Research Article On Full-State Hybrid Projective ...downloads.hindawi.com/archive/2014/983293.pdf · On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems

Research ArticleOn Full-State Hybrid Projective Synchronization ofGeneral Discrete Chaotic Systems

Adel Ouannas

LAMIS Laboratory Department of Mathematics and Computer Science University of Tebessa 12002 Tebessa Algeria

Correspondence should be addressed to Adel Ouannas ouannas adelyahoofr

Received 28 July 2014 Revised 27 September 2014 Accepted 28 September 2014 Published 14 October 2014

Academic Editor Ivo Petras

Copyright copy 2014 Adel OuannasThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problems of full-state hybrid projective synchronization (FSHPS) and inverse full-state hybrid projective synchronization(IFSHPS) for general discrete chaotic systems are investigated in 2D Based on nonlinear control method and Lyapunov stabilitytheory new controllers are designed to study FSHPS and IFSHPS respectively for 2D arbitrary chaotic systems in discrete-timeNumerical example and simulations are used to validate the main results of this paper

1 Introduction

Many mathematical models of biological processes physicalprocesses and chemical processes etc were dened usingchaotic dynamical systems in discrete-time Recently moreand more attention were paid to chaos synchronizationin discrete-time dynamical systems [1ndash3] Synchronizationof discrete chaotic dynamical systems has also potentialapplications in secure communication [4 5] Since the workof Pecora and Carroll [6] various powerful methods andtechniques have been proposed to investigate chaos synchro-nization in dynamical systems [7 8] and different types ofsynchronization have been reported [9ndash11]

Recently a novel type of synchronization known as full-state hybrid projective synchronization (FSHPS) has beenintroduced and applied to chaotic systems in continuous-time [12] which includes projective synchronization (PS) andhybrid projective synchronization (HPS) In FSHPS eachresponse system state synchronizes with a linear combinationof drive system states By the same procedure we candefine a new type of synchronization called inverse full-state hybrid projective synchronization (IFSHPS) when eachdrive system state synchronizes with a linear combination ofresponse system states

In this paper based on nonlinear control method in2D and discrete-time Lyapunov stability theory firstly anew synchronization controller is designed for full-statehybrid projective synchronization (FSHPS) of general chaotic

systems Secondly a new control scheme is proposed tostudy the problem of inverse full-state hybrid projectivesynchronization (IFSHPS) for arbitrary chaotic systems Thesynchronization criterions derived in this paper are estab-lished in the form of simple algebraic conditions about thelinear part of the response system and the drive systemrespectively which are very convenient to verify In orderto show the effectiveness of the proposed synchronizationschemes our approach is applied to the drive Fold discrete-time system and the controlled Lorenz discrete-time systemto achieve FSHPS and IFSHPS respectively

The remainder of this paper is organized as follows InSection 2 definitions of FSHPS and IFSHPS are introducedIn Section 3 a new controller is designed to study FSHPSIn Section 4 a new synchronization criterion for IFSHPS isderived In Section 5 numerical application and simulationsare given to show the use of the proposed control schemes andthe derived synchronization criterions Finally conclusion isfollowed in Section 6

2 Definitions of FSHPS and IFSHPS

Consider the following coupled chaotic systems

119909119894 (119896 + 1) = 119891

119894 (119883 (119896))

119910119894 (119896 + 1) = 119892

119894 (119884 (119896)) + 119906

119894

1 le 119894 le 119899

(1)

Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2014 Article ID 983293 6 pageshttpdxdoiorg1011552014983293

2 Journal of Nonlinear Dynamics

where 119883(119896) = (119909119894(119896))1le119894le119899

119884(119896) = (119910119894(119896))1le119894le119899

are the statevectors of drive system and response system respectively 119891

119894

R119899 rarr R 119892119894 R119899 rarr R (1 le 119894 le 119899) and 119906

119894 (1 le 119894 le 119899) are

controllersWe present the definition of full-state hybrid projective

synchronization (FSHPS) for the drive-response chaotic sys-tems (1)

Definition 1 The coupled drive-response chaotic systems (1)are in full-state hybrid projective synchronization (FSHPS)when for an initial condition there exist controllers 119906

119894 (1 le

119894 le 119899) and given real constants (120579119894119895) isin R119899times119899 such that the

synchronization errors

119890119894 (119896) = 119910

119894 (119896) minus

119899

sum

119894=1

120579119894119895119909119895 (119896) 1 le 119894 le 119899 (2)

satisfy that lim119896rarrinfin

119890119894 (119896) = 0 for 119894 = 1 2 119899

The definition of inverse full-state hybrid projectivesynchronization (IFSHPS) is given next

Definition 2 The coupled drive-response chaotic systems (1)are in inverse full-state hybrid projective synchronization(IFSHPS) when for an initial condition there exist con-trollers (119906

119894)1le119894le119899

and given real constant (120579119894119895) isin R119899times119899 such

that the synchronization errors

119890119894 (119896) = 119909

119894 (119896) minus

119899

sum

119894=1

120579119894119895119910119895 (119896) 1 le 119894 le 119899 (3)


119890119894 (119896) = 0 for 119894 = 1 2 119899

3 Controller Design for FSHPS in 2D

In this section we consider the drive system in the followingform

1199091 (119896 + 1) = 119891

1 (119883 (119896))

1199092 (119896 + 1) = 119891

2 (119883 (119896))

(4)

where119883 (119896) = (1199091 (119896) 1199092 (

119896))119879 is the state vector of the drive

system (4) 119891119894 R2 rarr R (119894 = 1 2)

As the response system we consider the following chaoticsystem

1199101 (119896 + 1) =

2

sum

119895=1

1198871119895119910119895 (119896) + 119892

1 (119884 (119896)) + 119906

1

1199102 (119896 + 1) =

2

sum

119895=1

1198872119895119910119895 (119896) + 119892

2 (119884 (119896)) + 119906

2

(5)

where 119884 (119896) = (1199101 (119896) 1199102 (

119896))119879 is the state vector of the

response system (5) (119887119894119895) isin R2times2 119892

119894 R2 rarr R (119894 = 1 2) are

nonlinear functions and (1199061 1199062)119879 is the vector controller

According to the definition of FSHPS the synchroniza-tion errors between the drive system (4) and the responsesystem (5) are defined as

1198901 (119896 + 1) = 119910

1 (119896 + 1) minus 120579

111199091 (119896 + 1) minus 120579

121199092 (119896 + 1)

1198902 (119896 + 1) = 119910

2 (119896 + 1) minus 120579

211199091 (119896 + 1) minus 120579

221199092 (119896 + 1)

(6)

where (120579119894119895) isin R2times2 are arbitrary scaling constants

Then the synchronization errors of FSHPS betweensystems (4) and (5) can be derived as

1198901 (119896 + 1) =

2

sum

119895=1

1198871119895119890119895 (119896) + 119871

1+ 1198731+ 1199061

1198902 (119896 + 1) =

2

sum

119895=1

1198872119895119890119895 (119896) + 119871

2+ 1198732+ 1199062

(7)

where

1198711=

2

sum

119895=1

1205821119895119909119895 (119896)

1198712=

2

sum

119895=1

1205822119895119909119895 (119896)

(8)

where

12058211= 1198871112057911+ 1198871212057921

12058212= 1198871112057912+ 1198871212057922

12058221= 1198872112057911+ 1198872212057921

12058222= 1198872112057912+ 1198872212057922

1198731= 1198921minus 120579111198911minus 120579121198912

1198732= 1198922minus 120579211198911minus 120579221198912

(9)

To achieve FSHPS between systems (4) and (5) we choosethe synchronization controllers as

1199061= (11988722minus 1198971) 1198901 (119896) + (119887

21minus 1198972) 1198902 (119896) minus 119871

1minus 1198731

1199062= (11988712minus 1198972) 1198901 (119896) minus (2119887

22+ 11988711minus 1198972) 1198902 (119896)

minus 1198712minus 1198732

(10)

where 1198971and 1198972are control constants to be determined

Theorem 3 If the control constants (119897119894)1le119894le2

are chosen suchthat

(11988711+ 11988722minus 1198971)2+ (11988721+ 11988712minus 1198972)2lt 1 (11)

then the drive system (4) and the response (5) are globally full-state hybrid projective synchronized

Journal of Nonlinear Dynamics 3

Proof By substituting the control law (10) into (7) thesynchronization errors can be written as

1198901 (119896 + 1) = (119887

11+ 11988722minus 1198971) 1198901 (119896)

+ (11988712+ 11988721minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119887

21+ 11988712minus 1198972) 1198901 (119896)

minus (11988722+ 11988711minus 1198971) 1198902 (119896)

(12)

Construct the candidate Lyapunov function in the form

119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (13)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)

= ((11988711+ 11988722minus 1198971)2+ (11988721+ 11988712minus 1198972)2minus 1)

sdot (1198902

1(119896) + 119890

2

2(119896))

+ 2 [(11988711+ 11988722minus 1198971) (11988712+ 11988722minus 1198972)

minus (11988721+ 11988712minus 1198972) (11988711+ 11988722minus 1198971)]

sdot 1198901 (119896) 1198902 (

119896)

(14)

By using (11) we obtain Δ119881 (119890 (119896)) lt 0 Thus by Lya-punov stability theory it is immediate that lim

119896rarrinfin119890119894(119896) =

0 (119894 = 1 2) and we conclude that the two systems (4) and (5)are globally full-state hybrid projective synchronized

4 New Criterion for IFSHPS in 2D

Now we consider the drive and the response chaotic systemsare in the following forms

119909119894 (119896 + 1) =

2

sum

119895=1

119886119894119895119909119895 (119896) + 119891

119894 (119883 (119896)) 1 le 119894 le 2 (15)

119910119894 (119896 + 1) = 119892

119894 (119884 (119896)) + 119906

119894 1 le 119894 le 2 (16)

where 119883(119896) = (1199091(119896) 1199092(119896))119879 119884 (119896) = (119910

1(119896) 1199102(119896))119879 are

the state vectors of drive system and the response systemsrespectively (119886

119894119895) isin R2times2 119891

119894 R2 rarr R (119894 = 1 2) are

nonlinear functions 119892119894 R2 rarr R (119894 = 1 2) and (119906

119894)1le119894le2

are synchronization controllersAccording to the definition of IFSHPS the synchroniza-

tion errors between the drive system (15) and the responsesystem (16) are defined as

1198901 (119896 + 1) = 119909

1 (119896 + 1) minus 120579

111199101 (119896 + 1) minus 120579

121199102 (119896 + 1)

1198902 (119896 + 1) = 119909

2 (119896 + 1) minus 120579

211199101 (119896 + 1) minus 120579

221199102 (119896 + 1)

(17)

where (120579119894119895) isin R2times2 are scaling constants

Then the synchronization errors of IFSHPS betweensystems (15) and (16) can be derived as

1198901 (119896 + 1) = (119886

21minus 1198971) 1198901 (119896) minus (119886

22+ 211988612minus 1198972) 1198902 (119896)

+ 1198771minus

2

sum

119895=1

1205791119895119906119895

1198902 (119896 + 1) = (119886

11minus 1198971) 1198901 (119896) + (119886

12minus 1198972) 1198902 (119896) + 119877

2

minus

2

sum

119895=1

1205792119895119906119895

(18)

where (119897119894)1le119894le2

are control constants and

1198771=

2

sum

119895=1

1205821119895119910119895 (119896) + 119891

1minus 120579111198921minus 120579121198922

1198772=

2

sum

119895=1

1205822119895119910119895 (119896) + 119891

2minus 120579211198921minus 120579221198922

(19)

where12058211= 1198861112057911+ 1198861212057921

12058212= 1198861112057912+ 1198861212057922

12058221= 1198862112057911+ 1198862212057921

12058222= 1198862112057912+ 1198862212057922

(20)

To achieve IFSHPS between systems (15) and (16) weassume that

1205791112057922

= 1205791212057921 (21)

and we choose the synchronization controllers as

1199061=

120579221198771minus 120579121198772

1205791212057921minus 1205791112057922

1199062=

120579111198772minus 120579211198771

1205791212057921minus 1205791112057922

(22)


are chosen suchthat

100381610038161003816100381611988611+ 11988621minus 1198971

1003816100381610038161003816lt

1

radic2

100381610038161003816100381611988622+ 11988612minus 1198972

1003816100381610038161003816lt

1

radic2

(23)

then the drive system (15) and the response (16) are globallyinverse full-state hybrid projective synchronized

Proof By substituting control law (22) into (18) the synchro-nization errors can be written as

1198901 (119896 + 1) = (119886

11+ 11988621minus 1198971) 1198901 (119896)

minus (11988612+ 11988622minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119886

21+ 11988611minus 1198971) 1198901 (119896)

+ (11988622+ 11988612minus 1198972) 1198902 (119896)

(24)



119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (25)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)

= (2(11988611+ 11988621minus 1198971)2minus 1) 119890

2

1(119896)

+ (2(11988622+ 11988612minus 1198972)2minus 1) 119890

2

1(119896)

+ 2 [(11988611+ 11988621minus 1198971) (11988622+ 11988612minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(26)


119896rarrinfin119890119894(119896) =

0 (119894 = 1 2) and we conclude that the two systems (15) and(16) are globally inverse full-state hybrid projective synchro-nized

5 Numerical Application and Simulations

We consider the discrete-time Fold system system as thedrive system and the controlled Lorenz discrete-time asthe response system The discrete-time Fold system can bedescribed as

1199091 (119896 + 1) = 120572119909

1 (119896) + 119909

2 (119896)

1199092 (119896 + 1) = 119909

2

1(119896) + 120573

(27)

which has a chaotic attractor for example when (120572 120573) =

(minus01minus17) [13] The discrete-time Fold system is shown inFigure 1

The controlled Lorenz discrete-time system can bedescribed as

1199101 (119896 + 1) = (1 + 119886119887) 1199101 (

119896) minus 1198871199101 (119896) 1199102 (

119896) + 1199061

1199102 (119896 + 1) = (1 minus 119887) 1199102 (

119896) + 1198871199102

1(119896) + 119906

2

(28)

where 119880 = (1199061 1199062)119879 is the vector controller The Lorenz

discrete-time has a chaotic attractor for example when(119886 119887) = (125 075) [13] The chaotic attractor of the Lorenzdiscrete-time system is shown in Figure 2

1

05

0

minus05

minus1

minus15

151050minus05minus1minus15

y1

y2

Figure 1 Chaotic attractor of Fold discrete-time system when(120572 120573) = (minus01 minus17)

4

3

2

1

0

210minus1minus2

y1

y2

Figure 2 Chaotic attractor of Lorenz discrete-time system when(119886 119887) = (125 075)

51 FSHPS between the Discrete-Time Fold System and theControlled Lorenz Discrete-Time System Her according toour approach presented in Section 3 we obtain

(

11988711

11988712

11988721

11988722

) = (

1 + 119886119887 0

0 1 minus 119887)

1198901 (119896 + 1) = [2 + (119886 minus 1) 119887 minus 119897

1] 1198901 (119896) minus 11989721198902 (119896)

1198902 (119896 + 1) = minus119897

21198901 (119896) minus [2 + (119886 minus 1) 119887 minus 119897

1] 1198902 (119896)

(29)


5 10 15 20 25

0

02

04

06

minus08

minus06

minus04

minus02

e1

e2

Figure 3 Time evolution of FSHPS errors between the drivediscrete-time Fold system (27) and the controlled Lorenz discrete-time system (28)

Corollary 5 The drive Fold discrete-time system and theresponse Lorenz discrete-time system are globally full-statehybrid projective synchronized if the control constants arechosen such that

(218 minus 1198971)2+ 1198972

2lt 1 (30)

If we take (1198971 1198972) = (2 05) and by using Matlab we get

the numeric result that is shown in Figure 3

52 Inverse FSHPS between the Discrete-Time Fold System andthe Controlled Lorenz Discrete-Time System Now accordingto our approach presented in Section 4 we obtain

(

11988611

11988612

11988621

11988622

) = (

120572 1

0 0)

1198901 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) minus (1 minus 119897

2) 1198902 (119896)

1198902 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) + (1 minus 119897

2) 1198902 (119896)

(31)

Corollary 6 The drive Fold discrete-time system and theresponse Lorenz discrete-time system are globally inverse full-state hybrid projective synchronized if the control constants arechosen such that

minus08 lt 1198971lt 06

03 lt 1198972lt 17

(32)

Finally if we take (1198971 1198972) = (05 1) and by using Matlab

we get the numeric result that is shown in Figure 4

6 Conclusion

In this paper to study FSHPS and IFSHPS between arbi-trary chaotic dynamical systems in 2D discrete-time a new

e1

e2

03

02

01

0

minus01

minus02

minus03

minus04

minus051110987654321

Figure 4 Time evolution of inverse FSHPS errors between the drivediscrete-time Fold system (27) and the controlled Lorenz discrete-time system (28)

nonlinear control method was presented It was shownthat the proposed synchronization criterions were based onsimple and effective results Firstly FSHPS is achieved bycontrolling the linear part of the response system SecondlyIFSHPS is guaranteedwhen the linear part of the drive systemis controlled Finally numerical example and simulationsresults were utilized to illustrate the effectiveness of theproposed schemes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W Liu Z M Wang and W D Zhang ldquoControlled synchro-nization of discrete-time chaotic systems under communicationconstraintsrdquo Nonlinear Dynamics vol 69 no 1-2 pp 223ndash2302012

[2] G Grassi ldquoArbitrary full-state hybrid projective synchroniza-tion for chaotic discrete-time systems via a scalar signalrdquoChinese Physics B vol 21 no 6 Article ID 060504 2012

[3] Z Yan ldquoQ-S synchronization in 3D Henon-like map andgeneralized Henon map via a scalar controllerrdquo Physics LettersA vol 342 no 4 pp 309ndash317 2005

[4] J G Lu and Y G Xi ldquoChaos communication based on synchro-nization of discrete-time chaotic systemsrdquo Chinese Physics vol14 no 2 pp 274ndash278 2005

[5] E Solak ldquoCryptanalysis of observer based discrete-time chaoticencryption schemesrdquo International Journal of Bifurcation andChaos inApplied Sciences andEngineering vol 15 no 2 pp 653ndash658 2005

[6] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990


[7] W Xiao-Qun and L Jun-An ldquoParameter identification andbackstepping control of uncertain Lu systemrdquo Chaos Solitonsand Fractals vol 18 no 4 pp 721ndash729 2003

[8] C-C Yang andC-L Lin ldquoRobust adaptive slidingmode controlfor synchronization of space-clamped FitzHugh-Nagumo neu-ronsrdquo Nonlinear Dynamics vol 69 no 4 pp 2089ndash2096 2012

[9] T Banerjee D Biswas and B C Sarkar ldquoComplete andgeneralized synchronization of chaos and hyperchaos in acoupled first-order time-delayed systemrdquo Nonlinear Dynamicsvol 71 no 1-2 pp 279ndash290 2013

[10] MMAl-Sawalha andM SMNoorani ldquoAnti-synchronizationbetween two different hyperchaotic systemsrdquo Journal of Uncer-tain Systems vol 3 no 3 pp 192ndash200 2009

[11] A Khan and R P Prasad ldquoProjective synchronization ofdifferent hyper-chaotic systems by active nonlinear controlrdquoJournal of Uncertain Systems vol 8 no 2 pp 90ndash100 2014

[12] M Hu Z Xu and R Zhang ldquoFull state hybrid projectivesynchronization in continuous-time chaotic (hyperchaotic)systemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 13 no 2 pp 456ndash464 2008

[13] Z Yan ldquoQ-S (complete or anticipated) synchronization back-stepping scheme in a class of discrete-time chaotic (hyper-chaotic) systems a symbolic-numeric computation approachrdquoChaos vol 16 no 1 Article ID 013119 11 pages 2006

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where 119883(119896) = (119909119894(119896))1le119894le119899

119884(119896) = (119910119894(119896))1le119894le119899

are the statevectors of drive system and response system respectively 119891

119894

R119899 rarr R 119892119894 R119899 rarr R (1 le 119894 le 119899) and 119906

119894 (1 le 119894 le 119899) are

controllersWe present the definition of full-state hybrid projective

synchronization (FSHPS) for the drive-response chaotic sys-tems (1)

Definition 1 The coupled drive-response chaotic systems (1)are in full-state hybrid projective synchronization (FSHPS)when for an initial condition there exist controllers 119906

119894 (1 le

119894 le 119899) and given real constants (120579119894119895) isin R119899times119899 such that the

synchronization errors

119890119894 (119896) = 119910

119894 (119896) minus

119899

sum

119894=1

120579119894119895119909119895 (119896) 1 le 119894 le 119899 (2)


119890119894 (119896) = 0 for 119894 = 1 2 119899

The definition of inverse full-state hybrid projectivesynchronization (IFSHPS) is given next

Definition 2 The coupled drive-response chaotic systems (1)are in inverse full-state hybrid projective synchronization(IFSHPS) when for an initial condition there exist con-trollers (119906

119894)1le119894le119899

and given real constant (120579119894119895) isin R119899times119899 such

that the synchronization errors

119890119894 (119896) = 119909

119894 (119896) minus

119899

sum

119894=1

120579119894119895119910119895 (119896) 1 le 119894 le 119899 (3)


119890119894 (119896) = 0 for 119894 = 1 2 119899

3 Controller Design for FSHPS in 2D

In this section we consider the drive system in the followingform

1199091 (119896 + 1) = 119891

1 (119883 (119896))

1199092 (119896 + 1) = 119891

2 (119883 (119896))

(4)

where119883 (119896) = (1199091 (119896) 1199092 (

119896))119879 is the state vector of the drive

system (4) 119891119894 R2 rarr R (119894 = 1 2)

As the response system we consider the following chaoticsystem

1199101 (119896 + 1) =

2

sum

119895=1

1198871119895119910119895 (119896) + 119892

1 (119884 (119896)) + 119906

1

1199102 (119896 + 1) =

2

sum

119895=1

1198872119895119910119895 (119896) + 119892

2 (119884 (119896)) + 119906

2

(5)

where 119884 (119896) = (1199101 (119896) 1199102 (

119896))119879 is the state vector of the

response system (5) (119887119894119895) isin R2times2 119892

119894 R2 rarr R (119894 = 1 2) are

nonlinear functions and (1199061 1199062)119879 is the vector controller

According to the definition of FSHPS the synchroniza-tion errors between the drive system (4) and the responsesystem (5) are defined as

1198901 (119896 + 1) = 119910

1 (119896 + 1) minus 120579

111199091 (119896 + 1) minus 120579

121199092 (119896 + 1)

1198902 (119896 + 1) = 119910

2 (119896 + 1) minus 120579

211199091 (119896 + 1) minus 120579

221199092 (119896 + 1)

(6)

where (120579119894119895) isin R2times2 are arbitrary scaling constants

Then the synchronization errors of FSHPS betweensystems (4) and (5) can be derived as

1198901 (119896 + 1) =

2

sum

119895=1

1198871119895119890119895 (119896) + 119871

1+ 1198731+ 1199061

1198902 (119896 + 1) =

2

sum

119895=1

1198872119895119890119895 (119896) + 119871

2+ 1198732+ 1199062

(7)

where

1198711=

2

sum

119895=1

1205821119895119909119895 (119896)

1198712=

2

sum

119895=1

1205822119895119909119895 (119896)

(8)

where

12058211= 1198871112057911+ 1198871212057921

12058212= 1198871112057912+ 1198871212057922

12058221= 1198872112057911+ 1198872212057921

12058222= 1198872112057912+ 1198872212057922

1198731= 1198921minus 120579111198911minus 120579121198912

1198732= 1198922minus 120579211198911minus 120579221198912

(9)

To achieve FSHPS between systems (4) and (5) we choosethe synchronization controllers as

1199061= (11988722minus 1198971) 1198901 (119896) + (119887

21minus 1198972) 1198902 (119896) minus 119871

1minus 1198731

1199062= (11988712minus 1198972) 1198901 (119896) minus (2119887

22+ 11988711minus 1198972) 1198902 (119896)

minus 1198712minus 1198732

(10)

where 1198971and 1198972are control constants to be determined


are chosen suchthat

(11988711+ 11988722minus 1198971)2+ (11988721+ 11988712minus 1198972)2lt 1 (11)

then the drive system (4) and the response (5) are globally full-state hybrid projective synchronized



1198901 (119896 + 1) = (119887

11+ 11988722minus 1198971) 1198901 (119896)

+ (11988712+ 11988721minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119887

21+ 11988712minus 1198972) 1198901 (119896)

minus (11988722+ 11988711minus 1198971) 1198902 (119896)

(12)


119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (13)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)


sdot (1198902

1(119896) + 119890

2

2(119896))

+ 2 [(11988711+ 11988722minus 1198971) (11988712+ 11988722minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(14)


119896rarrinfin119890119894(119896) =




119909119894 (119896 + 1) =

2

sum

119895=1

119886119894119895119909119895 (119896) + 119891

119894 (119883 (119896)) 1 le 119894 le 2 (15)

119910119894 (119896 + 1) = 119892

119894 (119884 (119896)) + 119906

119894 1 le 119894 le 2 (16)

where 119883(119896) = (1199091(119896) 1199092(119896))119879 119884 (119896) = (119910

1(119896) 1199102(119896))119879 are


119894119895) isin R2times2 119891

119894 R2 rarr R (119894 = 1 2) are


119894)1le119894le2



1198901 (119896 + 1) = 119909

1 (119896 + 1) minus 120579

111199101 (119896 + 1) minus 120579

121199102 (119896 + 1)

1198902 (119896 + 1) = 119909

2 (119896 + 1) minus 120579

211199101 (119896 + 1) minus 120579

221199102 (119896 + 1)

(17)



1198901 (119896 + 1) = (119886

21minus 1198971) 1198901 (119896) minus (119886

22+ 211988612minus 1198972) 1198902 (119896)

+ 1198771minus

2

sum

119895=1

1205791119895119906119895

1198902 (119896 + 1) = (119886

11minus 1198971) 1198901 (119896) + (119886

12minus 1198972) 1198902 (119896) + 119877

2

minus

2

sum

119895=1

1205792119895119906119895

(18)

where (119897119894)1le119894le2


1198771=

2

sum

119895=1

1205821119895119910119895 (119896) + 119891

1minus 120579111198921minus 120579121198922

1198772=

2

sum

119895=1

1205822119895119910119895 (119896) + 119891

2minus 120579211198921minus 120579221198922

(19)

where12058211= 1198861112057911+ 1198861212057921

12058212= 1198861112057912+ 1198861212057922

12058221= 1198862112057911+ 1198862212057921

12058222= 1198862112057912+ 1198862212057922

(20)


1205791112057922

= 1205791212057921 (21)


1199061=

120579221198771minus 120579121198772

1205791212057921minus 1205791112057922

1199062=

120579111198772minus 120579211198771

1205791212057921minus 1205791112057922

(22)


are chosen suchthat

100381610038161003816100381611988611+ 11988621minus 1198971

1003816100381610038161003816lt

1

radic2

100381610038161003816100381611988622+ 11988612minus 1198972

1003816100381610038161003816lt

1

radic2

(23)



1198901 (119896 + 1) = (119886

11+ 11988621minus 1198971) 1198901 (119896)

minus (11988612+ 11988622minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119886

21+ 11988611minus 1198971) 1198901 (119896)

+ (11988622+ 11988612minus 1198972) 1198902 (119896)

(24)



119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (25)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)

= (2(11988611+ 11988621minus 1198971)2minus 1) 119890

2

1(119896)

+ (2(11988622+ 11988612minus 1198972)2minus 1) 119890

2

1(119896)

+ 2 [(11988611+ 11988621minus 1198971) (11988622+ 11988612minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(26)


119896rarrinfin119890119894(119896) =




1199091 (119896 + 1) = 120572119909

1 (119896) + 119909

2 (119896)

1199092 (119896 + 1) = 119909

2

1(119896) + 120573

(27)




1199101 (119896 + 1) = (1 + 119886119887) 1199101 (

119896) minus 1198871199101 (119896) 1199102 (

119896) + 1199061

1199102 (119896 + 1) = (1 minus 119887) 1199102 (

119896) + 1198871199102

1(119896) + 119906

2

(28)



1

05

0

minus05

minus1

minus15


y1

y2


4

3

2

1

0

210minus1minus2

y1

y2



(

11988711

11988712

11988721

11988722

) = (

1 + 119886119887 0

0 1 minus 119887)

1198901 (119896 + 1) = [2 + (119886 minus 1) 119887 minus 119897

1] 1198901 (119896) minus 11989721198902 (119896)

1198902 (119896 + 1) = minus119897


1] 1198902 (119896)

(29)


5 10 15 20 25

0

02

04

06

minus08

minus06

minus04

minus02

e1

e2



(218 minus 1198971)2+ 1198972

2lt 1 (30)




(

11988611

11988612

11988621

11988622

) = (

120572 1

0 0)

1198901 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) minus (1 minus 119897

2) 1198902 (119896)

1198902 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) + (1 minus 119897

2) 1198902 (119896)

(31)


minus08 lt 1198971lt 06

03 lt 1198972lt 17

(32)



6 Conclusion


e1

e2

03

02

01

0

minus01

minus02

minus03

minus04

minus051110987654321





References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198901 (119896 + 1) = (119887

11+ 11988722minus 1198971) 1198901 (119896)

+ (11988712+ 11988721minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119887

21+ 11988712minus 1198972) 1198901 (119896)

minus (11988722+ 11988711minus 1198971) 1198902 (119896)

(12)


119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (13)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)


sdot (1198902

1(119896) + 119890

2

2(119896))

+ 2 [(11988711+ 11988722minus 1198971) (11988712+ 11988722minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(14)


119896rarrinfin119890119894(119896) =




119909119894 (119896 + 1) =

2

sum

119895=1

119886119894119895119909119895 (119896) + 119891

119894 (119883 (119896)) 1 le 119894 le 2 (15)

119910119894 (119896 + 1) = 119892

119894 (119884 (119896)) + 119906

119894 1 le 119894 le 2 (16)

where 119883(119896) = (1199091(119896) 1199092(119896))119879 119884 (119896) = (119910

1(119896) 1199102(119896))119879 are


119894119895) isin R2times2 119891

119894 R2 rarr R (119894 = 1 2) are


119894)1le119894le2



1198901 (119896 + 1) = 119909

1 (119896 + 1) minus 120579

111199101 (119896 + 1) minus 120579

121199102 (119896 + 1)

1198902 (119896 + 1) = 119909

2 (119896 + 1) minus 120579

211199101 (119896 + 1) minus 120579

221199102 (119896 + 1)

(17)



1198901 (119896 + 1) = (119886

21minus 1198971) 1198901 (119896) minus (119886

22+ 211988612minus 1198972) 1198902 (119896)

+ 1198771minus

2

sum

119895=1

1205791119895119906119895

1198902 (119896 + 1) = (119886

11minus 1198971) 1198901 (119896) + (119886

12minus 1198972) 1198902 (119896) + 119877

2

minus

2

sum

119895=1

1205792119895119906119895

(18)

where (119897119894)1le119894le2


1198771=

2

sum

119895=1

1205821119895119910119895 (119896) + 119891

1minus 120579111198921minus 120579121198922

1198772=

2

sum

119895=1

1205822119895119910119895 (119896) + 119891

2minus 120579211198921minus 120579221198922

(19)

where12058211= 1198861112057911+ 1198861212057921

12058212= 1198861112057912+ 1198861212057922

12058221= 1198862112057911+ 1198862212057921

12058222= 1198862112057912+ 1198862212057922

(20)


1205791112057922

= 1205791212057921 (21)


1199061=

120579221198771minus 120579121198772

1205791212057921minus 1205791112057922

1199062=

120579111198772minus 120579211198771

1205791212057921minus 1205791112057922

(22)


are chosen suchthat

100381610038161003816100381611988611+ 11988621minus 1198971

1003816100381610038161003816lt

1

radic2

100381610038161003816100381611988622+ 11988612minus 1198972

1003816100381610038161003816lt

1

radic2

(23)



1198901 (119896 + 1) = (119886

11+ 11988621minus 1198971) 1198901 (119896)

minus (11988612+ 11988622minus 1198972) 1198902 (119896)

1198902 (119896 + 1) = (119886

21+ 11988611minus 1198971) 1198901 (119896)

+ (11988622+ 11988612minus 1198972) 1198902 (119896)

(24)



119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (25)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)

= (2(11988611+ 11988621minus 1198971)2minus 1) 119890

2

1(119896)

+ (2(11988622+ 11988612minus 1198972)2minus 1) 119890

2

1(119896)

+ 2 [(11988611+ 11988621minus 1198971) (11988622+ 11988612minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(26)


119896rarrinfin119890119894(119896) =




1199091 (119896 + 1) = 120572119909

1 (119896) + 119909

2 (119896)

1199092 (119896 + 1) = 119909

2

1(119896) + 120573

(27)




1199101 (119896 + 1) = (1 + 119886119887) 1199101 (

119896) minus 1198871199101 (119896) 1199102 (

119896) + 1199061

1199102 (119896 + 1) = (1 minus 119887) 1199102 (

119896) + 1198871199102

1(119896) + 119906

2

(28)



1

05

0

minus05

minus1

minus15


y1

y2


4

3

2

1

0

210minus1minus2

y1

y2



(

11988711

11988712

11988721

11988722

) = (

1 + 119886119887 0

0 1 minus 119887)

1198901 (119896 + 1) = [2 + (119886 minus 1) 119887 minus 119897

1] 1198901 (119896) minus 11989721198902 (119896)

1198902 (119896 + 1) = minus119897


1] 1198902 (119896)

(29)


5 10 15 20 25

0

02

04

06

minus08

minus06

minus04

minus02

e1

e2



(218 minus 1198971)2+ 1198972

2lt 1 (30)




(

11988611

11988612

11988621

11988622

) = (

120572 1

0 0)

1198901 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) minus (1 minus 119897

2) 1198902 (119896)

1198902 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) + (1 minus 119897

2) 1198902 (119896)

(31)


minus08 lt 1198971lt 06

03 lt 1198972lt 17

(32)



6 Conclusion


e1

e2

03

02

01

0

minus01

minus02

minus03

minus04

minus051110987654321





References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119881 (119890 (119896)) = 1198902

1(119896) + 119890

2

2(119896) (25)

we get

Δ119881 (119890 (119896))

= 119881 (119890 (119896 + 1)) minus 119881 (119890 (119896))

=

2

sum

119894=1

1198902

119894(119896 + 1) minus

2

sum

119894=1

1198902

119894(119896)

= (2(11988611+ 11988621minus 1198971)2minus 1) 119890

2

1(119896)

+ (2(11988622+ 11988612minus 1198972)2minus 1) 119890

2

1(119896)

+ 2 [(11988611+ 11988621minus 1198971) (11988622+ 11988612minus 1198972)


sdot 1198901 (119896) 1198902 (

119896)

(26)


119896rarrinfin119890119894(119896) =




1199091 (119896 + 1) = 120572119909

1 (119896) + 119909

2 (119896)

1199092 (119896 + 1) = 119909

2

1(119896) + 120573

(27)




1199101 (119896 + 1) = (1 + 119886119887) 1199101 (

119896) minus 1198871199101 (119896) 1199102 (

119896) + 1199061

1199102 (119896 + 1) = (1 minus 119887) 1199102 (

119896) + 1198871199102

1(119896) + 119906

2

(28)



1

05

0

minus05

minus1

minus15


y1

y2


4

3

2

1

0

210minus1minus2

y1

y2



(

11988711

11988712

11988721

11988722

) = (

1 + 119886119887 0

0 1 minus 119887)

1198901 (119896 + 1) = [2 + (119886 minus 1) 119887 minus 119897

1] 1198901 (119896) minus 11989721198902 (119896)

1198902 (119896 + 1) = minus119897


1] 1198902 (119896)

(29)


5 10 15 20 25

0

02

04

06

minus08

minus06

minus04

minus02

e1

e2



(218 minus 1198971)2+ 1198972

2lt 1 (30)




(

11988611

11988612

11988621

11988622

) = (

120572 1

0 0)

1198901 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) minus (1 minus 119897

2) 1198902 (119896)

1198902 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) + (1 minus 119897

2) 1198902 (119896)

(31)


minus08 lt 1198971lt 06

03 lt 1198972lt 17

(32)



6 Conclusion


e1

e2

03

02

01

0

minus01

minus02

minus03

minus04

minus051110987654321





References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 10 15 20 25

0

02

04

06

minus08

minus06

minus04

minus02

e1

e2



(218 minus 1198971)2+ 1198972

2lt 1 (30)




(

11988611

11988612

11988621

11988622

) = (

120572 1

0 0)

1198901 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) minus (1 minus 119897

2) 1198902 (119896)

1198902 (119896 + 1) = (120572 minus 119897

1) 1198901 (119896) + (1 minus 119897

2) 1198902 (119896)

(31)


minus08 lt 1198971lt 06

03 lt 1198972lt 17

(32)



6 Conclusion


e1

e2

03

02

01

0

minus01

minus02

minus03

minus04

minus051110987654321





References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article On Full-State Hybrid Projective ...downloads.hindawi.com/archive/2014/983293.pdf · On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems