# Dynamics Feature and Synchronization of a Robust Fractional-Order Chaotic System

Authors: Xuan-Bing Yang; Yi-Gang He; Chun-Lai Li
Journal: Complexity
 (2018)

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## Abstract

Exploring the dynamics feature of robust chaotic system is an attractive yet recent topic of interest. In this paper, we introduce a three-dimensional fractional-order chaotic system. The important finding by analysis is that the position of signalx3 descends at the speed of 1/c as the parameterb increases, and the signal amplitude ofx1,x2 can be controlled by the parameterm in terms of the power function with the index −1/2. What is more, the dynamics remains constant with the variation of parametersb andm. Consequently, this system can provide rich encoding keys for chaotic communication. By considering the properties of amplitude and position modulation, the partial projective synchronization and partial phase synchronization are realized with linear control scheme. The distribution map of optimal synchronization region in the control-parameter space is charted by defining the power consumption of controller. Numerical simulations are executed to confirm the theoretical analysis.


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## Body

## 1. Introduction

Over the past few decades, the dynamics and property of chaotic system have been extensively studied from different points of views as an active topic [1–4]. The investigation of chaos has benefited the exploration of the complex behavior, intrinsic nonlinear structure of natural system, and the construction of chaotic system, as well as practical applications such as secure communications and signal detection [5–9]. Robust chaotic system can usually provide signal-amplitude modulation by controlling one or some of the parameters in the dynamical equations yet keep the Lyapunov exponents and power spectral density invariable [10–14]. Therefore, it is a type of chaotic system with potential applications in synchronization, signal processing, image encryption, chaotic radar, and chaotic communication [10, 12]. However, according to what we know, there is little information and variety about such system reported in the present literatures.Although fractional calculus has a history of more than 300 years, it was not really applied to physics, life sciences, and psychology until the last decade [15–18]. Fractional calculus can provide an excellent description of hereditary and memory properties in various materials and processes. The advantage of the fractional system is that it holds more degrees of freedom and that contains a “memory” in it. Therefore, it is of universal significance to study fractional dynamics and its applications [19–22]. Synchronization of integer-order chaotic system is investigated extensively and deeply, and many different types and methods have been presented since the landmark work by Pecora and Carroll [23–29]. However, due to the computational complexity of fractional systems, not all synchronization methods of the integer-order system are suitable for the fractional-order one [10]. In the existing synchronization types, projective synchronization can be interpreted as that the state trajectories of the drive and response systems synchronize to a constant proportion factor. This characteristic is usually used to extend binary digital communication to faster M-nary communication [30, 31]. Another synchronization type to be worth mentioning is phase synchronization, in which the controlled chaotic system adjusts the signal frequencies of dynamics to the rhythm of another chaotic system while the amplitudes go on varying in an irregular and uncorrelated fashion [32]. Phase synchronization has been observed in electrochemical oscillators [33], plasma discharge tubes [34], coupled HR neurons [35], fractional differential chaotic systems [36], etc.In this paper, we attempt to explore some new dynamics properties of robust chaotic system by constructing a three-dimensional fractional chaotic system and to further consider the synchronization problem. Analysis of the derived system shows that the parameterm can control the signal amplitude ofx1,x2 by the power function with the index −1/2 and that the parameterb can control the position of signalx3 at the descent velocity of 1/c. Then, by considering the vital property of amplitude and position modulation, a linear coupling scheme is designed to realize the partial projective synchronization and partial phase synchronization, respectively. Thus, the proportional factors in projective synchronization and the phase feature in phase synchronization are only determined by system parameters, which will improve the security of synchronous communication. The coupling-parameter range for synchronization is derived analytically and can be effectively narrowed down by appropriately selecting the auxiliary parameters. The distribution map of optimal synchronization region in the coupling-parameter space is further evaluated by defining the power consumption of controller. Numerical simulations are shown to further confirm the theoretical analysis.


## 2. Model of Fractional-Order Chaotic System

### 2.1. Model Description

The introduced fractional system is written in the form(1)Dqx1=x2-ax1Dqx2=-bx1-cx1x3Dqx3=-n+mx1x2In system (1), Dqx=dqx/dtq denotes the Caputo fractional derivative with the initial time t=0 [37]. With the positive parametersa,b,c,n, andm, two equilibrium points of system (1) are obtained as (2)E+nam,anm,-bc,E--nam,-anm,-bc.The corresponding characteristic equation for any equilibrium pointxE1,xE2,xE3 can be deduced as(3)φλ=-λ3-aλ2-b+cxE3+cmxE12λ-acmxE12-cmxE1xE2Specially, when selecting the parameter setP={a=2,b=1,c=1,n=30,m=1}, the corresponding equilibrium points and characteristic roots are (4)E+3.873,7.746,-1:λ1=-3.1912,λ2=0.5956+4.2950i,λ3=0.5956-4.2950iE--3.873,-7.746,-1:λ1=-3.1912,λ2=0.5956+4.2950i,λ3=0.5956-4.2950iWith the aid of stability theory of commensurate fractional system [38], the necessary condition for the chaos emergence is q>2/πatan⁡(Imλ∗/Reλ∗)=2/πatan⁡(4.295/0.5956)=0.9123. For the parameter setP and the fractional orderq=0.96, the 3D phase diagram and Poincaré map with dense dots are shown in Figure 1, revealing that system (1) is indeed chaotic.Figure 1
(a) 3D phase portrait; (b) Poincaré map withx3=0.6.
(a) (b)


### 2.2. Dynamics Feature of Amplitude and Position Modulation

Our analysis found that the introduced system has the robust chaos of constant Lyapunov exponents with the variation of parametersm andb. More specifically, the parameterm can control some of the signal amplitude of state variables, and the parameterb can control the position of one of the variables with a certain speed. The revealed feature is rare in the current literature and we believe it will stimulate an exploration boom.The linear transformation ofx1→x1/m, x2→x2/m, and x3→x3 is first considered to deduce system (1) to the normalized form about parameterm[12], as follows: (5)Dqx1=x2-ax1Dqx2=-bx1-cx1x3Dqx3=-n+x1x2Therefore, the control parameterm can modulate the signal amplitude of variablesx1,x2 according to 1/m, but the signalx3 keeps its amplitude constant.We substitute the nonzero equilibrium pointE+ or E- into characteristic equation (3) to obtain (6)φλ=-λ3-aλ2-cnaλ-2cnSince (6) is irrespective to parameterm, the Lyapunov exponent spectrum remains invariable whenm increases. The dynamics feature of amplitude modulation for system (1) is illustrated by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 2. In the figure, the bifurcation diagrams are plotted with the local maxima of variables versus control parameter, and the spectrum of Lyapunov exponents is calculated by the wolf method.Figure 2
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameterm.
(a) (b) (c) (d)When introducing the linear transformation ofx1→x1, x2→x2, and x3→x3+kp/c, we then deduce system (1) to (7)Dqx1=x2-ax1Dqx2=-b+kpx1-cx1x3Dqx3=-n+mx1x2Therefore, the position of signalx3 descends at the speed of 1/c as the parameterb increases, while the signalsx1,x2 keep their amplitude invariable. Similarly, it is known that (6) is also irrespective to parameterb. Thus, the spectrum of Lyapunov exponent remains constant whenb varies in field of real number. The phenomenon of position modulation for system (1) is described by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 3. And the speed of position modulation influenced by parameterc is interpreted in Figure 4 by the bifurcation diagram and the maximal Lyapunov exponent spectrum, which signifies that a smallerc will give rise to a larger descent velocity.Figure 3
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameterb with the parameter setPexcept forb.
(a) (b) (c) (d)Figure 4
Influence ofc on position modulation interpreted by (a) Bifurcation diagram and (b) Lyapunov exponent spectrum versusb withc=1 in magenta andc=0.5 in blue.
(a) (b)


## 2.1. Model Description

The introduced fractional system is written in the form(1)Dqx1=x2-ax1Dqx2=-bx1-cx1x3Dqx3=-n+mx1x2In system (1), Dqx=dqx/dtq denotes the Caputo fractional derivative with the initial time t=0 [37]. With the positive parametersa,b,c,n, andm, two equilibrium points of system (1) are obtained as (2)E+nam,anm,-bc,E--nam,-anm,-bc.The corresponding characteristic equation for any equilibrium pointxE1,xE2,xE3 can be deduced as(3)φλ=-λ3-aλ2-b+cxE3+cmxE12λ-acmxE12-cmxE1xE2Specially, when selecting the parameter setP={a=2,b=1,c=1,n=30,m=1}, the corresponding equilibrium points and characteristic roots are (4)E+3.873,7.746,-1:λ1=-3.1912,λ2=0.5956+4.2950i,λ3=0.5956-4.2950iE--3.873,-7.746,-1:λ1=-3.1912,λ2=0.5956+4.2950i,λ3=0.5956-4.2950iWith the aid of stability theory of commensurate fractional system [38], the necessary condition for the chaos emergence is q>2/πatan⁡(Imλ∗/Reλ∗)=2/πatan⁡(4.295/0.5956)=0.9123. For the parameter setP and the fractional orderq=0.96, the 3D phase diagram and Poincaré map with dense dots are shown in Figure 1, revealing that system (1) is indeed chaotic.Figure 1
(a) 3D phase portrait; (b) Poincaré map withx3=0.6.
(a) (b)


## 2.2. Dynamics Feature of Amplitude and Position Modulation

Our analysis found that the introduced system has the robust chaos of constant Lyapunov exponents with the variation of parametersm andb. More specifically, the parameterm can control some of the signal amplitude of state variables, and the parameterb can control the position of one of the variables with a certain speed. The revealed feature is rare in the current literature and we believe it will stimulate an exploration boom.The linear transformation ofx1→x1/m, x2→x2/m, and x3→x3 is first considered to deduce system (1) to the normalized form about parameterm[12], as follows: (5)Dqx1=x2-ax1Dqx2=-bx1-cx1x3Dqx3=-n+x1x2Therefore, the control parameterm can modulate the signal amplitude of variablesx1,x2 according to 1/m, but the signalx3 keeps its amplitude constant.We substitute the nonzero equilibrium pointE+ or E- into characteristic equation (3) to obtain (6)φλ=-λ3-aλ2-cnaλ-2cnSince (6) is irrespective to parameterm, the Lyapunov exponent spectrum remains invariable whenm increases. The dynamics feature of amplitude modulation for system (1) is illustrated by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 2. In the figure, the bifurcation diagrams are plotted with the local maxima of variables versus control parameter, and the spectrum of Lyapunov exponents is calculated by the wolf method.Figure 2
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameterm.
(a) (b) (c) (d)When introducing the linear transformation ofx1→x1, x2→x2, and x3→x3+kp/c, we then deduce system (1) to (7)Dqx1=x2-ax1Dqx2=-b+kpx1-cx1x3Dqx3=-n+mx1x2Therefore, the position of signalx3 descends at the speed of 1/c as the parameterb increases, while the signalsx1,x2 keep their amplitude invariable. Similarly, it is known that (6) is also irrespective to parameterb. Thus, the spectrum of Lyapunov exponent remains constant whenb varies in field of real number. The phenomenon of position modulation for system (1) is described by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 3. And the speed of position modulation influenced by parameterc is interpreted in Figure 4 by the bifurcation diagram and the maximal Lyapunov exponent spectrum, which signifies that a smallerc will give rise to a larger descent velocity.Figure 3
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameterb with the parameter setPexcept forb.
(a) (b) (c) (d)Figure 4
Influence ofc on position modulation interpreted by (a) Bifurcation diagram and (b) Lyapunov exponent spectrum versusb withc=1 in magenta andc=0.5 in blue.
(a) (b)


## 3. Preliminaries for Synchronization of Fractional Chaotic System

In this section, some concepts and techniques are recalled for the stability analysis of fractional system.Definition 1 (see [39]).
When functionα:R+→R+ is continuous, strictly increasing, and α(0)=0, it is said to be of class κ. If α∈κ and satisfies α(t)→∞ with t→∞, it is said that α is of class κ∞.Lemma 2 (see [40, 41]).
Letz(t)∈R be a continuous and derivable function. Then, for any time instant t≥0, it is held: (8)0.5Dqz2t≤ztDqzt,∀q∈0,1Lemma 3 (see [42, 43]).
Letz(t)∈Rn be a real-valued continuous and derivable vector function. Then, for any time instant t≥0, one will always hold the following inequality: (9)0.5DqzTtPzt≤zTtPDqzt,∀q∈0,1,P=diagp1,p2,⋯,pn>0Theorem 4 (Lyapunov stability and uniform stability of fractional system [42]).
Considering the following fractional-order system with the Caputo definition(10)Dqzt=fz,t,q∈0,1Letz=0 be an equilibrium point of system (10). If there exists a continuous Lyapunov function V(z,t) and a scalar class-K function α1 satisfying (11)Vz,t≥α1ztand (12)DqVz,t≤0,q∈0,1for any z≠0, then system (10) is Lyapunov stable at z=0.
Furthermore, if there exists another scalar class-K function α2 such that (13)Vz,t≤α2ztthen the origin of system (10) is said to be Lyapunov uniformly stable.


## 4. Partial Projective Synchronization of Fractional Chaotic System

### 4.1. Synchronization Scheme

The partial projective synchronization of the proposed system is studied here by taking the advantage of the property of amplitude modulation.Let system (1) be as the drive system, and the response system is expressed as (14)Dqy1=y2-ay1Dqy2=-by1-cy1y3+u2Dqy3=-n+m1y1y2+u3whereu2 andu3 are the controllers to be determined.Assumption 5.
The state variables of systems (1) and (14) are all bounded, and there exist three positive constants σ1, σ2, and σ3, such that x1,y1≤σ1, x2,y2≤σ2, and x3,y3≤σ3.The synchronization errors are set ase1=y1-m/m1x1, e2=y2-m/m1x2, and e3=y3-x3, by taking the property of amplitude modulation considered. Then we obtain the error dynamical system (15)Dqe1=e2-ae1Dqe2=-be1-cy1e3-ce1x3+u2Dqe3=m1y2e1+m1mm1x1e2+u3Theorem 6.
For the drive system (1) and response system (14), the controllers are designed as u2=-k2e2 and u3=-k3e3. If the coupling parametersk2 andk3 satisfy (16)k2>p1-bp2+cp2σ324ap1p2(17)k3>m1p3σ2p1-bp2+cp2σ3m1p3m/m1σ1+cp2σ1+k2p2m1p3σ22+ap1m1p3m/m1σ1+cp2σ12p34ap1p2k2-p1-bp2+cp2σ32wherep1,p2, andp3 are positive auxiliary parameters to narrow down the values range of the coupling parameters, then the partial projective synchronization is realized with Lyapunov uniformly stability.Proof.
Let us propose the Lyapunov functionV(t)=0.5p1e12+p2e22+p3e32, which satisfies α1(e)≤V(t)≤α2(e) for α1(e)=λmin(P)e2, α2(e)=λmax(P)e2, and P=diagp1,p2,p3. Takingq-order fractional derivative with respect to timet along the trajectories of (15), it yields (18)
D
q
V
t
=
0.5
D
q
p
1
e
1
2
+
0.5
D
q
p
2
e
2
2
+
0.5
D
q
p
3
e
3
2
≤
p
1
e
1
D
q
e
1
+
p
2
e
2
D
q
e
2
+
p
3
e
3
D
q
e
3
=
-
a
p
1
e
1
2
-
k
2
p
2
e
2
2
-
k
3
p
3
e
3
2
+
p
1
-
b
p
2
-
c
p
2
x
3
e
1
e
2
+
m
1
p
3
m
m
1
x
1
-
c
p
2
y
1
e
2
e
3
+
m
1
p
3
y
2
e
1
e
3
≤
-
a
p
1
e
1
2
-
k
2
p
2
e
2
2
-
k
3
p
3
e
3
2
+
p
1
-
b
p
2
+
c
p
2
σ
3
e
1
e
2
+
m
1
p
3
m
m
1
σ
1
+
c
p
2
σ
1
e
2
e
3
+
m
1
p
3
σ
2
e
1
e
3
=
e
T
Π
ewith (19)e=e1,e2,e3T,Π=-ap1p1-bp2+cp2σ32m1p3σ22p1-bp2+cp2σ32-p2k2m1p3m/m1σ1+cp2σ12m1p3σ22m1p3m/m1σ1+cp2σ12-p3k3.
It requiresΠ≤0 for catering to DqV(t)≤0. Thus we have(20)Δ1=-ap1≤0Δ2=det-ap1p1-bp2+cp2σ32p1-bp2+cp2σ32-p2k2≥0Δ3=det-ap1p1-bp2+cp2σ32m1p3σ22p1-bp2+cp2σ32-p2k2m1p3m/m1σ1+cp2σ12m1p3σ22m1p3m/m1σ1+cp2σ12-p3k3≤0
Therefore, we finally obtain inequalities (16) and (17). This completes the proof.


### 4.2. Simulation Analysis

The appropriate selection of parametersp1,p2, andp3 can effectively narrow down the values range of the coupling parameters, which is useful in actual synchronization process. However, it is a complicated relation between the coupling strengths (k2,k3) and (p1,p2,p3). To optimize the synchronization scheme, we will evaluate the distribution map of optimal synchronization region in the coupling-parameter space, by coopting the idea introduced by Ma [44]. We first define the power consumption of the synchronous systems as pd=x1x3 and pr=y1y3, respectively. Thus, the power consumption for evaluating the synchronous controller can be defined as (21)psyn=y1y3-mm1x1x3And for the sake of simplicity, we only consider the maximal power consumption of synchronous controller, as below(22)pmsyn=max⁡psynTherefore, the optimal synchronization region will be the coupling-parameter region (k2,k3) with the minimum pmsyn.In the numerical analysis, we choose the parameter values asa=2,b=1,c=1,n=30,m=1,m1=16, andq=0.96 and set the initial values of drive system and the response system asx(0)=(-0.2, 0.1, 0.5),y(0)=(0.6, -0.5, -0.2). The distribution about the maximal power consumption of synchronous controller in the coupling-parameter space (k2,k3) is illustrated in Figure 5. It is known that the power consumption pmsyn is mainly determined by the coupling parameterk2, and the optimal synchronization region appears alternately ask2 increases. But to get shorter synchronous transition time, larger values ofk2 andk3 are more appropriate.Figure 5
The distribution of the maximal power consumption of synchronous controller.The synchronization result withk2=3.3,k3=0.6 is depicted in Figure 6, which provides the proportional factors α1=m/m1=0.25, α2=m/m1=0.25, and α3=1. Figure 7 shows the synchronization result with the same proportional factors, whenk2=6,k3=8. The comparative analysis shows that larger parametersk2 andk3 will lead to shorter synchronous transition time.Figure 6
Partial projective synchronization withk2=3.3;k3=0.6: (a) time response; (b) synchronization error.
(a) (b)Figure 7
Partial projective synchronization withk2=6;k3=8: (a) time response; (b) synchronization error.
(a) (b)


## 4.1. Synchronization Scheme

The partial projective synchronization of the proposed system is studied here by taking the advantage of the property of amplitude modulation.Let system (1) be as the drive system, and the response system is expressed as (14)Dqy1=y2-ay1Dqy2=-by1-cy1y3+u2Dqy3=-n+m1y1y2+u3whereu2 andu3 are the controllers to be determined.Assumption 5.
The state variables of systems (1) and (14) are all bounded, and there exist three positive constants σ1, σ2, and σ3, such that x1,y1≤σ1, x2,y2≤σ2, and x3,y3≤σ3.The synchronization errors are set ase1=y1-m/m1x1, e2=y2-m/m1x2, and e3=y3-x3, by taking the property of amplitude modulation considered. Then we obtain the error dynamical system (15)Dqe1=e2-ae1Dqe2=-be1-cy1e3-ce1x3+u2Dqe3=m1y2e1+m1mm1x1e2+u3Theorem 6.
For the drive system (1) and response system (14), the controllers are designed as u2=-k2e2 and u3=-k3e3. If the coupling parametersk2 andk3 satisfy (16)k2>p1-bp2+cp2σ324ap1p2(17)k3>m1p3σ2p1-bp2+cp2σ3m1p3m/m1σ1+cp2σ1+k2p2m1p3σ22+ap1m1p3m/m1σ1+cp2σ12p34ap1p2k2-p1-bp2+cp2σ32wherep1,p2, andp3 are positive auxiliary parameters to narrow down the values range of the coupling parameters, then the partial projective synchronization is realized with Lyapunov uniformly stability.Proof.
Let us propose the Lyapunov functionV(t)=0.5p1e12+p2e22+p3e32, which satisfies α1(e)≤V(t)≤α2(e) for α1(e)=λmin(P)e2, α2(e)=λmax(P)e2, and P=diagp1,p2,p3. Takingq-order fractional derivative with respect to timet along the trajectories of (15), it yields (18)
D
q
V
t
=
0.5
D
q
p
1
e
1
2
+
0.5
D
q
p
2
e
2
2
+
0.5
D
q
p
3
e
3
2
≤
p
1
e
1
D
q
e
1
+
p
2
e
2
D
q
e
2
+
p
3
e
3
D
q
e
3
=
-
a
p
1
e
1
2
-
k
2
p
2
e
2
2
-
k
3
p
3
e
3
2
+
p
1
-
b
p
2
-
c
p
2
x
3
e
1
e
2
+
m
1
p
3
m
m
1
x
1
-
c
p
2
y
1
e
2
e
3
+
m
1
p
3
y
2
e
1
e
3
≤
-
a
p
1
e
1
2
-
k
2
p
2
e
2
2
-
k
3
p
3
e
3
2
+
p
1
-
b
p
2
+
c
p
2
σ
3
e
1
e
2
+
m
1
p
3
m
m
1
σ
1
+
c
p
2
σ
1
e
2
e
3
+
m
1
p
3
σ
2
e
1
e
3
=
e
T
Π
ewith (19)e=e1,e2,e3T,Π=-ap1p1-bp2+cp2σ32m1p3σ22p1-bp2+cp2σ32-p2k2m1p3m/m1σ1+cp2σ12m1p3σ22m1p3m/m1σ1+cp2σ12-p3k3.
It requiresΠ≤0 for catering to DqV(t)≤0. Thus we have(20)Δ1=-ap1≤0Δ2=det-ap1p1-bp2+cp2σ32p1-bp2+cp2σ32-p2k2≥0Δ3=det-ap1p1-bp2+cp2σ32m1p3σ22p1-bp2+cp2σ32-p2k2m1p3m/m1σ1+cp2σ12m1p3σ22m1p3m/m1σ1+cp2σ12-p3k3≤0
Therefore, we finally obtain inequalities (16) and (17). This completes the proof.


## 4.2. Simulation Analysis

The appropriate selection of parametersp1,p2, andp3 can effectively narrow down the values range of the coupling parameters, which is useful in actual synchronization process. However, it is a complicated relation between the coupling strengths (k2,k3) and (p1,p2,p3). To optimize the synchronization scheme, we will evaluate the distribution map of optimal synchronization region in the coupling-parameter space, by coopting the idea introduced by Ma [44]. We first define the power consumption of the synchronous systems as pd=x1x3 and pr=y1y3, respectively. Thus, the power consumption for evaluating the synchronous controller can be defined as (21)psyn=y1y3-mm1x1x3And for the sake of simplicity, we only consider the maximal power consumption of synchronous controller, as below(22)pmsyn=max⁡psynTherefore, the optimal synchronization region will be the coupling-parameter region (k2,k3) with the minimum pmsyn.In the numerical analysis, we choose the parameter values asa=2,b=1,c=1,n=30,m=1,m1=16, andq=0.96 and set the initial values of drive system and the response system asx(0)=(-0.2, 0.1, 0.5),y(0)=(0.6, -0.5, -0.2). The distribution about the maximal power consumption of synchronous controller in the coupling-parameter space (k2,k3) is illustrated in Figure 5. It is known that the power consumption pmsyn is mainly determined by the coupling parameterk2, and the optimal synchronization region appears alternately ask2 increases. But to get shorter synchronous transition time, larger values ofk2 andk3 are more appropriate.Figure 5
The distribution of the maximal power consumption of synchronous controller.The synchronization result withk2=3.3,k3=0.6 is depicted in Figure 6, which provides the proportional factors α1=m/m1=0.25, α2=m/m1=0.25, and α3=1. Figure 7 shows the synchronization result with the same proportional factors, whenk2=6,k3=8. The comparative analysis shows that larger parametersk2 andk3 will lead to shorter synchronous transition time.Figure 6
Partial projective synchronization withk2=3.3;k3=0.6: (a) time response; (b) synchronization error.
(a) (b)Figure 7
Partial projective synchronization withk2=6;k3=8: (a) time response; (b) synchronization error.
(a) (b)


## 5. Partial Phase Synchronization of Fractional Chaotic System

### 5.1. Synchronization Scheme

Select the drive system (1) and the following response system: (23)Dqy1=y2-ay1Dqy2=-b1y1-cy1y3+u2Dqy3=-n+my1y2+u3And when considering the property of position modulation, the errors of partial phase synchronization are expressed ase1=y1-x1, e2=y2-x2, and e3=y3-x3+(b1-b)/c, then the error dynamical system is obtained as (24)Dqe1=e2-ae1Dqe2=-be1-cy1e3-ce1x3+u2Dqe3=me1y2+mx1e2+u3Theorem 7.
For the drive system (1) and response system (23), the controllers are designed as u2=-k2e2, u3=-k3e3. If the coupling parametersk2 andk3 satisfy the condition (25)k2>p1-bp2+cp2σ324ap1p2(26)k3≥mp3σ2mp3σ1+cp2σ1p1-bp2+cp2σ3+k2p2mp3σ22+ap1mp3σ1+cp2σ124ap1p2p3k2-p3p1-bp2+cp2σ32wherep1,p2, andp3are positive constants, then the partial phase synchronization is realized with Lyapunov uniformly stability.Here we skip the proof process for brevity since it resembles the one of Theorem6.


### 5.2. Simulation Analysis

Likewise, to effectively evaluate the optimal synchronization region, we consider the distribution of coupling parameters by defining the maximal power consumption of synchronous controller, as follows:(27)pmsyn=max⁡y1y3-x1x3-b1-bcIn the numerical analysis, we choose the parameter values asa=2,b=1,c=1,n=30,m=1,b1=12, andq=0.96 and set the initial values of drive system and the response system asx(0)=(-0.2, 0.1, 0.5),y(0)=(0.6, -0.5, -0.2). The distribution map about the maximal power consumption of synchronous controller in the coupling-parameter space (k2,k3) is illustrated in Figure 8. It is found that the distribution of pmsyn is complicated whenk2 is less than 4, and there exists an optimized coupling-parameter region neark2=1. Whenk2 continues to increase with the condition of k3>1, the value of pmsyn gradually decreases, and finally it keeps in the optimal synchronization region with k2>9.Figure 8
The distribution of the maximal power consumption of synchronous controller.As the examples to explain our analysis, we select two sets of coupling parameters (k2=1,k3=4) and (k2=10,k3=4) to realize the optimal synchronization. It can be found that we can realize the partial phase synchronization of fractional chaotic system with the selected coupling parameters, but larger parametersk2 andk3 will lead to shorter synchronous transition time, as illustrated in Figures 9 and 10. Further, to analyze the influence ofc on the difference between various states of synchronized systems, the representative process of phase synchronization whenk2=10,k3=4, andc=0.5 is considered, as shown in Figure 11. The graphs in Figures 10 and 11 signify that a smallerc will lead to a larger difference between the system variables of the phase synchronization.Figure 9
Partial phase synchronization withk2=1;k3=4: (a) time response; (b) synchronization error.
(a) (b)Figure 10
Partial phase synchronization withk2=10,k3=4, andc=1: (a) time response; (b) synchronization error.
(a) (b)Figure 11
Partial phase synchronization withk2=10,k3=4, andc=0.5: (a) time response; (b) synchronization error.
(a) (b)


## 5.1. Synchronization Scheme

Select the drive system (1) and the following response system: (23)Dqy1=y2-ay1Dqy2=-b1y1-cy1y3+u2Dqy3=-n+my1y2+u3And when considering the property of position modulation, the errors of partial phase synchronization are expressed ase1=y1-x1, e2=y2-x2, and e3=y3-x3+(b1-b)/c, then the error dynamical system is obtained as (24)Dqe1=e2-ae1Dqe2=-be1-cy1e3-ce1x3+u2Dqe3=me1y2+mx1e2+u3Theorem 7.
For the drive system (1) and response system (23), the controllers are designed as u2=-k2e2, u3=-k3e3. If the coupling parametersk2 andk3 satisfy the condition (25)k2>p1-bp2+cp2σ324ap1p2(26)k3≥mp3σ2mp3σ1+cp2σ1p1-bp2+cp2σ3+k2p2mp3σ22+ap1mp3σ1+cp2σ124ap1p2p3k2-p3p1-bp2+cp2σ32wherep1,p2, andp3are positive constants, then the partial phase synchronization is realized with Lyapunov uniformly stability.Here we skip the proof process for brevity since it resembles the one of Theorem6.


## 5.2. Simulation Analysis

Likewise, to effectively evaluate the optimal synchronization region, we consider the distribution of coupling parameters by defining the maximal power consumption of synchronous controller, as follows:(27)pmsyn=max⁡y1y3-x1x3-b1-bcIn the numerical analysis, we choose the parameter values asa=2,b=1,c=1,n=30,m=1,b1=12, andq=0.96 and set the initial values of drive system and the response system asx(0)=(-0.2, 0.1, 0.5),y(0)=(0.6, -0.5, -0.2). The distribution map about the maximal power consumption of synchronous controller in the coupling-parameter space (k2,k3) is illustrated in Figure 8. It is found that the distribution of pmsyn is complicated whenk2 is less than 4, and there exists an optimized coupling-parameter region neark2=1. Whenk2 continues to increase with the condition of k3>1, the value of pmsyn gradually decreases, and finally it keeps in the optimal synchronization region with k2>9.Figure 8
The distribution of the maximal power consumption of synchronous controller.As the examples to explain our analysis, we select two sets of coupling parameters (k2=1,k3=4) and (k2=10,k3=4) to realize the optimal synchronization. It can be found that we can realize the partial phase synchronization of fractional chaotic system with the selected coupling parameters, but larger parametersk2 andk3 will lead to shorter synchronous transition time, as illustrated in Figures 9 and 10. Further, to analyze the influence ofc on the difference between various states of synchronized systems, the representative process of phase synchronization whenk2=10,k3=4, andc=0.5 is considered, as shown in Figure 11. The graphs in Figures 10 and 11 signify that a smallerc will lead to a larger difference between the system variables of the phase synchronization.Figure 9
Partial phase synchronization withk2=1;k3=4: (a) time response; (b) synchronization error.
(a) (b)Figure 10
Partial phase synchronization withk2=10,k3=4, andc=1: (a) time response; (b) synchronization error.
(a) (b)Figure 11
Partial phase synchronization withk2=10,k3=4, andc=0.5: (a) time response; (b) synchronization error.
(a) (b)


## 6. Conclusion

Robust chaotic system is a recent research interest yet a promising candidate for signal processing, image encryption, chaotic radar, and chaotic communication. Therefore, it is worthwhile to explore the dynamics feature of robust chaotic system. In this paper, we introduced a robust fractional-order chaotic system and revealed the significant dynamics of position modulation and amplitude modulation.The linear control scheme is designed to realize the partial projective synchronization and partial phase synchronization, by considering the property of amplitude and position modulation. The distribution of optimal synchronization region in the control-parameter space is evaluated by searching the minimum power consumption of the linear controller. Numerical experiments are executed to confirm the theoretical analysis. Since the relation of synchronization variables only depends on the system parameters, it is not easy to attack and accurately reconstruct the drive system. Therefore, it could be significant in secure communication for masking signals.



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*Source: 8797314-2018-12-02.xml*