Synchronization of dynamical systems of different orders and different dimensions

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https://doi.org/10.26637/MJM0602/0009

Synchronization of dynamical systems of different

orders and different dimensions

Ayub Khan

1

, Mridula Budhraja

2

and Aysha Ibraheem

3

*

Abstract

A method of tracking control is proposed to achieve synchronization between the systems of fractional order and integer order. This article presents two cases of synchronization, in first case synchronization for three dimensional integer order Cai system and a four dimensional fractional order hyperchaotic Gao system is achieved and in second case synchronization for three dimensional fractional order Newton-Leipnik system and four dimensional hyperchaotic Pang-Liu system is achieved by tracking control method. Computational results shows that the controllers designed are useful to synchronize the considered master and slave systems in both the cases. In order to execute numerical results Matlab software is used.

Keywords

Synchronization, fractional order system, integer order system, tracking Control.

AMS Subject Classification 37B25, 37D45, 37N35, 70K99.

1_{Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India.} 2_{Department of Mathematics, Shivaji College, New Delhi, 110027, India.} 3_{Department of Mathematics, University of Delhi, Delhi, 110007, India.}

*Corresponding author:1_{[email protected];} 2_{[email protected];}3_{[email protected]}

Article History: Received17November2017; Accepted19February2018 2018 MJM.c

1. Introduction

Chaotic phenomenon has been present in the nature since a long time but when the Lorenz system [1] was found in 1963, this field arose as a new attraction in non linear sciences. After that pioneer invention of Lorenz a lot of literature is available on chaotic and hyperchaotic systems. Chaos is the property of typical dynamics and highly unpredictable behavior of a

system. Having a 300 years old history fractional calculus has always been a branch of intrinsic interest for researchers. Chaos in fractional order systems have also been the field of new and application based researches. Several chaotic sys-tems with fractional order are discovered and studied by the researchers. Some of the well known fractional order systems are Lorenz, Rossler, Chen, Liu, Volta fractional order systems [2]-[6]. Importance of fractional order can be seen as there are so many systems which can be modeled in a better and more accurate manner as a fractional order system in comparison of integer order. Different methods of solving fractional order systems and stability of these methods have been analyzed by some researchers [7,8]. Several other significant analysis have also been done on fractional differential equations [9 ]-[11].

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syn-chronization can be seen with significant developments in the field of secure communication [19,20]. In secure communi-cation, although the information signal is always masked by a chaotic signal but when it comes to the security of trans-mitted message, role of chaos synchronization becomes vital. Synchronization has been an integral part of chaotic lasers [21] due to its applicability in optical secure communication. Several methods have been found for security of transmitted signal in secure communication including chaos masking. But it is found that information signal masked with chaotic signal is not much secure, therefore to enhance the grade of secu-rity of information signal several types of synchronizations were suggested among which projective synchronization was found much effective for secure communication because of its unpredictable multiple factor. Several methods like active control [22], adaptive control [23], feedback control [24], pin-ning control [25], sliding mode control [26], fuzzy control [27] etc have been studied and found effective to obtain the desired synchronization. Some work have also been done on synchronization of fractional order and integer order system [28,29], but still a very few work have been done on different order and different dimensional systems. A very particular case was considered by Ping et. al [28] to achieve complete synchronization which is extended in this paper in a more gen-eralized way to achieve different kinds of synchronizations. This paper is categorized in five sections. Section2presents synchronization scheme, fractional order derivatives and sta-bility of fractional order systems. Section3presents the brief discussion of chaotic and hyperchaotic nature of all four sys-tems considered. In section4construction of controllers and numerical results are presented to justify the theoretical ap-proach and the last section5is conclusion where the main findings of this paper are highlighted.

2. Fractional order derivatives, stability

of fractional order system and

synchronization scheme

2.1 Fractional order derivative [8]

For differentiation and integration, in fractional calculus the operatorbDtβ is defined wherebandt are the limits of the

operator andβ is the non integer order. This operator is defined in the following manner

bDβt =



   

   

dβ

dtβ i fβ>0

0 i fβ=0

t

R

b

(dτ)β i fβ<0

(2.1)

Three definitions for fractional integro differential operator are Riemann-Liouville definition, Grunwald Letnikov definition and Caputo’s definition.

The Riemann-Liouville definition is defined by

bDβt f(t) =

1

Γ(n−β)

dn dtn

t

Z

b

(t−τ)n−β−1f(τ)d(τ), (2.2)

forn−1<β <n. The Caputo’s fractional derivative is given

by

bDβt =

1

Γ(n−β)

Z t

b

fn(τ)

(t−τ)β+1−ndτ, :n−1<β<n (2.3)

whereΓ(.)is the Gamma function. The Grunwald Letnikov definition is

bD β

t f(t) =lim h→0

1

hβ

[(t−β) h ]

∑

j=0

(−1)j

β

j

f(t−jh) (2.4)

where[.]denotes greatest integer part.

2.2 Stability of fractional order system[28]

Theorem:Suppose a fractional order system is considered as

Dβ_x₌_Bx, _x₍₀_{) =}_x 0

HereB∈Rp×p, andβ= (β1,β2, ...,βp),(0<βi≤1). The

system is asymptotically stable if and only if |arg(νi)|>

β π/2 is satisfied for all eigenvaluesνiof the matrixB.

Fur-thermore this system is stable if and only if|arg(νi)| ≥β π/2

is satisfied for all eigenvaluesνiof the matrixBand the

crit-ical eigenvalues for which the condition|arg(νi)|=β π/2

have geometric multiplicity one. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e., number of linearly independent eigenvectors with that eigenvalue.

2.3 Synchronization Scheme

A brief discussion is presented in this subsection to achieve synchronization between two different order systems. Sup-pose a p-dimensional system is considered as master system

dη_m

dtη =h(m) (2.5)

wherehis a differentiable function fromRp_to_Rp_,_m_∈_Rp_,

η= (η1,η2, ...,ηp)is the order and 0<ηi≤1 for all i=1,2....p.Suppose the slave system with controller is

dκ_s

dtκ =l(s) +U (2.6)

wherel is a differentiable function fromRp_to_Rp_,_s_∈_Rp_,

κ= (κ1,κ2, ...,κp)is the order and 0<ηi≤1 for all i=1,2....p.U is the controller which will be decomposed in two subcontrollersU1,U2asU=U1+U2and

U1=χd

κ_m

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whereχ= (χ1,χ2, ...,χp)is a 1×pvector of scaling

fac-tors by choosing of which different types of synchronization could be achieved andm= (m1,m2, ...,mp)t is ap×1

vec-tor. HereU1is a compensation controller andU2is feedback

controller. If errors are defined as

e=s−χm (2.8)

The goal is to design an effective controller so that error will tend to zero and the desired synchronization could be achieved. As the slave systems considered in this paper is of greater dimension than master system so the last dimension of slave system will be synchronized to zero.

3. Description of chaotic and

hyperchaotic systems

−30 −20 −10 0

10 20 30 40

−40 −20 0 20 40 60

0 20 40 60 80

m11 m21

m

3

1

−30 −20 −10 0 10 20 30 40 −40

−30 −20 −10 0 10 20 30 40 50 60

m11

m

2

1

Figure 1.Chaotic attractor of Cai system in three dimensional space and inm11−m31space

Dynamics of integer order Cai [30] system is given below

m11=λ1(m21−m11)

m21=µ1m11+ρ1m21−m11m31 m31=m211−σ1m31

(3.1)

−40 −20

0 20

40 60

−40 −20 0 20 400 20 40 60 80 100 120

s11 s21

s3

1

−40 −30 −20 −10 0 10 20 30 40 −40

−30 −20 −10 0 10 20

s21 s4

1

Figure 2.Hyperchaotic attractor of Gao system in three dimensional space and ins11−s41space

which shows chaotic behavior for λ1=20,µ1=14,ρ1=

10.6,σ1=2.8 as shown in Figure (1). Dynamics of fractional

order Gao system [31] is

dκ1_s 11

dtκ1 =−λ2s11+µ2s21 dκ2_s

21

dtκ2 =ρ2s11−s11s31−s21+s41 dκ3_s

31 dtκ3 =s

2

11−σ2(s11+s31) dκ4_s

41

dtκ4 =−k2s11

(3.2)

which shows hyperchaotic behavior for parameter valuesλ2=

25,µ2=60,ρ2=40,σ2=4,k2=5 and orderκ1=0.95,κ2=

0.95,κ3=0.95,κ4=0.95. Hyperchaotic attractors of Gao

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−40 −30 −20 −10

0 10 20 30

40

0 20 40 60 80 100 120 −40 −30 −20 −10 0 10 20

s21 s31

s4

1

−400 −30 −20 −10 0 10 20 30 40 20

40 60 80 100 120

s21

s3

1

Figure 3.Hyperchaotic attractor of Gao system in three dimensional space and ins21−s31space

which is given by

dκ1_m 12

dtκ1 =−λ3m12+m22+10m22m32 dκ2_m

22

dtκ2 =−m12−0.4m22+5m12m32 dκ3_m

32

dtκ3 =µ3m32−5m12m22

(3.3)

which shows chaotic behavior for parameter values λ3=

0.4,µ3=0.175 and orderκ1=0.95,κ2=0.95,κ3=0.95.

Chaotic attractors of fractional order Newton-Leipnik system are shown in Figure (4).

Integer order hyperchaotic Pang-Liu system [33] is given by

s12=λ4(s22−s12) s22=ρ4s22−s12s32+s42 s32=−µ4s32+s12s22 s42=−h4s12−k4s22

(3.4)

which exhibits hyperchaotic behavior for the parameter values

λ4=36,µ4=3,ρ4=20,h4=2,k4=2.Hyperchaotic

attrac-tors of Pang-Liu system are shown in Figure (5) and Figure (6) .

−0.3 −0.2 −0.1 0

0.1 0.2 0.3 0.4

−0.4 −0.2 0 0.2 0.4 0.6 −0.4 −0.3 −0.2 −0.1 0 0.1

m12 m22

m

3

2

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

m12

m

2

Figure 4.Chaotic attractor of fractional order Newton Leipnik system in three dimensional space and inm12−m32

space

4. Synchronization of systems of

different order and different dimension

and numerical results

Two cases are considered in this paper. In first case Cai sys-tem is considered as master syssys-tem and fractional order Gao system is considered as slave system and in second case frac-tional order Newton-Leipnik system is considered as master system and integer order Pang Liu system is considered as slave system.

Case 1Suppose scaling vector is taken asχ= (χ1,χ2,χ3,χ4).

Then errors are

e11=s11−χ1m11 e21=s21−χ2m21 e31=s31−χ3m31 e41=s41−χ4m41

(4.1)

To synchronize three dimensional Cai system and the four dimensional fractional order hyperchaotic Gao system via tracking control,s41 has to be synchronized to 0. So the

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5 10 15 20 25 30 35 40 −30 −20 −10 0 10 20 30 −40 −20 0 20 s32 s22 s4 2

−505 −40 −30 −20 −10 0 10 20 30 10 15 20 25 30 35 40 s42 s3 2

Figure 5.Hyperchaotic attractor of Pang-Liu system in three dimensional space and ins32−s42space

Theorem 1Hyperchaotic fractional order Gao system (3.2)

will attain synchronization with chaotic Cai system (3.1) for the errors defined by (4.1), if the compensation controllerV1

and feedback controllerV2are

V1=χ



   

dκ₁_m 11 dtκ1 dκ2m21

dtκ₂

dκ₃_m 31 dtκ3 dκ4m41

dtκ4

     −    

−λ2χ1m11+µ2χ2m21

ρ2χ1m11−χ1m11χ3m31−χ2m21+χ4m41

χ₁2m2₁₁−σ2(χ1m11+χ3m31)

−k2χ1m11

    (4.2) and

V2=

Qe¯11−G1(m0,e¯11,e¯21) Se¯21−G3(m0,e¯11,e¯21)

(4.3)

where Q,S are suitably chosen matrices, e¯11 =e11,e¯21=

(e21,e31,e41)andm0represents terms containing master

sys-tem’s state variables.

ProofSuppose fractional order Gao system with controller is



   

dκ1s11 dtκ₁

dκ2s21 dtκ2 dκ₃_s

31 dtκ₃

dκ₄_s 41 dtκ4

     =    

−λ2s11+µ2s21

ρ2s11−s11s31−s21+s41 s2₁₁−σ2(s11+s31)

−k2s11



  

+V (4.4)

whereVis the controller. SupposeV=V1+V2whereV1is

compensation andV2is the feedback controller. Then by using

equation (4.2) in the previous equation



   

dκ1s11 dtκ1 dκ₂_s

21 dtκ2 dκ3s31

dtκ3 dκ4s41

dtκ₄

     =    

−λ2s11+µ2s21

(ρ2−s31)s11−s21+s41

s2₁₁−σ2(s11+s31)

−k2s11



  

+V2+χ



   

dκ1m11 dtκ1 dκ₂_m 21 dtκ2 dκ3m31

dtκ3 dκ4m41

dtκ₄

     −    

−λ2χ1m11+µ2χ2m21

(ρ2χ1−χ1χ3m31)m11−χ2m21+χ4m41

χ₁2m2₁₁−σ2(χ1m11+χ3m31)

−k2χ1m11



  

(4.5)

Then error dynamical system will be



   

dκ₁_e 11 dtκ1 dκ2e21

dtκ₂

dκ₃_e 31 dtκ3 dκ4e41

dtκ4

     =    

−λ2s11+µ2s21

ρ2s11−s11s31−s21+s41 s2₁₁−σ2(s11+s31)

−k2s11



  

+V2−



  

−λ2χ1m11+µ2χ2m21

ρ2χ1m11−χ1m11χ3m31−χ2m21+χ4m41

χ₁2m2₁₁−σ2(χ1m11+χ3m31)

−k2χ1m11

    (4.6) Hence     

dκ₁_e 11 dtκ1 dκ2e21

dtκ2 dκ₃_e 31 dtκ₃

dκ4e41 dtκ4



   

=V2+



  

−λ2e11+µ2e21

(ρ2−e31−χ3m31)e11−e31χ1m11−e21+e41

e11(e11+2χ1m11)−σ2(e11+e31)

−k2e11



  

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Our aim is to design controllerV2so that a stable error

dy-namical system can be achieved.

dκ₁_e_¯ 11 dtκ1 dκ2e21¯

dtκ₂

!

=

Pe11¯ +G1(m0,e¯11,e¯21)

Re¯21+G2(m0,e¯11,e¯21) +G3(m0,e¯11,e¯21)

+V2

(4.8)

where,P= (−λ2),R=





−1 0 0

0 −σ2 0

0 0 0



, ¯e11=e11,e¯21=



         

         

G1(m0,e¯11,e21¯ ) =µ2e21

G2(m0,e¯11,e21¯ ) =





ρ2e11−e11e31−e11χ3m31 e11(e11+2χ1m11)−σ2e11

−k2e11





G3(m0,e¯11,e21¯ ) =





−e31χ1m11+e41

−σ2e31

0





(4.9)

Hence by defining

V2=

Qe11¯ −G1(m0,e¯11,e¯21) Se¯21−G3(m0,e¯11,e¯21)

(4.10)

The error dynamical system will be

dκ₁_e_¯ 11 dtκ1 dκ2e¯21

dtκ₂

!

=

(P+Q)e¯11

(R+S)e¯21+G2(m0,e11¯ ,e¯21)

(4.11)

where lime11¯→0G2(m 0_,_e_¯

11,e¯21) =0. By choosing suitable Q,Smatrices condition of eigenvalues of the system can be made suitable so that asymptotical stability condition will be satisfied.

For numerical simulation of first case to achieve anti-synchroni-zation scaling vector is chosen as(χ1,χ2,χ3,χ4) = (−1,−1,

−1,−1). Parameters of master and slave system are cho-sen asλ1=20,µ1=14,ρ1=10.6,σ1=2.8,λ2=36,µ2=

3,ρ2=20,h4=2,k4=2. Fractional order of slave system is

κ1=κ2=κ3=κ4=0.95.Initial values for master system

and slave systems are(−3,−8,−4)and(2,6,9,2). Matrices

QandSare chosen in such a way that eigenvalues ofP+Q

is−1 and eigenvalue ofQ+S are−3,−2,−4. Hence the condition|arg(νi)| ≥β π/2 will be satisfied. Initial

condi-tions for the error systems will be(−1,−2,5,2)and order of error dynamical system isκ1=κ2=κ3=κ4=0.95. Errors

converging to zero are shown in Figure (7).

Case2

In this case fractional order Newton Leipnik system is consid-ered as master system and integer order Pang Liu system is

−25 −20 −15 −10 −5 0

5 10 15 20 25

5 10 15 20 25 30 35 40 −30 −20 −10 0 10 20 30

s22 s32

s1

2

5 10 15 20 25 30 35 40 −30

−20 −10 0 10 20 30

s32 s1

2

Figure 6.Hyperchaotic attractor of Pang-Liu system in three dimensional space and ins12−s32space

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2

−1 0 1 2 3 4 5

Time (Sec)

e11 e21 e31 e41

Figure 7.Convergence of anti-synchronization errors to zero for integer order Cai system and fractional order Gao system

considered as slave system. Errors are defined as

e12=s12−χ1m12 e22=s22−χ2m22 e32=s32−χ3m32 e42=s42−χ4m42

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Theorem 2Hyperchaotic integer order Pang-Liu system (3.4) will attain synchronization with chaotic fractional order Newton-Leipnik system (3.3) for the errors defined by (4.12), if the compensation controllerV1and feedback controllerV2are

V1=χ



   

dm12 dt dm22

dt dm32

dt dm42

dt



   

−



  

λ4(χ2m22−χ1m12)

ρ4χ2m22−χ1χ3m12m32+χ4m42

−µ4χ3m32+m12χ2m22

−h4χ1m12−k4χ2m22



  

(4.13)

and

V2=

Q0e¯12−H1(m0,e¯12,e¯22) S0e¯22−H3(m0,e¯12,e¯22)

(4.14)

whereQ0,S0 are suitably chosen matrices, ¯e12=e12,e¯22=

ProofOn applying the same procedure which is described in

previous theorem an error dynamical system will be obtained in the following form



   

de12 dt de22

dt de32

dt de42

dt



   

= 

  

λ4(e22−e12)

ρ4e22−e12e32−e12χ3m32−e32χ1m12−e42

−µ4e32+e12e22+e12χ2m22+e22χ1m12

−h4e12−k4e22



  

+V2

(4.15)

de12¯ dt de22¯

dt

=

P0e¯12+H1(m0,e12¯ ,e¯22)

R0e¯22+H2(m0,e¯12,e¯22) +H3(m0,e¯12,e¯22)

+V2

(4.16)

whereP0= (−λ4),R0=





ρ4 0 0

0 −µ4 0

0 0 0



, ¯e12=e12,e¯22=

(e22,e32,e42)andmrepresents terms containing master



         

         

H1(m0,e¯12,e¯22) =λ4e22

H2(m0,e¯12,e¯22) =





−e12e32−e12χ3m32 e12e22+e12χ2e22

−k4e12





H3(m0,e¯12,e¯22) =





−e32χ1m12−e42 e22χ1m12

−k4e22





(4.17)

Hence by defining

V2=

Q0e¯12−H1(m0,e¯12,e¯22) S0e¯22−H3(m0,e¯12,e¯22)

(4.18)

Hence error dynamical system will be

de12¯ dt de22¯

dt

=

(P0+Q0)e¯12

(R0+S0)e¯22+H2(m0,e¯12,e¯22)

(4.19)

where lime¯12→0H2(s,e¯12,e¯22) =0. Similarly, by choosing

suitableQ0,S0an asymptotically stable error dynamical sys-tem can be obtained.

For the second case to achieve hybrid projective synchroniza-tion scaling vector is chosen as(χ1,χ2,χ3,χ4) = (1,−1,1,−1).

Parameters of master and slave system are chosen asλ3=

0.4,µ3=0.175,λ4=36,µ4=3,ρ4=20,h4=2,k4=2.

Frac-tional order of master system isκ1=κ2=κ3=0.95.Initial

values for master system and slave systems are(0.349,0,−0.16) and(6,7,8,4). MatricesQ0andS0are chosen in such a way that eigenvalues ofP+Qis−1 and eigenvalue ofQ+Sare

−1,−2,−3,−4. Initial conditions for error systems will be (5.651,7,8.16,4). Errors converging to zero are shown in Figure (8).

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9

Time(sec)

e12 e22 e32 e42

Figure 8.Convergence of hybrid projective synchronization errors to zero for fractional order Newton Leipnik system and integer order Pang Liu system

5. Conclusion

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? ? ? ? ? ? ? ? ? ISSN(P):2319−3786

Malaya Journal of Matematik

Synchronization of dynamical systems of different orders and different dimensions