendual combination combination multi switching synchronization changes tomulti switching combination combination synchronization problem of chaoticsystems.

Remark 3. If C1 = C2 = 0, or D1 = D2 = 0, then dual combinationcombination multi switching synchronization changes to dual combinationmulti switching synchronization of chaotic systems.

Remark 4. If A1 = B1 = C1 = D1 = 0, and C2 = 0 or D2 = 0, orA2 = B2 = C2 = D2 = 0, and C1 = 0 or D1 = 0, then dual combina-tion combination multi switching synchronization changes to multi switchingcombination synchronization of chaotic systems.

Remark 5. Using suitable values for the scaling factors A1, A2, B1, B2, C1,C2, D1, and D2, multi switching dual projective synchronization and multiswitching projective synchronization may also be obtained by the proposedscheme.

3. Synchronization Theory

To achieve the dual combination combination multi switching synchro-nization among four chaotic drive systems and four chaotic response systems,let the control functions be defined as{

U1m = a1if1i + b1jg1j − c1lh1l − d1mk1m + e1m(ijlm), (i, j, l,m = 1, 2, ..., n)

U2m = a2if2i + b2jg2j − c2lh2l − d2mk2m + e2m(ijlm), (i, j, l,m = 1, 2, ..., n)

(17)where

{

U1m = c1lu1l + d1mu3m, (l, m = 1, 2, ..., n)

U2m = c2lu2l + d2mu4m, (l, m = 1, 2, ..., n)(18)

and f1 = (f11, f12, ..., f1n)T , f2 = (f21, f22, ..., f2n)

T , g1 = (g11, g12, ..., g1n)T ,

g2 = (g21, g22, ..., g2n)T , h1 = (h11, h12, ..., h1n)

T , h2 = (h21, h22, ..., h2n)T , k1 =

(k11, k12, ..., k1n)T , and k2 = (k21, k22, ..., k2n)

T .

Theorem 1. The drive systems (1), (2), (4), and (5) achieve dual combina-tion combination multi switching synchronization with response systems (7),(8), (10), and (11) if the control functions are chosen as given in (17).

7

Proof. Using (13) the error dynamical system can be written as

e =

[

e1e2

]

=

[

A1x1 +B1y1 − C1z1 −D1w1

A2x2 +B2y2 − C2z2 −D2w2

]

(19)

which can be further written as[

e1e2

]

=

[

A1f1 +B1g1 − C1(h1 + u1)−D1(k1 + u3)A2f2 +B2g2 − C2(h2 + u2)−D2(k2 + u4)

]

(20)

From this we obtain

e1m(ijlm)=a1if1i + b1jg1j − c1l(h1l + u1l)− d1m(k1m + u3m),

(i, j, l,m = 1, 2, ..., n)

e2m(ijlm)=a2if2i + b2jg2j − c2l(h2l + u2l)− d2m(k2m + u4m),

(i, j, l,m = 1, 2, ..., n)

(21)

where the indices (ijlm) satisfies one of the generic conditions given in Defi-nition 2.Let the Lyapunov function be defined as

V =1

2eT e

=1

2

n∑

m=1

(e1m(ijlm))2 +

1

2

n∑

m=1

(e2m(ijlm))2

The derivative V is obtained as

V =

n∑

m=1

e1m(ijlm)e1m(ijlm)

+

n∑

m=2

e2m(ijlm)e2m(ijlm)

(22)

Using (17) and (21) in the above equation we get

V =

n∑

m=1

e1m(ijlm)[a1if1i + b1jg1j − c1lh1l − d1mk1m − U1m]

+

n∑

m=1

e2m(ijlm)[a2if2i + b2jg2j − c2lh2l − d2mk2m − U2m]

=n

∑

m=1

e1m(ijlm)(−e1m(ijlm)

) +n

∑

m=1

e2m(ijlm)(−e2m(ijlm)

) (Using(17))

8

=− eT e

Thus we see that V is negative definite. Using Lyapunov stability theory, weget limt→∞ ‖e‖ = 0, which gives us limt→∞ ‖e1‖ = 0 and limt→∞ ‖e2‖ = 0.This means that the drive systems (1), (2), (4), and (5) achieve dual combi-nation combination multi switching synchronization with response systems(7), (8), (10), and (11).

The following corollaries are easily obtained from Theorem 1 and theirproofs are omitted here.

Corollary 1. (i) If a1i = b1j = c1l = d1m = 0, i, j, l,m = 1, 2, ..., n then thedrive systems (2) and (5) achieve multi switching combination combinationsynchronization with the response systems (8) and (11) provided the controlfunction is chosen as

U2m = a2if2i + b2jg2j − c2lh2l − d2mk2m + e2m(ijlm), (i, j, l,m = 1, 2, ..., n)

(ii) If a2i = b2j = c2l = d2m = 0, i, j, l,m = 1, 2, ..., n then the drive systems(1) and (4) achieve multi switching combination combination synchronizationwith the response systems (7) and (10) provided the control function is chosenas

U2m = a2if2i + b2jg2j − c2lh2l − d2mk2m + e2m(ijlm), (i, j, l,m = 1, 2, ..., n)

Corollary 2. (i) If c1l = c2l = 0, l = 1, 2, ..., n then the drive systems (1),(2), (4) and (5) achieve dual combination multi switching synchronizationwith the response systems (10) and (11) provided the control functions arechosen as

u3m =d−11ma1if1i + d−1

1mb1jg1j − k1m + d−11me1m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

u4m =d−12ma2if2i + d−1

2mb2jg2j − k2m + d−12me2m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

(ii) If d1m = d2m = 0, m = 1, 2, ..., n then the drive systems (1), (2), (4)and (5) achieve dual combination multi switching synchronization with theresponse systems (7) and (8) provided the control functions are chosen as

u1l =c−11l a1if1i + c−1

1l b1jg1j − h1l + c−11l e1m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

u2l =c−12l a2if2i + c−1

2l b2jg2j − h2l + c−12l e2m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

9

Corollary 3. (i) If a1i = b1j = c1l = d1m = 0, and c2m = 0, i, j, l,m =1, 2, ..., n then the drive systems (2) and (5) achieve multi switching com-bination synchronization with the response system (11) provided the controlfunction is chosen as

u4m = d−12ma2if2i + d−1

2mb2jg2j − k2m + d−12me2m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

(ii) If a1i = b1j = c1l = d1m = 0, and d2m = 0, i, j, l,m = 1, 2, ..., n thenthe drive systems (2) and (5) achieve multi switching combination synchro-nization with the response system (8) provided the control function is chosenas

u2l = c−12l a2if2i + c−1

2l b2jg2j − h2l + c−12l e2m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

(iii) If a2i = b2j = c2l = d2m = 0, and c1m = 0, i, j, l,m = 1, 2, ..., n then thedrive systems (1) and (4) achieve multi switching combination synchroniza-tion with the response system (10) provided the control function is chosenas

u3m = d−11ma1if1i + d−1

1mb1jg1j − k1m + d−11me1m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

(iv) If a2i = b2j = c2l = d2m = 0, and d1m = 0, i, j, l,m = 1, 2, ..., n thenthe drive systems (1) and (4) achieve multi switching combination synchro-nization with the response system (7) provided the control function is chosenas

u1l = c−11l a1if1i + c−1

1l b1jg1j − h1l + c−11l e1m(ijlm)

, (i, j, l,m = 1, 2, ..., n)

4. Illustration of the synchronization scheme

In this section we realize the dual combination combination multi switch-ing synchronization among eight chaotic systems and perform numerical sim-ulations to show the validity and effectiveness of the proposed scheme. Asan example we consider Genesio Tesi system and Lu system to demonstratethe method. let the first two drive systems be given as

x11 =x12

x12 =x13

x13 =− 6x11 − 2.92x12 − 1.2x13 + x211

(23)

10

x21 =36(x22 − x21)

x22 =− x21x23 + 20x22

x23 =x21x22 − 3x23

(24)

The next two drive systems are considered as

y11 =y12

y12 =y13

y13 =− 6y11 − 2.92y12 − 1.2y13 + y211

(25)

y21 =36(y22 − y21)

y22 =− y21y23 + 20y22

y23 =y21y22 − 3y23

(26)

The first two response system are described as

z11 =z12 + u11

z12 =z13 + u12

z13 =− 6z11 − 2.92z12 − 1.2z13 + z211 + u13

(27)

z21 =36(z22 − z21) + u21

z22 =− z21z23 + 20z22 + u22

z23 =z21z22 − 3z23 + u23

(28)

and the next two response systems are taken as

w11 =w12 + u31

w12 =w13 + u32

w13 =− 6w11 − 2.92w12 − 1.2w13 + w211 + u33

(29)

w21 =36(w22 − w21) + u41

w22 =− w21w23 + 20w22 + u42

w23 =w21w22 − 3w23 + u43

(30)

By the conditions on indices i, j, l,m = 1, 2, 3 stated in Definition 2, severalmulti switching combination exist for defining the error e = (e1, e2)

T . Wewill present results for one randomly selected error space vector combination

11

formed out of several possibilities. Let us define e1 = (e112131 , e121322 , e133213),and e2 = (e213221 , e221332 , e232113) where

e112131 =a12x12 + b11y11 − c13z13 − d11w11

e121322 =a11x11 + b13y13 − c12z12 − d12w12

e133213 =a13x13 + b12y12 − c11z11 − d13w13

e213221 =a23x23 + b22y22 − c22z22 − d21w21

e221332 =a21x21 + b23y23 − c23z23 − d22w22

e232113 =a22x22 + b21y21 − c21z21 − d23w23

under the assumption that A1 = diag(a11, a12, a13), A2 = diag(a21, a22, a23),B1 = diag(b11, b12, b13), B2 = diag(b21, b22, b23), C1 = diag(c11, c12, c13), C2 =diag(c21, c22, c23), D1 = diag(d11, d12, d13), and D2 = diag(d21, d22, d23). As-suming A1 = A2 = B1 = B2 = C1 = C2 = D1 = D2 = I, the controllers arechosen as

U11 =x13 + y12 − (−6z11 − 2.92z12 − 1.2z13 + z211)− w12 + e112131

U12 =x12 + (−6y11 − 2.92y12 − 1.2y13 + y211)− z13 − w13 + e121322

U13 =(−6x11 − 2.92x12 − 1.2x13 + x211) + y13 − z12

− (−6w11 − 2.92w12 − 1.2w13 + w211) + e133213

(31)

U21 =x21x22 − 3x23 + (−y21y23 + 20y22)− (−z21z23 + 20z22)− 36(w22 − w21) + e213221

U22 =36(x22 − x21) + (y21y22 − 3y23)− (z21z22 − 3z23)− (−w21w23 + 20w22) + e221332

U23 =(−x21x23 + 20x22) + 36(y22 − y21)− 36(z22 − z21)− (w21w22 − 3w23) + e232113(32)

where U11 = u13 + u31, U12 = u12 + u32, U13 = u11 + u33, U21 = u22 + u41,U22 = u23 + u42, and U23 = u21 + u43.

These controllers (31) and (32) are designed in accordance with Theorem1 in order to realise the desired synchronization. In the numerical simulationsprocess the initial conditions of the drive and response systems are chosenas (x11, x12, x13) = (2,−3, 1), (x21, x22, x23) = (−2.5, 1,−3), (y11, y12, y13) =(1, 0,−1), (y21, y22, y23) = (−1.5, 2, 1.5), (z11, z12, z13) = (4,−3.5, 3), (z21, z22, z23) =(−0.5, 1.5, 0), (w11, w12, w13) = (1,−1.5,−2), and (w21, w22, w23) = (−1, 1.5, 3).Figures (1) − (6) illustrates the time response of synchronized states andFigure (7) displays time response of synchronization errors. We can see thatthe desired dual combination combination multi switching synchronizationis achieved with the controllers we designed.

12

0 2 4 6 8 10 12 14 16 18 20t

-4

-2

0

2

4

6x

12+y

11

z13+w

11

Figure 1: Response for states x12 + y11 and z13 + w11 for drive systems (23), (25) andresponse systems (27), (29).

0 2 4 6 8 10 12 14 16 18 20t

-10

-5

0

5

10

15x

11+y

13

z12

+w12

Figure 2: Response for states x11 + y13 and z12 + w12 for drive systems (23), (25) andresponse systems (27), (29).

0 2 4 6 8 10 12 14 16 18 20t

-10

-5

0

5

10x

13+y

12

z11

+w13

Figure 3: Response for states x13 + y12 and z11 + w13 for drive systems (23), (25) andresponse systems (27), (29).

13

0 1 2 3 4 5 6 7 8 9 10t

-10

0

10

20

30

40

50x

23+y

22

z22

+w21

Figure 4: Response for states x23 + y22 and z22 + w21 for drive systems (24), (26) andresponse systems (28), (30).

0 1 2 3 4 5 6 7 8 9 10t

-20

0

20

40

60

80x

21+y

23

z23

+w22

Figure 5: Response for states x21 + y23 and z23 + w22 for drive systems (24), (26) andresponse systems (28), (30).

0 1 2 3 4 5 6 7 8 9 10t

-40

-20

0

20

40

60x

22+y

21

z21

+w23

Figure 6: Response for states x22 + y21 and z21 + w23 for drive systems (24), (26) andresponse systems (28), (30).

14

0 2 4 6 8 10 12 14 16 18 20t

-6

-4

-2

0

2

4

6e

11(2131)

e12

(1322)

e13

(3213)

e21

(3221)

e22

(1332)

e23

(2113)

Figure 7: Time response of synchronization errors

5. Conclusions

The main purpose of this paper is to propose a novel scheme for syn-chronization involving eight chaotic systems. The proposed scheme dualcombination combination multi switching synchronization achieves synchro-nization between four chaotic drive systems and four chaotic response systemsin a multi switching manner. The main advantages of this scheme can besummarised as

i) The complexity of signal achieved by multiple combination enhances thesecurity of transmitted signal, as the dynamic behaviour of resultant signalis so complex that it becomes very difficult, for the intruder, to separate theinformation signal from the transmitted signal. Thus in the context of securecommunication applications, this scheme may provide improved performanceand better resistance.

ii) The concept of multi switching in this scheme further strengthens theanti attack ability of the transmitted signals from drive systems because,for an intruder, determining the correct combination for error space vector isextremely difficult due to large number of possible synchronization directions.

iii) Several other synchronization schemes such as multi switching com-bination combination synchronization, dual combination multi switching syn-chronization, multi switching combination synchronization, dual multi switch-ing projective synchronization, multi switching projective synchronizationare obtained as special cases of dual combination combination multi switch-ing synchronization scheme.

iv) The proposed scheme theoretically guarantees a good control perfor-mance.

v) The significant outcome may form the basis of several future studies.Using Lyapunov stability theory, sufficient conditions are obtained for

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achieving dual combintaion combination multi switching synchronization.Numerical simulations has been demonstrated using four Genesio Tesi sys-tems and four Lu systems to show the effectiveness and validity of themethod. Using fractional order drive and response systems, or utilising thescheme to implement in secure communication applications, and designingthe controllers in the presence of uncertain factors and disturbances in thesystem are some interesting directions for future work.

Acknowledgements

The work of the third author is supported by the Senior Research Fel-lowship of Council of Scientific and Industrial Research, India(Grant no.09/045(1319)/2014-EMR-I).

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