Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2014, Article ID 490364, 7 pages http://dx.doi.org/10.1155/2014/490364
Research Article One Adaptive Synchronization Approach for Fractional-Order Chaotic System with Fractional-Order 1 < π < 2 Ping Zhou1,2 and Rongji Bai1,2 1 2
Center of System Theory and Its Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Key Laboratory of Network Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Correspondence should be addressed to Ping Zhou;
[email protected] Received 19 May 2014; Accepted 11 August 2014; Published 27 August 2014 Academic Editor: Jianquan Lu Copyright Β© 2014 P. Zhou and R. Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on a new stability result of equilibrium point in nonlinear fractional-order systems for fractional-order lying in 1 < π < 2, one adaptive synchronization approach is established. The adaptive synchronization for the fractional-order Lorenz chaotic system with fractional-order 1 < π < 2 is considered. Numerical simulations show the validity and feasibility of the proposed scheme.
1. Introduction Fractional-order differential equations can be more accurately described in the real-world physical systems [1β3]. Many fractional-order systems create chaotic attractor. Many fractional-order chaotic attractors have been reported in recent years, for example, the fractional-order Lorenz chaotic attractor [1, 4, 5], the fractional-order Chen chaotic attractor [5], the fractional-order Lu chaotic attractor [2, 4], the fractional-order Chua chaotic attractor [5], the fractionalorder Duffing chaotic attractor [6], the fractional-order RΒ¨ossler chaotic attractor [7, 8], and so on. On the other hand, synchronization of chaotic systems has been given more attention [2, 9β13]. This is due to its applications in the field of engineering and science. Over the last two decades, many scholars have proposed various synchronization schemes. It is well known that many chaotic systems in practical situations are usually with fully or partially unknown parameters. In order to estimate the unknown parameters, a synchronization scheme named adaptive synchronization has been proposed. Now, the adaptive synchronization [13β16] has attracted more and more attention. This is due to its effectiveness in many practical chaos applications.
However, many adaptive synchronization approaches on fractional-order chaotic systems reported previously [2, 9β 13] were considered the fractional-order lying in 0 < π < 1. To the best of our knowledge, there are a few results about adaptive synchronization on fractional-order chaotic systems with fractional-order 1 < π < 2. In fact, there are many fractional-order systems with fractional-order 1 < π < 2 in real-world physical systems, for example, the fractional diffusion-wave equation [17], the fractional telegraph equation [18], the time fractional heat conduction equation [19], and so forth. So, an interesting question is how to realize the adaptive synchronization for fractional-order lying in 1 < π < 2? This question is of practical importance as well as academic significance. In this paper, a positive answer is given for the above question. Inspired by the above-mentioned discussion, one adaptive synchronization approach for a class of fractional-order chaotic system with 1 < π < 2 is established. This approach is based on a new stability result of equilibrium point in nonlinear fractional-order systems for fractional-order lying in 1 < π < 2 [1]. The adaptive synchronization for the fractional-order Lorenz chaotic system with fractional-order 1 < π < 2 is considered. Numerical simulations show the validity and feasibility of the proposed scheme.
2
The Scientific World Journal
2. Preliminaries and Main Results In our paper, the πth Caputo derivative for function π(π‘) is shown as π·π π (π‘) =
π‘ π(π) (π) 1 ππ, β« Ξ (π β π) 0 (π‘ β π)π+1βπ
π β 1 < π < π, (1)
π
where π· denote the Caputo derivative, π is the smallest integer larger than π, π(π) (π‘) is the π th derivative in the usual sense, and Ξ is the gamma function. Now, consider the following fractional-order chaotic system: π·π π₯ = π (π₯) = ππ₯ + π (π₯) ,
(2)
where fractional-order 1 < π < 2, π₯ β π
πΓ1 , and π(π₯) β π
πΓ1 . π β π
πΓπ is a constant matrix. π(π₯) β π
πΓ1 is the nonlinear part of system (2). The system (2) can be rewritten as follows: π₯ π· π₯ = πΏ ( ) + π (π₯, π0 ) , π0 π
(3) πΓ1
where π0 β π
is the system parameter. π(π₯, π0 ) β π
is the nonlinear part, and all the terms with system parameter π0 are contained in π(π₯, π0 ). πΏ β π
πΓ(π+1) is a constant matrix, and matrix element πΏ π,π+1 = 0 (π = 1, 2, . . . , π). In this paper, we focus on a class of fractional-order chaotic system in which the equation π(π¦, π0 ) β π(π₯, π0 ) = πππ (π₯)(π¦ β π₯) + πππ (π¦ β π₯, π₯) holds. Here, variable π¦ β π
πΓ1 is real number. Vector πππ (π₯)(π¦ β π₯) and vector πππ (π¦ β π₯, π₯) are the linear part and nonlinear part with respect to (π¦ β π₯), respectively. In fact, the nonlinear term π(π₯, π0 ) in many fractional-order chaotic systems meet this equation, for example, the fractional-order Lorenz chaotic system, fractional-order Chen chaotic system, fractional-order Lu chaotic system, fractional-order RΒ¨ossler chaotic system, the fractional-order Chuaβs chaotic system and its modified chaotic system, the fractional-order Duffing chaotic system, the fractional-order Arneodo chaotic system, the fractionalorder Sprott chaotic system, and so forth. Next, the adaptive synchronization for fractional-order chaotic system (3) is proposed. Select system (3) as drive system; the response systems with parameter update law are shown as follows: π¦ π·π π¦ = πΏ ( ) + π (π¦, π) + π’ (π₯, π¦, π) , π
(4)
π
Lemma 1 (see [1]). For the nonlinear part π(π₯) of systems (2), if (i) π(π₯)|π₯=0 = 0, limπ₯ β 0 (βπ(π₯)β/βπ₯β) = 0; (ii) Re[π(π)] < 0, β max[Re π(π)] > [Ξ(π)]1/π . Then, the zero solution of fractional-order chaotic system (2) is asymptotically stable. Based on this lemma, the following main results are given. Theorem 2. If the controller is selected as π’ (π₯, π¦, π) = [πΉ β πππ (π₯)] π
(5)
and the following conditions are satisfied: (i) ( πππ (π,π₯) ) |π=0 = 0, limπ β 0 (β ( πππ (π,π₯) ) β/βπβ) = 0 for 0 0 any π₯, 1/π πΏ+πΉ (ii) Re[π ( πΏ+πΉ Ξ© )] < 0, β max[Re π ( Ξ© )] > [Ξ(π)] ,
then the adaptive synchronization between fractional-order chaotic system (3) and fractional-order system (4) can be arrived, where π(π¦, π) β π(π₯, π0 ) = πππ (π₯)π + πππ (π, π₯), πππ (π₯) β π
πΓ(π+1) , and πππ (π, π₯) β π
πΓ1 . πΉ β π
πΓ(π+1) is a suitable constant matrix. Proof. The error system between systems (4) and (3) can be shown as π¦ π₯ π·π (π¦ β π₯) = πΏ (( ) β ( )) + π (π¦, π) π π0 β π (π₯, π0 ) + π’ (π₯, π¦, π) .
(6)
π·π π = Ξ©π. Due to π = (π1 , . . . , ππ , ππ+1 )T , ππ = π¦π β π₯π (π = 1, 2, . . . , π), and ππ+1 = ππ = (π β π0 ), system (6) can be rewritten as π·π (π¦ β π₯) = πΏπ + π (π¦, π) β π (π₯, π0 ) + π’ (π₯, π¦, π) , π·π π = Ξ©π.
(7)
Using π(π¦, π) β π(π₯, π0 ) = πππ (π₯)π + πππ (π, π₯), π·π π0 = 0, and π·π π = π·π (π β π0 ) = π·π ππ , error system (7) can be changed as
π· π = Ξ©π,
π·π (π¦ β π₯) = πΏπ + πππ (π₯) π + πππ (π, π₯) + π’ (π₯, π¦, π) ,
where π¦ β π
πΓ1 is state vector, π’(π₯, π¦, π) β π
πΓ1 is a controller, Ξ© β π
1Γ(π+1) is real constant matrix, and parameter π is unknown in response system (4). The true value of the βunknownβ parameter π is selected as π0 . The parameter update law is π·π π = Ξ©π. The adaptive synchronization errors are π = (π1 , . . . , ππ , ππ+1 )T β π
(π+1)Γ1 , ππ = (π¦π β π₯π ) β π
(π = 1, 2, . . . , π), and ππ+1 = ππ = (π β π0 ) β π
.
π·π ππ = Ξ©π.
(8)
Since π’(π₯, π¦, π) = [πΉ β πππ (π₯)]π and π = (π1 , . . . , ππ , ππ+1 )T , therefore, (8) can be changed as π (π, π₯) πΏ+πΉ ). π·π π = ( ) π + ( ππ Ξ© 0
(9)
The Scientific World Journal
3
50
x3
x3
50
0 40 0 β20
β10
0 x1
10
20
20
0 x2
β20
β40
β20
0
20 x1
Figure 1: The attractor of fractional-order Lorenz system (12) for π = 1.05.
Due to ( πππ (π,π₯) ) |π=0 = 0, limπ β 0 (β ( πππ (π,π₯) ) β/βπβ) = 0 0 0 for any π₯, Re[π ( πΏ+πΉ )] < 0, and β max[Re π ( πΏ+πΉ Ξ© Ξ© )] > 1/π [Ξ(π)] . According to the above-mentioned lemma, the zero solution of fractional-order system (9) is asymptotically stable. So, the following result holds: lim βπβ = 0.
π‘ β +β
π₯1 0 0 0 0 π·π π₯1 π₯ π 28 β1 0 0 ) ( 2) (π· π₯2 ) = ( 8 π₯3 π π· π₯3 0 0 β 0 π0 3
(10)
+(
(13)
π0 (π₯2 β π₯1 ) βπ₯1 π₯3 ) , π₯1 π₯2
where π0 = 10. So
It implies the following: σ΅© σ΅© lim σ΅©σ΅©π¦ β π₯σ΅©σ΅©σ΅© = 0, π‘ β +β σ΅©
The fractional-order Lorenz system (12) can be rewritten as
lim (π β π0 ) = 0.
π‘ β +β
(11)
Therefore, the adaptive synchronization between fractional-order chaotic system (3) and fractional-order system (4) can be arrived. The proof is completed.
3. Illustrative Example In this section, to show the effectiveness of the adaptive synchronization approach in this paper, the adaptive synchronization for the fractional-order Lorenz chaotic system [4] with fractional-order 1 < π < 2 is considered. Numerical simulations show the validity and feasibility of the proposed scheme. The fractional-order Lorenz system is described by π·π π₯1 π0 (π₯2 β π₯1 ) (π·π π₯2 ) = (π0 π₯1 β π₯2 β π₯1 π₯3 ) , π·π π₯3 π₯1 π₯2 β π0 π₯3
0 0 0 0 28 β1 0 0) . (14) πΏ=( 8 0 0 β 0 3 According to Section 2, the response system with unknown parameter π can be given as π¦1 0 0 0 0 π·π π¦1 π¦ π 28 β1 0 0 (π· π¦2 ) = ( ) ( 2) 8 π¦3 π π· π¦3 0 0 β 0 π 3 π1 π (π¦2 β π¦1 ) π + ( βπ¦1 π¦3 ) + (πΉ β πππ (π₯)) ( 2 ) π3 π¦1 π¦2 ππ
(15)
and the parameter update law is T
π·π π = Ξ©(π1 π2 π3 ππ ) .
(16)
It is easy to obtain the following: (12)
where π0 , π0 , and π0 are system parameters. Let π0 = 10, π0 = 28, π0 = 8/3, and π = 1.05; the system (12) creates chaotic attractor. The chaotic attractor is shown in Figure 1. Case 1 (parameter π is unknown in response system). Now, assume the system parameter π in in response system is the unknown parameter. The true value of the βunknownβ parameter π is selected as π0 .
π (π¦2 β π¦1 ) π0 (π₯2 β π₯1 ) ( βπ¦1 π¦3 ) β ( βπ₯1 π₯3 ) π¦1 π¦2 π₯1 π₯2 π1 βπ0 π0 0 π₯2 β π₯1 π 0 ) ( 2) = (βπ₯3 0 βπ₯1 π3 0 π₯2 π₯1 0 ππ +(
ππ (π2 β π1 ) βπ1 π3 ) . π1 π2
(17)
4
The Scientific World Journal 20
So βπ0 π0 0 π₯2 β π₯1 0 ), πππ (π₯) = (βπ₯3 0 βπ₯1 0 π₯2 π₯1 0 πππ (π, π₯) = (
ππ (π2 β π1 ) βπ1 π3 ) . π1 π2
e1 , e2 , e3
(18)
2
2
e3 0
2
4 t
6
8
β€β
2 2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 ππ2 (σ΅¨σ΅¨σ΅¨π2 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π1 σ΅¨σ΅¨σ΅¨) + (π1 π3 ) + (π1 π2 ) π12 + π22 + π32 + ππ2
σ΅¨ σ΅¨ σ΅¨ σ΅¨2 π2 (σ΅¨π σ΅¨ + σ΅¨π σ΅¨) β π σ΅¨σ΅¨ 2 σ΅¨σ΅¨ σ΅¨σ΅¨ 1 σ΅¨σ΅¨ π12 + π22 + π32 + ππ2
10
2
ππ2 (π2 β π1 ) + (π1 π3 ) + (π1 π2 ) π12 + π22 + π32 + ππ2
2
a
7
2 2 σ΅¨ σ΅¨2 + σ΅¨σ΅¨σ΅¨π1 σ΅¨σ΅¨σ΅¨) (π1 π3 ) + (π1 π2 ) + ππ2 π12 2
(19)
σ΅©σ΅© πππ (π,π₯) σ΅©σ΅© σ΅©σ΅©( )σ΅©σ΅©σ΅© 0 lim σ΅© πβ0 βπβ πβ0
π (π, π₯) σ΅¨σ΅¨σ΅¨σ΅¨ ( ππ )σ΅¨σ΅¨ = 0. 0 σ΅¨σ΅¨π=0
Figure 2: Simulation results of the adaptive synchronization for the fractional-order Lorenz chaotic system.
The fractional-order Lorenz system (12) can be rewritten
Therefore, the first condition in the above-mentioned theorem holds. Now, choose suitable real constant matrix Ξ© β π
1Γπ and πΉ β π
πΓ(π+1) such that
(20)
So the second condition in the above-mentioned theorem holds. According to the theorem in Section 2, the adaptive synchronization between drive system (13) and response system (15) with parameter update law (16) can be achieved. β1 0 0 0 For example, let πΉ = ( β28 0 0 0 ) and Ξ© = (0 0 0 β1). 0 0 0 β1 0 0 0 β8/3 0 ). 0 0 β1
15
Case 2 (parameter π is unknown in response system). Now, assume that the system parameter π in response system is the unknown parameter. The true value of the βunknownβ parameter π is selected as π0 .
2 2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 β€ lim β(σ΅¨σ΅¨σ΅¨π2 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π1 σ΅¨σ΅¨σ΅¨) + (π2 ) + (π3 ) = 0,
πΏ+πΉ 1/π β max [Re π ( )] > [Ξ (π)] . Ξ©
10
1/π π 4 = β8/3, and β max[Re π ( πΏ+πΉ = 0.9722, Ξ© )] = 1 > [Ξ(π)] respectively. Simulation results are shown in Figure 2. Here, π(0) = 7, and all the initial conditions in this paper are (π₯10 , π₯20 , π₯30 ) = (10, 20, 30), and (π¦10 , π¦20 , π¦30 ) = (1, 2, 5), respectively.
2
πΏ+πΉ Re [π ( )] < 0, Ξ©
5 t
σ΅¨ σ΅¨ σ΅¨ σ΅¨ = β(σ΅¨σ΅¨σ΅¨π2 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π1 σ΅¨σ΅¨σ΅¨) + (π2 ) + (π3 ) , 2
0
2
(π1 π3 ) + (π1 π2 ) π12 + π22 + π32 + ππ2
+
9 8
σ΅¨ σ΅¨ π2 (σ΅¨π σ΅¨ β π σ΅¨σ΅¨ 2 σ΅¨σ΅¨
β1 0 0 0
e2
11
=β
So ( πΏ+πΉ Ξ© ) = (
β10
β30
σ΅©σ΅© πππ (π,π₯) σ΅©σ΅© σ΅©σ΅©( )σ΅©σ΅©σ΅© 0 σ΅© βπβ
β€
0
β20
Now, it is easy to verify the following:
=
e1
10
0 000
Therefore, π π = β1 (π = 1, 2, 3),
as π₯1 β10 10 0 0 0 π·π π₯1 π₯ π 0 β1 0 0 ) ( 2 ) + (π0 π₯1 β π₯1 π₯3 ) , (π· π₯2 ) = ( 8 π₯3 π·π π₯3 π₯1 π₯2 0 0 β 0 π0 3 (21) where π0 = 28. So β10 10 πΏ = ( 0 β1 0
0 0 0 0) . 8 0 β 0 3
(22)
The Scientific World Journal
5
According to Section 2, the response system with unknown parameter π can be given as e1 , e2 , e3
π¦1 β10 10 0 0 π·π2 π¦1 π¦ π2 0 β1 0 0 (π· π¦2 ) = ( ) ( 2) 8 π¦3 π2 π· π¦3 0 0 β 0 π 3
10
π1 0 π2 + (ππ¦1 β π¦1 π¦3 ) + (πΉ β πππ (π₯)) ( ) π3 π¦1 π¦2 ππ
e3
0 β10
e1
e2
(23) β20 0
5
10
15
10
15
t
30
and the parameter update law is 25
T
π·π π = Ξ©(π1 π2 π3 ππ ) .
(24) b
20
It is easy to obtain the following: 15
0 0 (ππ¦1 β π¦1 π¦3 ) β (π0 π₯1 β π₯1 π₯3 ) π¦1 π¦2 π₯1 π₯2
10
π1 0 0 0 0 π = (π0 β π₯3 0 βπ₯1 0) ( 2 ) π3 π₯1 0 0 π₯2 ππ
0
5 t
(25)
Figure 3: Simulation results of the adaptive synchronization for the fractional-order Lorenz chaotic system.
σ΅©σ΅© πππ (π,π₯) σ΅©σ΅© σ΅©σ΅©( )σ΅©σ΅©σ΅© 2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 0 lim σ΅© β€ lim β(σ΅¨σ΅¨σ΅¨π3 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨ππ σ΅¨σ΅¨σ΅¨) + (π2 ) = 0, πβ0 πβ0 βπβ
0 + (βπ1 π3 + ππ π1 ) . π1 π2
π (π, π₯) σ΅¨σ΅¨σ΅¨σ΅¨ )σ΅¨σ΅¨ = 0. ( ππ 0 σ΅¨σ΅¨π=0
So 0 0 0 0 πππ (π₯) = (π0 β π₯3 0 βπ₯1 0) , π₯1 0 0 π₯2 0 πππ (π, π₯) = (βπ1 π3 + ππ π1 ) . π1 π2
(27) (26)
Therefore, the first condition in the above-mentioned theorem holds. Now, choose suitable real constant matrix Ξ© β π
1Γπ and πΉ β π
πΓ(π+1) such that πΏ+πΉ Re [π ( )] < 0, Ξ©
Now, it is easy to verify the following:
(28) πΏ+πΉ 1/π β max [Re π ( )] > [Ξ (π)] . Ξ©
σ΅©σ΅© πππ (π,π₯) σ΅©σ΅© σ΅©σ΅©( )σ΅©σ΅©σ΅© 0 σ΅© βπβ =β
2
2
(βπ1 π3 + ππ π1 ) + (π1 π2 ) π12 + π22 + π32 + ππ2
2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 (σ΅¨σ΅¨π σ΅¨σ΅¨ + σ΅¨σ΅¨π σ΅¨σ΅¨) π2 + (π π ) β€ β σ΅¨ 3 σ΅¨ 2 σ΅¨ π σ΅¨2 1 2 12 2 π1 + π2 + π3 + ππ 2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 (σ΅¨σ΅¨π σ΅¨σ΅¨ + σ΅¨σ΅¨π σ΅¨σ΅¨) π2 + (π1 π2 ) β€ β σ΅¨ 3σ΅¨ σ΅¨ πσ΅¨ 2 1 π1 2 σ΅¨ σ΅¨ σ΅¨ σ΅¨2 = β(σ΅¨σ΅¨σ΅¨π3 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨ππ σ΅¨σ΅¨σ΅¨) + (π2 ) ,
So the second condition in the above-mentioned theorem holds. According to the theorem in Section 2, the adaptive synchronization between drive system (21) and response system (23) with parameter update law (24) can be achieved. 0 000 For example, let πΉ = ( β10 0 0 0 ) and Ξ© = (0 0 0 β1). So
( πΏ+πΉ Ξ© )
=
β10 ( β10 0 0
0 000 10 0 0 β1 0 0 0 β8/3 0 ). Therefore, 0 0 β1
π Β± = β5.5 Β± 8.9303π,
π 3 = β8/3, π 4 = β1, and β max[Re π ( πΏ+πΉ Ξ© )] = 1 > 1/π [Ξ(π)] = 0.9722, respectively. Simulation results are shown in Figure 3. Here, π(0) = 10. Case 3 (parameter π is unknown in response system). Now, assume that the system parameter π in response system is
6
The Scientific World Journal
the unknown parameter. The true value of the βunknownβ parameter π is selected as π0 . The fractional-order Lorenz system (12) can be rewritten as π₯1 π· π₯1 β10 10 0 0 0 π₯ (π·π π₯2 ) = ( 28 β1 0 0) ( 2 ) + ( βπ₯1 π₯3 ) . π₯3 π·π π₯3 π₯1 π₯2 β π0 π₯3 0 0 0 0 π0 (29) π
So β10 10 0 0 πΏ = ( 28 β1 0 0) . 0 0 0 0
(30)
According to Section 2, the response system is given as
Now, it is easy to verify the following: σ΅©σ΅© πππ (π,π₯) σ΅©σ΅© σ΅©σ΅©( )σ΅©σ΅©σ΅© 0 σ΅© βπβ =β
2
2
(π1 π2 β ππ π3 ) + (π1 π3 ) π12 + π22 + π32 + ππ2
2 σ΅¨ σ΅¨σ΅¨ σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨ 2 (σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ + σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨) + (π1 π3 ) β€ β σ΅¨ 1 σ΅¨ σ΅¨ 2 σ΅¨2 σ΅¨ 2π σ΅¨ σ΅¨ 32σ΅¨ π1 + π2 + π3 + ππ2 2 σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ (π π ) 2 σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ = ( 2σ΅¨ 1 σ΅¨ σ΅¨2 2 σ΅¨ σ΅¨ 23 σ΅¨ σ΅¨ π2σ΅¨ + 2 2π 3 2 2 π1 + π2 + π3 + ππ π1 + π2 + π3 + ππ 2
2
(π π ) + (π π ) + 21 2 2 2 1 3 2 ) π1 + π2 + π3 + ππ
1/2
2 2 2 σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ (π π ) + (π π ) 2 σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ (π π ) β€ β 2σ΅¨ 1 σ΅¨ σ΅¨2 2 σ΅¨ σ΅¨ 23 σ΅¨ σ΅¨ π2σ΅¨ + π 23 + 1 2 2 1 3 ππ π1 + π2 + π3 + ππ π1
π¦1 β10 10 0 0 π·π π¦1 π¦ π (π· π¦2 ) = ( 28 β1 0 0) ( 2 ) π¦3 π π· π¦3 0 0 0 0 π π1 0 π2 + ( βπ¦1 π¦3 ) + (πΉ β πππ (π₯)) ( ) π3 π¦1 π¦2 β ππ¦3 ππ
(31)
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ 2 σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ = β 2σ΅¨ 1 σ΅¨ σ΅¨2 2 σ΅¨ σ΅¨ 23 σ΅¨ σ΅¨ π2σ΅¨ + 2π32 + π22 π1 + π2 + π3 + ππ σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ 2 σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨σ΅¨π σ΅¨σ΅¨ β€ β σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ 1σ΅¨σ΅¨ σ΅¨ σ΅¨ σ΅¨σ΅¨ 2 σ΅¨ σ΅¨ 3σ΅¨σ΅¨ σ΅¨ σ΅¨ σ΅¨σ΅¨ πσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ + 2π32 + π22 2 σ΅¨σ΅¨π1 σ΅¨σ΅¨ σ΅¨σ΅¨π2 σ΅¨σ΅¨ + 2 σ΅¨σ΅¨π3 σ΅¨σ΅¨ σ΅¨σ΅¨ππ σ΅¨σ΅¨ β1
and the parameter update law is T
π·π π = Ξ©(π1 π2 π3 ππ ) .
(32)
It is easy to obtain the following: 0 0 ( βπ¦1 π¦3 ) β ( βπ₯1 π₯3 ) π¦1 π¦2 β ππ¦3 π₯1 π₯2 β π0 π₯3 π1 0 0 0 0 π = (βπ₯3 0 βπ₯1 0 ) ( 2 ) π3 π₯2 π₯1 βπ0 βπ₯3 ππ
1 1 = β ( σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ + σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ) σ΅¨σ΅¨π3 σ΅¨σ΅¨ σ΅¨σ΅¨ππ σ΅¨σ΅¨ σ΅¨σ΅¨π1 σ΅¨σ΅¨ σ΅¨σ΅¨π2 σ΅¨σ΅¨ σ΅©σ΅©σ΅©( πππ (π,π₯) )σ΅©σ΅©σ΅© σ΅© σ΅©σ΅© 0 lim σ΅© πβ0 βπβ
+ 2π32 + π22 ,
β1
(33)
1 1 β€ lim β ( σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ + σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ) + 2π32 + π22 = 0, πβ0 σ΅¨σ΅¨π3 σ΅¨σ΅¨ σ΅¨σ΅¨ππ σ΅¨σ΅¨ σ΅¨σ΅¨π1 σ΅¨σ΅¨ σ΅¨σ΅¨π2 σ΅¨σ΅¨ π (π, π₯) σ΅¨σ΅¨σ΅¨σ΅¨ ( ππ )σ΅¨σ΅¨ = 0. 0 σ΅¨σ΅¨π=0
Therefore, the first condition in the above-mentioned theorem holds. Now, choose suitable real constant matrix Ξ© β π
1Γπ and πΉ β π
πΓ(π+1) such that
0 + ( βπ1 π3 ) . π1 π2 β ππ π3
πΏ+πΉ Re [π ( )] < 0, Ξ©
So 0 0 0 0 πππ (π₯) = (βπ₯3 0 βπ₯1 0 ) , π₯2 π₯1 βπ0 βπ₯3 0 πππ (π, π₯) = ( βπ1 π3 ) . π1 π2 β ππ π3
(35)
πΏ+πΉ 1/π β max [Re π ( )] > [Ξ (π)] . Ξ© (34)
(36)
So, the second condition in the above-mentioned theorem holds. According to the theoremin Section 2, the adaptive synchronization between drive system (29) and response system (31) with parameter update law (32) can be achieved.
The Scientific World Journal 10
7
References
e3
e1 , e2 , e3
0 e1
β10 β20 β30
e2 0
1
2
3
4
5
6
8
10
t 10 8 c
6 4 2
2
0
4 t
Figure 4: Simulation results of the adaptive synchronization for the fractional-order Lorenz chaotic system.
0 0 0 0 0 0) 0 0 β1 0
For example, let πΉ = ( β38 0 So ( πΏ+πΉ Ξ© ) =
β10 ( β10 0 0
10 β1 0 0
0 0 β1 0
0 0 ). 0 β1
and Ξ© = (0 0 0 β1).
Therefore, π π = β1 (π = 1, 2),
π Β± = β5.5 Β± 8.9303π, and β max[Re π ( πΏ+πΉ Ξ© )] = 1 > [Ξ(π)]1/π = 0.9722, respectively. Simulation results are shown in Figure 4. Here, π(0) = 10.
4. Conclusions One adaptive synchronization scheme for a class of fractional-order chaotic system with fractional-order 1 < π < 2 is suggested in this paper. This synchronization approach is based on a new stability result of equilibrium point in nonlinear fractional-order systems for fractionalorder lying in 1 < π < 2. In order to verify the effectiveness of the adaptive synchronization approach, the adaptive synchronization for the fractional-order Lorenz chaotic system with fractional-order 1 < π < 2 is considered. Numerical simulations show the validity and feasibility of the proposed scheme. The current results in this paper can be extended to several unknown parameters of the fractionalorder chaotic systems. Moreover, this synchronization approach can be applied to other fractional-order chaotic systems.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] L. Chen, Y. He, Y. Chai, and R. Wu, βNew results on stability and stabilization of a class of nonlinear fractional-order systems,β Nonlinear Dynamics, vol. 75, no. 4, pp. 633β641, 2014. [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [3] M. Naber, βTime fractional SchrΒ¨odinger equation,β Journal of Mathematical Physics, vol. 45, no. 8, pp. 3339β3352, 2004. [4] I. Grigorenko and E. Grigorenko, βChaotic dynamics of the fractional Lorenz system,β Physical Review Letters, vol. 91, Article ID 034101, 2003. [5] P. Zhou, R. Ding, and Y. X. Cao, βMulti drive-one response synchronization for fractional-order chaotic systems,β Nonlinear Dynamics, vol. 70, no. 2, pp. 1263β1271, 2012. [6] Z. M. Ge and C. Y. Ou, βChaos in a fractional order modified Duffing system,β Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 262β291, 2007. [7] D. Cafagna and G. Grassi, βHyperchaos in the fractional-order Roπssler system with lowest-order,β International Journal of Bifurcation and Chaos, vol. 19, no. 1, pp. 339β347, 2009. [8] P. Zhou and F. Yang, βHyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points,β Nonlinear Dynamics, vol. 76, no. 1, pp. 473β480, 2014. [9] J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, Oxford, UK, 2003. [10] G. R. Chen and X. Dong, βFrom chaos to order perspectives,β in Methodologies and Applications, World scientific, Singapore, 1998. [11] X. Y. Wang and J. M. Song, βSynchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,β Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3351β3357, 2009. [12] P. Zhou and W. Zhu, βFunction projective synchronization for fractional-order chaotic systems,β Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 811β816, 2011. [13] R. Zhang and S. Yang, βAdaptive synchronization of fractionalorder chaotic systems via a single driving variable,β Nonlinear Dynamics, vol. 66, no. 4, pp. 831β837, 2011. [14] L. L. Li and J. D. Cao, βCluster synchronization in an array of coupled stochastic delayed neural networks via pinning control,β Neurocomputing, vol. 74, no. 5, pp. 846β856, 2011. [15] J. Q. Lu and J. D. Cao, βAdaptive synchronization in tree-like dynamical networks,β Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1252β1260, 2007. [16] J. Q. Lu and J. D. Cao, βAdaptive synchronization of uncertain dynamical networks with delayed coupling,β Nonlinear Dynamics, vol. 53, no. 1-2, pp. 107β115, 2008. [17] F. Mainardi, βThe fundamental solutions for the fractional diffusion-wave equation,β Applied Mathematics Letters, vol. 9, no. 6, pp. 23β28, 1996. [18] L. Beghin and E. Orsingher, βThe telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation,β Fractional Calculus & Applied Analysis, vol. 6, no. 2, pp. 187β204, 2003. [19] W. Zhang, X. Cai, and S. Holm, βTime-fractional heat equations and negative absolute temperatures,β Computers & Mathematics with Applications, vol. 67, no. 1, pp. 164β171, 2014.