# Fractional-Order Sliding Mode Synchronization for Fractional-Order Chaotic Systems

Authors: Chenhui Wang
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

Some sufficient conditions, which are valid for stability check of fractional-order nonlinear systems, are given in this paper. Based on these results, the synchronization of two fractional-order chaotic systems is investigated. A novel fractional-order sliding surface, which is composed of a synchronization error and its fractional-order integral, is introduced. The asymptotical stability of the synchronization error dynamical system can be guaranteed by the proposed fractional-order sliding mode controller. Finally, two numerical examples are given to show the feasibility of the proposed methods.


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## Body

## 1. Introduction

In the past two decades, synchronization of chaotic systems (CSs) has received more and more attention, and a lot of interesting works have been done, which have potential application values in secret communications, signal processing, and complex systems [1–9]. Recently, control and synchronization of fractional-order chaotic systems (FOCSs), which can be seen as a generalization of the integer-order CSs, have been studied extensively. A lot of controllers have been implemented such as active control [10], feedback control [11], sliding mode control [12, 13], adaptive control, [14, 15], and adaptive fuzzy control [8, 9, 16].It is well known that sliding mode control (SMC) is a very effective control method to cope with system uncertainties and external disturbances [17–27]. Consequently, it has been used to synchronize FOCSs. For example, a novel FOCS and its SMC have been studied in [28]; SMC of a 3D FOCS using a fractional-order switching type controller is investigated in [29]. Using a hierarchical fuzzy neural network, [30] proposed a new adaptive SMC method for the synchronization of uncertain FOCSs. On the other hand, it is well known that, in stability analysis of nonlinear systems, quadratic Lyapunov functions are most commonly used. However, [31, 32] show that it is not realistic to use quadratic Lyapunov functions in the stability analysis of fractional-order nonlinear systems due to the complicated infinite series produced by differentiating the squared Lyapunov function with fractional order. It should be mentioned that, in most aforementioned works, the stability analysis is given based on fractional Lyapunov methods. How to establish some stability analysis methods according to the model of FOCSs is a meaningful work.In control theory, stability analysis is an essential aspect. With respect to fractional-order linear systems, the stability condition was firstly investigated in [33]. Then, using LMI, some sufficient conditions are given in [34]. The related results on the stability analysis of fractional-order nonlinear systems can be seen in [35–41] and the references therein. It should be pointed out that the stability criterion for fractional-order nonlinear systems requires further study. Thus, proposing some new stability criterion for FOCSs is necessary. In this paper, we will give two sufficient conditions for the stability of a class of FOCSs. Based on these theorems, a fractional-order SMC will be given. The contributions of this paper are concluded as follows: (1) two sufficient conditions are proposed to check the stability of the fractional-order nonlinear system and (2) a novel fractional-order SMC is given, and the stability of the closed-loop system is proven rigorously.


## 2. Preliminaries

In this section, we will give some properties of fractional calculus. Theqth fractional-order integral is expressed as [42] (1)Dqft=1Γq∫0tt-τq-1fτdτ.The Caputo fractional derivative is given by (2)Dqft=1Γn-q∫0tt-τn-q-1fnτdτ,where q is the fractional order satisfying n-1≤q<n.The Laplace transform of Caputo fractional derivative is given as [42] (3)∫0∞e-stDqxtdt=sqFs-∑k=0n-1sq-k-1xk0,where Fs=Lft. In the next section, we will use the following results.The Mittag-Leffler function is given by(4)Eβ1,β2z=∑k=0∞zkΓβ1k+β2,where β1,β2>0 and z∈C. The Laplace transform of (4) is (5)Ltβ2-1Eβ1,β2-atβ1=sβ1-β2sβ1+a.Lemma 1 (see [42]).
LetA∈Rn×n, 0<α≤1, β be an arbitrary real number, and b>0 be a real constant; then, (6)Eα,βA≤b1+A,where μ≤argeigA≤π with μ∈R satisfying πα/2<μ<min⁡π,πα.Lemma 2 (see [43]).
Lett∈0,T and (7)xt≤ht+∫0tkτxτdτ,where k(t)≥0. Then, one has (8)xt≤ht+∫0tkτhτexp⁡∫τtkududτ.Lemma 3 (see [42, 44]).
Let0<α<2. β is a complex number, and μ is a real number. If (9)πα2<μ<min⁡π,πα,then, for an arbitrary integer n≥1, the following expansion holds: (10)Eα,βz=-∑j=1n1Γβ-αjzj+∘1zn+1.


## 3. Main Results

### 3.1. Some Sufficient Conditions for the Stability Analysis of Fractional-Order Systems

Consider a class of fractional-order systems described by(11)Dqxjt=cjxjt+∑i=1naijfixit,or equivalently (12)Dqxt=Cxt+Afxt,where j=1,2,…,n, 0<q<1, and x(t)=x1t,x2t,…,xntT∈Rn is the state vector; fxt=f1x1t,f2x2t,…,fnxntT∈Rn represents a smooth nonlinear function, A=aij, i=1,2,…,n, j=1,2,…,n, and C=diag⁡ci are two matrices. Then, we have the following results.Theorem 4.
Ifci<0 and the nonlinear function is bounded, that is, there exists a constant mi>0 such that (13)fixit≤mi,then there exist two positive constants t0 and M such that (14)xit≤Mfor all t>t0.Proof.
It follows from (11) that (15)Xis=sq-1sq-cixi0+1sq-ci∑j=1naijLfjxjt.
Using (5), one solves (15) as (16)xit=xi0Eq,1citq+∑j=1naij∫0tt-τq-1Eq,qcit-τqfjxjτdτ.
Thus, according to (13), one has (17)xit≤xi0Eq,1citq+∑j=1naijmj∫0tt-τq-1Eq,qcit-τqdτ.
Noting that the Laplace transform of a Mittag-Leffler function is(18)∫0tτβ-1Eβ1,β2-kτβ1dτ=tβ2Eβ1,β2+1-ktβ1,then one has (19)xit≤xi0Eq,1citq+AitqEq,q+1citq,where Ai=∑j=1n‍aijmj is a positive constant.
It follows from Lemma3 that (20)AitqEq,q+1-citq≤Ai-ci.
Consequently, for large enough timet, one has (21)limt→∞⁡xit≤M,where M=max1≤i≤nAi/ci. This ends the proof of Theorem 4.It should be pointed out that Theorem4 can only drive xi(t) to a small region of zero. To discuss the asymptotic stability, one needs the following assumptions.Assumption 5.
The equilibrium point of system (11) is the origin.Assumption 6.
f
x
t is a Lipshitz continuous function; that is, the following inequality holds: (22)fxt-fyt≤lxt-yt,where l>0 is a Lipshitz constant.Remark 7.
It should be mentioned that Assumptions5 and 6 are reasonable. In fact, every equilibrium point of system (11) can be moved to the origin by some linear transformations. In many FOCSs, the nonlinear functions are smooth and Lipshitz continuous, for example, fractional-order Lorenz system, fractional-order Chen system, fractional-order Lü system, fractional-order financial system, and fractional Volta system [45].Theorem 8.
Consider system (12). Under Assumption 6, if lb/cA<q, where c=max1≤i≤n-ci, then the asymptotical stability of system (12) can be guaranteed.Proof.
Suppose thatx(t),yt∈Rn are two arbitrary solutions of (12). Denote e(t)=x(t)-yt; then, one has (23)Dqet=Cet+Afxt-fyt.
It follows from (23) that (24)sqEs=sq-1e0+CEs+ALfxt-fyt,where E(s)=Let.
After some straightforward manipulators, one has(25)Es=Isq-C-1sq-1e0+ALfxt-fyt.Solving (25) yields (26)et=Eq,1Ctqe0+A∫0tt-τq-1Eq,qCt-τqLfxt-fytdτ.According to Assumption 6 and Lemma 1, one can find a constant b>0 such that (27)et≤be01+Ctq+lbA∫0tt-τq-11+Ct-τqeτdτ.
Using Lemma2, one has (28)xt≤be01+Ctq+∫0tlbAt-τα-1e01+Ct-τq1+Cτqexp⁡∫τtlbAt-uq-11+Ct-uqdudτ=be01+Ctq+∫0tblAt-τq-1e01+Cτq1+Ct-τq1-b/qCdτ≤be01+Ctq+lbAe0ClbA/qC-2∫0tt-τlbA/C-1τ-qdτ=be01+Ctq+lbAe0ClbA/qC-2ΓlbA/CΓ1-qΓ1+lbA/C-qtlbA/C-q.Noting that lb/cA<q, where c=C=max1≤i≤n-ci, then according to (28) one has (29)limt→∞⁡et=0,which completes the proof.


### 3.2. Synchronization Controller Design

The master and slave FOCSs are defined, respectively, as(30)Dqζt=Pζt+Qħζt,(31)Dqζ^t=Pζ^t+Qħζ^t+Gut,where ζ(t),ζ^t∈Rn are the state vectors of the master FOCS and slave FOCS, respectively, P,Q,G∈Rn×n are three constant matrices, G is a positive definite control gain matrix, and ut∈Rn represents the control input.Define the synchronization errorεt=ζt-ζ^t. The objective of this section is to design a proper control input u(t) such that ε(t) converges to zero eventually. To proceed, let us give the following assumption first.Assumption 9.
ħ is a Lipshitz continuous function; that is, the following inequality holds: (32)ħζt-ħζ^t≤a0ζt-ζ^t,where a0>0 is a constant.To meet the synchronization object, let us construct the following fractional-order sliding mode surface:(33)st=Λεt-1ΓqΛP+K∫0tt-τq-1ετdτ,where Λ,K∈Rn×n are two design matrices. Then, it follows from (30), (31), and (33) that (34)Dqst=ΛPεt+ΛQħζt-ħζ^t-Gut-ΛP+Kεt=ΛQħζt-ħζ^t-Gut-ΛKεt.Consequently, let Dqst=0; the control input can be given as (35)ut=G-1ΛQħζt-ħζ^t-G-1ΛKεt.Now, we can give the following results.Theorem 10.
Consider the master FOCS (30) and the slave FOCS (31) under Assumption 9. Suppose that the sliding surface is given by (33) and the control input is designed as (35). If the design matrices satisfy P-ΛK<0 and a0Q-ΛQ≤λ_q, where λ_ is the smallest eigenvalue of ΛK-P, then one can conclude that the synchronization error converges to the origin asymptotically.Proof.
It follows from (30) and (31) that (36)Dqεt=Pεt+Qħħζt-ħζ^t-Gut.
Substituting (35) into (36) yields (37)Dqεt=P-ΛKεt+Q-ΛQħħζt-ħζ^t.
Noting thatP-ΛK<0 and a0Q-ΛQ≤λ_q, it follows from (37), Assumption 9, and Theorem 8 that limt→∞ε(t)=0. This completes the proof of Theorem 10.


## 3.1. Some Sufficient Conditions for the Stability Analysis of Fractional-Order Systems

Consider a class of fractional-order systems described by(11)Dqxjt=cjxjt+∑i=1naijfixit,or equivalently (12)Dqxt=Cxt+Afxt,where j=1,2,…,n, 0<q<1, and x(t)=x1t,x2t,…,xntT∈Rn is the state vector; fxt=f1x1t,f2x2t,…,fnxntT∈Rn represents a smooth nonlinear function, A=aij, i=1,2,…,n, j=1,2,…,n, and C=diag⁡ci are two matrices. Then, we have the following results.Theorem 4.
Ifci<0 and the nonlinear function is bounded, that is, there exists a constant mi>0 such that (13)fixit≤mi,then there exist two positive constants t0 and M such that (14)xit≤Mfor all t>t0.Proof.
It follows from (11) that (15)Xis=sq-1sq-cixi0+1sq-ci∑j=1naijLfjxjt.
Using (5), one solves (15) as (16)xit=xi0Eq,1citq+∑j=1naij∫0tt-τq-1Eq,qcit-τqfjxjτdτ.
Thus, according to (13), one has (17)xit≤xi0Eq,1citq+∑j=1naijmj∫0tt-τq-1Eq,qcit-τqdτ.
Noting that the Laplace transform of a Mittag-Leffler function is(18)∫0tτβ-1Eβ1,β2-kτβ1dτ=tβ2Eβ1,β2+1-ktβ1,then one has (19)xit≤xi0Eq,1citq+AitqEq,q+1citq,where Ai=∑j=1n‍aijmj is a positive constant.
It follows from Lemma3 that (20)AitqEq,q+1-citq≤Ai-ci.
Consequently, for large enough timet, one has (21)limt→∞⁡xit≤M,where M=max1≤i≤nAi/ci. This ends the proof of Theorem 4.It should be pointed out that Theorem4 can only drive xi(t) to a small region of zero. To discuss the asymptotic stability, one needs the following assumptions.Assumption 5.
The equilibrium point of system (11) is the origin.Assumption 6.
f
x
t is a Lipshitz continuous function; that is, the following inequality holds: (22)fxt-fyt≤lxt-yt,where l>0 is a Lipshitz constant.Remark 7.
It should be mentioned that Assumptions5 and 6 are reasonable. In fact, every equilibrium point of system (11) can be moved to the origin by some linear transformations. In many FOCSs, the nonlinear functions are smooth and Lipshitz continuous, for example, fractional-order Lorenz system, fractional-order Chen system, fractional-order Lü system, fractional-order financial system, and fractional Volta system [45].Theorem 8.
Consider system (12). Under Assumption 6, if lb/cA<q, where c=max1≤i≤n-ci, then the asymptotical stability of system (12) can be guaranteed.Proof.
Suppose thatx(t),yt∈Rn are two arbitrary solutions of (12). Denote e(t)=x(t)-yt; then, one has (23)Dqet=Cet+Afxt-fyt.
It follows from (23) that (24)sqEs=sq-1e0+CEs+ALfxt-fyt,where E(s)=Let.
After some straightforward manipulators, one has(25)Es=Isq-C-1sq-1e0+ALfxt-fyt.Solving (25) yields (26)et=Eq,1Ctqe0+A∫0tt-τq-1Eq,qCt-τqLfxt-fytdτ.According to Assumption 6 and Lemma 1, one can find a constant b>0 such that (27)et≤be01+Ctq+lbA∫0tt-τq-11+Ct-τqeτdτ.
Using Lemma2, one has (28)xt≤be01+Ctq+∫0tlbAt-τα-1e01+Ct-τq1+Cτqexp⁡∫τtlbAt-uq-11+Ct-uqdudτ=be01+Ctq+∫0tblAt-τq-1e01+Cτq1+Ct-τq1-b/qCdτ≤be01+Ctq+lbAe0ClbA/qC-2∫0tt-τlbA/C-1τ-qdτ=be01+Ctq+lbAe0ClbA/qC-2ΓlbA/CΓ1-qΓ1+lbA/C-qtlbA/C-q.Noting that lb/cA<q, where c=C=max1≤i≤n-ci, then according to (28) one has (29)limt→∞⁡et=0,which completes the proof.


## 3.2. Synchronization Controller Design

The master and slave FOCSs are defined, respectively, as(30)Dqζt=Pζt+Qħζt,(31)Dqζ^t=Pζ^t+Qħζ^t+Gut,where ζ(t),ζ^t∈Rn are the state vectors of the master FOCS and slave FOCS, respectively, P,Q,G∈Rn×n are three constant matrices, G is a positive definite control gain matrix, and ut∈Rn represents the control input.Define the synchronization errorεt=ζt-ζ^t. The objective of this section is to design a proper control input u(t) such that ε(t) converges to zero eventually. To proceed, let us give the following assumption first.Assumption 9.
ħ is a Lipshitz continuous function; that is, the following inequality holds: (32)ħζt-ħζ^t≤a0ζt-ζ^t,where a0>0 is a constant.To meet the synchronization object, let us construct the following fractional-order sliding mode surface:(33)st=Λεt-1ΓqΛP+K∫0tt-τq-1ετdτ,where Λ,K∈Rn×n are two design matrices. Then, it follows from (30), (31), and (33) that (34)Dqst=ΛPεt+ΛQħζt-ħζ^t-Gut-ΛP+Kεt=ΛQħζt-ħζ^t-Gut-ΛKεt.Consequently, let Dqst=0; the control input can be given as (35)ut=G-1ΛQħζt-ħζ^t-G-1ΛKεt.Now, we can give the following results.Theorem 10.
Consider the master FOCS (30) and the slave FOCS (31) under Assumption 9. Suppose that the sliding surface is given by (33) and the control input is designed as (35). If the design matrices satisfy P-ΛK<0 and a0Q-ΛQ≤λ_q, where λ_ is the smallest eigenvalue of ΛK-P, then one can conclude that the synchronization error converges to the origin asymptotically.Proof.
It follows from (30) and (31) that (36)Dqεt=Pεt+Qħħζt-ħζ^t-Gut.
Substituting (35) into (36) yields (37)Dqεt=P-ΛKεt+Q-ΛQħħζt-ħζ^t.
Noting thatP-ΛK<0 and a0Q-ΛQ≤λ_q, it follows from (37), Assumption 9, and Theorem 8 that limt→∞ε(t)=0. This completes the proof of Theorem 10.


## 4. Simulation Results

In this section, two examples will be given to show the effectiveness of the proposed method.


### 4.1. Synchronizing Two 2D Fractional-Order Duffing Systems

The fractional-order Duffing system is described by [46] (38)Dqζ1t=ζ2t,Dqζ2t=ζ1t-ζ13t-0.15ζ2t+0.3cos⁡t.The Jacobian matrix of system (38) for the equilibrium point E1∗=(ζ1∗,ζ2∗) is (39)JE∗=011-3ζ1∗-0.15.It is easy to know that system (38) has three equilibria: E1=1.0729,0,  E2=-0.9062,0, and E3=-0.1667,0. For equilibrium E1, we get the eigenvalues λ1≈-0.0750+1.4876i and λ2≈-0.0750-1.4876i. For equilibrium E2, the eigenvalues are λ1≈1.8548 and λ2≈-2.0048. For equilibrium E3, we obtain the eigenvalues λ1≈1.1521 and λ2≈-1.3021. According to these eigenvalues, we can conclude that a minimal commensurate order to obtain the chaotic behavior of system (38) is [45] (40)q>2πarctan⁡1.48760.0750=0.9679.Under the initial conditionsζ10=2 and  ζ20=-1 and the fractional order q=0.98, FOCS (38) shows a chaotic behavior, which is depicted in Figure 1.Figure 1
Phase attractor of FOCS (38).According to (30) and (38), it is easy to know that (41)P=0110,Q=0001,ħt=0-ζ13t-0.15ζ2t+0.3cos⁡t.Since system (38) is a chaotic system, then we know that both signals ζ1(t) and ζ2(t) are bounded (from Figure 1, one knows that |ζ1(t)|<2 and  |ζ2(t)|<2). Thus, Assumption 9 is satisfied with a0=3.In the simulation, the initial condition for the slave FOCS isζ^1(0)=-2 and  ζ^2(0)=4. Suppose that G=diag⁡1,1. The design matrices are chosen as (42)K=-733452-12,Λ=0.10.20.20.3.Thus, we have that Q-ΛQ=0.728,  λ_=3.1, and the two conditions P-ΛK<0 and a0Q-ΛQ≤λ_q in Theorem 10 are satisfied.The simulation results are presented in Figures2–5. The results where the state variables of the slave FOCS track the master system’s states are presented in Figures 2 and 3. The time response of the synchronization errors is depicted in Figure 4. From these pictures, we can see that the synchronization controller works well, and the synchronization errors converge to the origin fast. From (35), we know that the synchronization control input is a continuous function. The smoothness of the control input is given in Figure 5, from which we can see that the proposed controller has small fluctuation.Figure 2
Synchronization betweenζ1t and ζ^1t.Figure 3
Synchronization betweenζ2t and ζ^2t.Figure 4
Time response of synchronization errorse1t and e2t.Figure 5
Time response of control inputsu1t and u2t.


### 4.2. Synchronizing Two 3D Fractional-Order Chaotic Neural Networks

Let us consider the following fractional-order chaotic neural networks expressed by [15] (43)Dqζ1t=-ζ1t+2tanh⁡ζ1t-1.2tanh⁡ζ2t,Dqζ2t=-ζ2t+2tanh⁡ζ1t+1.71tanh⁡ζ2t+1.15tanh⁡ζ3t,Dqζ3t=-ζ3t-4.75tanh⁡ζ1t+1.10tanh⁡ζ3t.Suppose thatq=0.95 and the initial condition is ζ10=-0.3,  ζ20=0.4, and  ζ30=0.3. The dynamical behavior of FOCS (43) is given in Figure 6.Figure 6
Dynamical behavior of system (43) in (a) 3D space, (b) ζ1t-ζ2t plane, (c) ζ1t-ζ3t plane, and (d) ζ2t-ζ3t plane.
(a)
(b)
(c)
(d)It is easy to know in the master chaotic system (43) that P=diag⁡-1,-1,-1, ħt=tanh⁡ζ1t,tanh⁡ζ2t,tanh⁡ζ3tT, (44)Q=2.00-1.200.002.001.711.15-4.750.001.10.Thus, we have Q=1.235 and ħ(t) satisfy the Lipshitz condition. The Lipshitz constant a0 can be chosen as 1.The initial condition of the slave FOCS isζ^10=3.2,  ζ^20=-4, and  ζ^30=-3.5. Let G=diag⁡1,1,1. The design matrices are chosen as (45)K=1.2484-0.14010.01270.01271.1210-1.1019-0.1146-0.08920.9172,Λ=0.80.1000.90.10.10.11.1.Thus, we know that Q-ΛQ=0.7115,  λ_=2, and the two conditions P-ΛK<0 and a0Q-ΛQ≤λ_q in Theorem 10 are satisfied.The simulation results are given in Figures7 and 8. Just like the results in Figures 2–5, we know that good synchronization performance has been obtained.Figure 7
Simulation results in (a) synchronization betweenζ1t and ζ^1t, (b) synchronization between ζ2t and ζ^2t, (c) synchronization between ζ3t and ζ^3(t), and (d) synchronization errors e1t,  e2t, and e3t.
(a)
(b)
(c)
(d)Figure 8
Time response of control inputsu1(t),  u2(t), and u3(t).


## 4.1. Synchronizing Two 2D Fractional-Order Duffing Systems

The fractional-order Duffing system is described by [46] (38)Dqζ1t=ζ2t,Dqζ2t=ζ1t-ζ13t-0.15ζ2t+0.3cos⁡t.The Jacobian matrix of system (38) for the equilibrium point E1∗=(ζ1∗,ζ2∗) is (39)JE∗=011-3ζ1∗-0.15.It is easy to know that system (38) has three equilibria: E1=1.0729,0,  E2=-0.9062,0, and E3=-0.1667,0. For equilibrium E1, we get the eigenvalues λ1≈-0.0750+1.4876i and λ2≈-0.0750-1.4876i. For equilibrium E2, the eigenvalues are λ1≈1.8548 and λ2≈-2.0048. For equilibrium E3, we obtain the eigenvalues λ1≈1.1521 and λ2≈-1.3021. According to these eigenvalues, we can conclude that a minimal commensurate order to obtain the chaotic behavior of system (38) is [45] (40)q>2πarctan⁡1.48760.0750=0.9679.Under the initial conditionsζ10=2 and  ζ20=-1 and the fractional order q=0.98, FOCS (38) shows a chaotic behavior, which is depicted in Figure 1.Figure 1
Phase attractor of FOCS (38).According to (30) and (38), it is easy to know that (41)P=0110,Q=0001,ħt=0-ζ13t-0.15ζ2t+0.3cos⁡t.Since system (38) is a chaotic system, then we know that both signals ζ1(t) and ζ2(t) are bounded (from Figure 1, one knows that |ζ1(t)|<2 and  |ζ2(t)|<2). Thus, Assumption 9 is satisfied with a0=3.In the simulation, the initial condition for the slave FOCS isζ^1(0)=-2 and  ζ^2(0)=4. Suppose that G=diag⁡1,1. The design matrices are chosen as (42)K=-733452-12,Λ=0.10.20.20.3.Thus, we have that Q-ΛQ=0.728,  λ_=3.1, and the two conditions P-ΛK<0 and a0Q-ΛQ≤λ_q in Theorem 10 are satisfied.The simulation results are presented in Figures2–5. The results where the state variables of the slave FOCS track the master system’s states are presented in Figures 2 and 3. The time response of the synchronization errors is depicted in Figure 4. From these pictures, we can see that the synchronization controller works well, and the synchronization errors converge to the origin fast. From (35), we know that the synchronization control input is a continuous function. The smoothness of the control input is given in Figure 5, from which we can see that the proposed controller has small fluctuation.Figure 2
Synchronization betweenζ1t and ζ^1t.Figure 3
Synchronization betweenζ2t and ζ^2t.Figure 4
Time response of synchronization errorse1t and e2t.Figure 5
Time response of control inputsu1t and u2t.


## 4.2. Synchronizing Two 3D Fractional-Order Chaotic Neural Networks

Let us consider the following fractional-order chaotic neural networks expressed by [15] (43)Dqζ1t=-ζ1t+2tanh⁡ζ1t-1.2tanh⁡ζ2t,Dqζ2t=-ζ2t+2tanh⁡ζ1t+1.71tanh⁡ζ2t+1.15tanh⁡ζ3t,Dqζ3t=-ζ3t-4.75tanh⁡ζ1t+1.10tanh⁡ζ3t.Suppose thatq=0.95 and the initial condition is ζ10=-0.3,  ζ20=0.4, and  ζ30=0.3. The dynamical behavior of FOCS (43) is given in Figure 6.Figure 6
Dynamical behavior of system (43) in (a) 3D space, (b) ζ1t-ζ2t plane, (c) ζ1t-ζ3t plane, and (d) ζ2t-ζ3t plane.
(a)
(b)
(c)
(d)It is easy to know in the master chaotic system (43) that P=diag⁡-1,-1,-1, ħt=tanh⁡ζ1t,tanh⁡ζ2t,tanh⁡ζ3tT, (44)Q=2.00-1.200.002.001.711.15-4.750.001.10.Thus, we have Q=1.235 and ħ(t) satisfy the Lipshitz condition. The Lipshitz constant a0 can be chosen as 1.The initial condition of the slave FOCS isζ^10=3.2,  ζ^20=-4, and  ζ^30=-3.5. Let G=diag⁡1,1,1. The design matrices are chosen as (45)K=1.2484-0.14010.01270.01271.1210-1.1019-0.1146-0.08920.9172,Λ=0.80.1000.90.10.10.11.1.Thus, we know that Q-ΛQ=0.7115,  λ_=2, and the two conditions P-ΛK<0 and a0Q-ΛQ≤λ_q in Theorem 10 are satisfied.The simulation results are given in Figures7 and 8. Just like the results in Figures 2–5, we know that good synchronization performance has been obtained.Figure 7
Simulation results in (a) synchronization betweenζ1t and ζ^1t, (b) synchronization between ζ2t and ζ^2t, (c) synchronization between ζ3t and ζ^3(t), and (d) synchronization errors e1t,  e2t, and e3t.
(a)
(b)
(c)
(d)Figure 8
Time response of control inputsu1(t),  u2(t), and u3(t).


## 5. Conclusion

In this paper, two stability criteria for fractional-order nonlinear systems are given. Based on these theorems, the synchronization of two identical FOCSs is addressed. A fractional-order sliding surface, which contains a fractional-order integral of the synchronization errors, is given. The proposed controller can guarantee the asymptotical stability of the closed-loop systems. However, in the controller design, we need to know the exact value of the Lipchitz constant. How to reduce this condition is one of our future research directions.



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*Source: 3545083-2018-01-17.xml*