# Complex Dynamics of the Fractional-Order Rössler System and Its Tracking Synchronization Control

Authors: Huihai Wang; Shaobo He; Kehui Sun
Journal: Complexity
 (2018)

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## Abstract

Numerical analysis of fractional-order chaotic systems is a hot topic of recent years. The fractional-order Rössler system is solved by a fast discrete iteration which is obtained from the Adomian decomposition method (ADM) and it is implemented on the DSP board. Complex dynamics of the fractional-order chaotic system are analyzed by means of Lyapunov exponent spectra, bifurcation diagrams, and phase diagrams. It shows that the system has rich dynamics with system parameters and the derivative orderq. Moreover, tracking synchronization controllers are theoretically designed and numerically investigated. The system can track different signals including chaotic signals from the fractional-order master system and constant signals. It lays a foundation for the application of the fractional-order Rössler system.


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## Body

## 1. Introduction

Some studies have shown that fractional-order chaotic systems have more complex dynamic properties than the corresponding integer-order systems. The differential orderq of the fractional-order chaotic system equation is one of the bifurcation parameters of the system [1, 2]. It is necessary to study the corresponding fractional-order chaotic system based on the existing integer-order chaotic system. Since the well-known Rössler system was derived in 1976 [3], it has been studied as one of typical chaotic systems [4–6]. The properties and characteristics of its attractor depend on the control parametersa,b, and c. Some researchers investigated the chaotic dynamics of fractional-order Rössler systems [7–9]. In [7], Li and Chen analyzed the dynamics of fractional-order Rössler systems by employing frequency-domain method [10] based on the approximation of the transition function 1/sq and discussed only cases ofq=0.7, 0.8, 0.9. For this approximation, in general, q∈[0.1,0.9] with the step size of 0.1 is considered because of complicated mathematical transformations [11]. Obviously, the step size of order is too large. So, it is not competent when q changes continuously. In addition, whether this approximation accurately reflects the chaos characteristics of a fractional-order nonlinear system was questioned in [12, 13]. The Adams-Bashforth-Moulton predictor-corrector approaches [14–16] were used to solve the fractional-order Rössler systems in [8, 9]. However, along with the computation, the algorithm consumes too much computer resources, and the speed of calculation is very slow. So, this algorithm is not suitable for engineering applications.Adomian decomposition method (ADM) [17] is capable of dealing with linear and nonlinear problems in time domain. In [18], the fractional-order Chen system was investigated, and it shows that ADM provides the solution in a closed form and preserves the system nonlinearity. In [19], the fractional-order Lorenz system, the fractional-order Chua-Hartley system, and the fractional-order Lü system are analyzed by using ADM, and the results reveal that the method is very effective, and it leads to accurate, approximately convergent solutions. The fractional-order Lorenz-Stenflo system was analyzed by adopting ADM, and some different dynamics were provided with different order q compared with its integer-order counterpart [1]. The chaos range of the fractional-order simplified Lorenz system solved by adopting ADM is wider than that by using Adams-Bashforth-Moulton predictor-corrector approach [2]. Further, the two fractional-order chaotic systems in [1, 2] are implemented on DSP platform. Heet al. [20] concluded the characteristics of ADM which has the advantages of high accuracy, fast convergence, and less computer resource consumption. So, it is worth studying whether the fractional-order Rössler system solved by employing ADM provides different dynamics compared with its integrate-order counterpart and by using others methods.On the other hand, the synchronization of chaotic systems has always been a hot topic [21–23]. The synchronization of the fractional-order chaotic systems has just begun to attract some attentions due to its potential applications in secure communications and control processing [24]. A variety of approaches have been proposed for the synchronization of fractional-order chaotic systems [25–28]. However, numerical solutions of the above reports are obtained by employing the Adams-Bashforth-Moulton algorithm and the frequency-domain method. As far as we know, there are no articles dealing with synchronization of fractional-order Rössler systems based on ADM. Thus, it is still a novel topic to investigate the synchronization of fractional-order Rössler systems by employing ADM.The outline of this paper is as follows. In Section2, the ADM is introduced briefly, and the iterative algorithm and Lyapunov exponent spectra algorithm of the fractional-order Rössler system are presented. In Section 3, the dynamics of this fractional-order chaotic system are analyzed by Lyapunov exponent spectra and bifurcation diagram. In Section 4, the synchronization of fractional-order Rössler systems is investigated theoretically and numerically. It includes tracking constant signals and tracking chaotic signal from the master system. Finally, the results are summarized.


## 2. Numerical Analysis Methods Based on ADM

### 2.1. The Adomian Decomposition Method

For a given fractional-order differential equation​∗Dt0qx(t)=f(x(t)),f (x(t)) can be separated into three terms [29, 30],(1)​∗Dt0qxt=Lxt+Nxt+gt.Here, ​∗Dt0q is the Caputo derivative operator of orderq (m-1<q ≤m,m∈N) [31], and t0 represents the starting time,t>t0.x(t) = [x1(t), x2(t), …, xn(t)]T are state variables.L and N represent linear and nonlinear operator, respectively, andg(t)=[g1(t), g2(t), …, gn(t)]T are constants in the autonomous system. x(k)(t0+)=bk, k=0,⋯,m-1, are initial states of the equation. Jt0q is R-L fractional integral operator with orderq.Definition 1 (see [31]).
The Caputo definition is defined as(2)​∗Dt0qft=1Γ1-q∫t0tf˙τt-τqdτ,where 0<q≤1 and Γ· is the gamma function.
The fundamental properties of the integral operatorJt0q are described by [31](3)​∗Dt0qJt0qxt=xt,(4)Jt0q​∗Dt0qxt=xt-∑k=0m-1xkt0+t-t0kk!,(5)Jt0qt-t0γ=Γγ+1Γγ+1+qt-t0γ+q,(6)Jt0qC=CΓq+1t-t0q,(7)Jt0qJt0rxt=Jt0q+rxt.Here,r≥0, γ>-1, and C is a real constant. According to (4), the following equation is obtained by applying the operator Jt0q to both sides of (1) [32]:(8)x=Jt0qLx+Jt0qNx+Jt0qg+∑k=0m-1bkt-t0kk!.Based on ADM, the nonlinear terms in (8) are expressed by Adomian polynomials [32](9)Nx=∑i=0∞Aix0,x1,⋯,xi.Here,xi is the No.i(i = 0, 1, …, ∞) of Adomian polynomials, and Aji is obtained according to (10)Aji=1i!didλiNvjiλλ=0vjiλ=∑k=0iλkxjk,wherej= 1, 2, …,n (the system dimension). Then, the solution of (1) is expressed as x=∑i=0∞xi, wherexi is derived from(11)x0=Jt0qg+∑k=0m-1bkt-t0kk!x1=Jt0qLx0+Jt0qA0x0x2=Jt0qLx1+Jt0qA1x0,x1⋮xi=Jt0qLxi-1+Jt0qAi-1x0,x1,⋯,xi-1⋮.


### 2.2. Numerical Solution of Fractional-Order Rössler System

The fractional-order Rössler system is [7](12)​∗Dt0qx=-y-z​∗Dt0qy=x+ay​∗Dt0qz=b+zx-c,wherea,b, and c are the control parameters. By employing ADM and according to (5)-(7), the exact numerical solution of fractional-order Rössler system can be obtained, but it is an infinite length iteration. Considering the fast convergence performance of ADM [33], the first five terms of the iteration are truncated as approximate solution as shown in (13)-(18).(13)xm+1ym+1zm+1=x10+x11+x12+x13+x14y10+y11+y12+y13+y14z10+z11+z12+z13+z14,where(14)x10=xmy10=ymz10=zm+bhqΓq+1C10=x10C20=y10C30=z0,(15)C11=-C20-C30C110=-bx11=C11hqΓq+1+C110h2qΓ2q+1C21=C10+aC20y11=C21hqΓq+1C31=C10C30-cC30C310=bC10-bcz11=C31hqΓq+1+C310h2qΓ2q+1,(16)C12=-C21-C31C120=-C310x12=C12h2qΓ2q+1+C120h3qΓ3q+1C22=C11+aC21C220=C110y12=C22h2qΓ2q+1+C220h3qΓ3q+1C32=C11C30+C10C31-cC31C320=bC11Γ2q+1Γ2q+1+C30C10+C10C310-cC310C321=bC110Γ3q+1Γq+1Γ2q+1z12=C32h2qΓ2q+1+C320h3qΓ3q+1+C321h4qΓ4q+1,(17)C13=-C22-C32C130=-C120-C320C131=-C321x13=C13h3qΓ3q+1+C130h4qΓ4q+1+C131h5qΓ5q+1C23=C12+aC22C230=C120+aC220y13=C23h3qΓ3q+1+C230h4qΓ4q+1C33=C12C30+C10C32-cC32+C11C31Γ2q+1Γ2q+1C330=C30C120+C10C320+bC12+C11C310+C31C110Γ3q+1Γ2q+1Γq+1-cC320C331=bC120Γ4q+1Γ3q+1Γq+1+C10C321+C110C310Γ4q+1Γ22q+1-cC321z13=C33h3qΓ3q+1+C330h4qΓ4q+1+C331h5qΓ5q+1,(18)C14=-C23-C33C140=-C230-C330C141=-C331x14=C14h4qΓ4q+1+C140h5qΓ5q+1+C141h6qΓ6q+1C24=C13+aC23C240=C130+aC230y14=C24h4qΓ4q+1+C240h5qΓ5q+1+C131h6qΓ6q+1C340=C13C30+C10C33+C11C32+C12C31Γ3q+1Γ2q+1Γq+1-cC33C341=C30C130+C10C330-cC330+bC13+C11C320+C31C120Γ4q+1Γ3q+1Γq+1+C110C32+C12C310Γ4q+1Γ22q+1C342=C30C331+C10C331-cC331+bC130+C11C321Γ5q+1Γ4q+1Γq+1+C110C320+C120C310Γ5q+1Γ3q+1Γ2q+1C343=bC131Γ6q+1Γ5q+1Γq+1+C110C321Γ6q+1Γ2q+1Γ4q+1z14=C340h4qΓ4q+1+C341h5qΓ5q+1+C342h6qΓ6q+1+C343h7qΓ7q+1.Here h is iteration step size. In this paper, we seth=0.01.


### 2.3. Lyapunov Exponent Spectra

According to (13)-(18), the chaotic sequences of the fractional-order Rössler system are obtained with initial values [x0, y0, z0],q, and appropriate parametersa,b,c. So, the bifurcation diagrams of the fractional-order chaotic system are plotted. In addition, if we obtain the solution of the fractional-order chaotic system, the Lyapunov exponent spectra can be calculated by adopting QR-factorization [34] (19)qrJmJm-1...J1=qrJmJm-1...J2J1Q0=qrJmJm-1...J3J2Q1R1=qrJmJm-1...J3Q2R2R1=...=qrJmJm-1...JiQi-1Ri-1Ri-2...R2R1=...=QmRm...R2R1=QmR.Here,qr[.] denotes the QR-factorization process.J is the Jacobin matrix of the iteration (13).Q is the orthogonal matrix, andRare the diagonal elements.m is the iteration number. Then, the Lyapunov exponent spectra are calculated by(20)λk=1mh∑i=1mlnRik,k,where the system dimensionk = 1, 2, …,n. Dynamics of system (12) are investigated with the variation of derivative q and system parametera.


## 2.1. The Adomian Decomposition Method

For a given fractional-order differential equation​∗Dt0qx(t)=f(x(t)),f (x(t)) can be separated into three terms [29, 30],(1)​∗Dt0qxt=Lxt+Nxt+gt.Here, ​∗Dt0q is the Caputo derivative operator of orderq (m-1<q ≤m,m∈N) [31], and t0 represents the starting time,t>t0.x(t) = [x1(t), x2(t), …, xn(t)]T are state variables.L and N represent linear and nonlinear operator, respectively, andg(t)=[g1(t), g2(t), …, gn(t)]T are constants in the autonomous system. x(k)(t0+)=bk, k=0,⋯,m-1, are initial states of the equation. Jt0q is R-L fractional integral operator with orderq.Definition 1 (see [31]).
The Caputo definition is defined as(2)​∗Dt0qft=1Γ1-q∫t0tf˙τt-τqdτ,where 0<q≤1 and Γ· is the gamma function.
The fundamental properties of the integral operatorJt0q are described by [31](3)​∗Dt0qJt0qxt=xt,(4)Jt0q​∗Dt0qxt=xt-∑k=0m-1xkt0+t-t0kk!,(5)Jt0qt-t0γ=Γγ+1Γγ+1+qt-t0γ+q,(6)Jt0qC=CΓq+1t-t0q,(7)Jt0qJt0rxt=Jt0q+rxt.Here,r≥0, γ>-1, and C is a real constant. According to (4), the following equation is obtained by applying the operator Jt0q to both sides of (1) [32]:(8)x=Jt0qLx+Jt0qNx+Jt0qg+∑k=0m-1bkt-t0kk!.Based on ADM, the nonlinear terms in (8) are expressed by Adomian polynomials [32](9)Nx=∑i=0∞Aix0,x1,⋯,xi.Here,xi is the No.i(i = 0, 1, …, ∞) of Adomian polynomials, and Aji is obtained according to (10)Aji=1i!didλiNvjiλλ=0vjiλ=∑k=0iλkxjk,wherej= 1, 2, …,n (the system dimension). Then, the solution of (1) is expressed as x=∑i=0∞xi, wherexi is derived from(11)x0=Jt0qg+∑k=0m-1bkt-t0kk!x1=Jt0qLx0+Jt0qA0x0x2=Jt0qLx1+Jt0qA1x0,x1⋮xi=Jt0qLxi-1+Jt0qAi-1x0,x1,⋯,xi-1⋮.


## 2.2. Numerical Solution of Fractional-Order Rössler System

The fractional-order Rössler system is [7](12)​∗Dt0qx=-y-z​∗Dt0qy=x+ay​∗Dt0qz=b+zx-c,wherea,b, and c are the control parameters. By employing ADM and according to (5)-(7), the exact numerical solution of fractional-order Rössler system can be obtained, but it is an infinite length iteration. Considering the fast convergence performance of ADM [33], the first five terms of the iteration are truncated as approximate solution as shown in (13)-(18).(13)xm+1ym+1zm+1=x10+x11+x12+x13+x14y10+y11+y12+y13+y14z10+z11+z12+z13+z14,where(14)x10=xmy10=ymz10=zm+bhqΓq+1C10=x10C20=y10C30=z0,(15)C11=-C20-C30C110=-bx11=C11hqΓq+1+C110h2qΓ2q+1C21=C10+aC20y11=C21hqΓq+1C31=C10C30-cC30C310=bC10-bcz11=C31hqΓq+1+C310h2qΓ2q+1,(16)C12=-C21-C31C120=-C310x12=C12h2qΓ2q+1+C120h3qΓ3q+1C22=C11+aC21C220=C110y12=C22h2qΓ2q+1+C220h3qΓ3q+1C32=C11C30+C10C31-cC31C320=bC11Γ2q+1Γ2q+1+C30C10+C10C310-cC310C321=bC110Γ3q+1Γq+1Γ2q+1z12=C32h2qΓ2q+1+C320h3qΓ3q+1+C321h4qΓ4q+1,(17)C13=-C22-C32C130=-C120-C320C131=-C321x13=C13h3qΓ3q+1+C130h4qΓ4q+1+C131h5qΓ5q+1C23=C12+aC22C230=C120+aC220y13=C23h3qΓ3q+1+C230h4qΓ4q+1C33=C12C30+C10C32-cC32+C11C31Γ2q+1Γ2q+1C330=C30C120+C10C320+bC12+C11C310+C31C110Γ3q+1Γ2q+1Γq+1-cC320C331=bC120Γ4q+1Γ3q+1Γq+1+C10C321+C110C310Γ4q+1Γ22q+1-cC321z13=C33h3qΓ3q+1+C330h4qΓ4q+1+C331h5qΓ5q+1,(18)C14=-C23-C33C140=-C230-C330C141=-C331x14=C14h4qΓ4q+1+C140h5qΓ5q+1+C141h6qΓ6q+1C24=C13+aC23C240=C130+aC230y14=C24h4qΓ4q+1+C240h5qΓ5q+1+C131h6qΓ6q+1C340=C13C30+C10C33+C11C32+C12C31Γ3q+1Γ2q+1Γq+1-cC33C341=C30C130+C10C330-cC330+bC13+C11C320+C31C120Γ4q+1Γ3q+1Γq+1+C110C32+C12C310Γ4q+1Γ22q+1C342=C30C331+C10C331-cC331+bC130+C11C321Γ5q+1Γ4q+1Γq+1+C110C320+C120C310Γ5q+1Γ3q+1Γ2q+1C343=bC131Γ6q+1Γ5q+1Γq+1+C110C321Γ6q+1Γ2q+1Γ4q+1z14=C340h4qΓ4q+1+C341h5qΓ5q+1+C342h6qΓ6q+1+C343h7qΓ7q+1.Here h is iteration step size. In this paper, we seth=0.01.


## 2.3. Lyapunov Exponent Spectra

According to (13)-(18), the chaotic sequences of the fractional-order Rössler system are obtained with initial values [x0, y0, z0],q, and appropriate parametersa,b,c. So, the bifurcation diagrams of the fractional-order chaotic system are plotted. In addition, if we obtain the solution of the fractional-order chaotic system, the Lyapunov exponent spectra can be calculated by adopting QR-factorization [34] (19)qrJmJm-1...J1=qrJmJm-1...J2J1Q0=qrJmJm-1...J3J2Q1R1=qrJmJm-1...J3Q2R2R1=...=qrJmJm-1...JiQi-1Ri-1Ri-2...R2R1=...=QmRm...R2R1=QmR.Here,qr[.] denotes the QR-factorization process.J is the Jacobin matrix of the iteration (13).Q is the orthogonal matrix, andRare the diagonal elements.m is the iteration number. Then, the Lyapunov exponent spectra are calculated by(20)λk=1mh∑i=1mlnRik,k,where the system dimensionk = 1, 2, …,n. Dynamics of system (12) are investigated with the variation of derivative q and system parametera.


## 3. Numerical Analysis of the Fractional-Order Rössler System

### 3.1. Dynamics with Variation of Derivativeq

Fora=0.55,b=2, andc=4, the bifurcation diagrams versusq∈[0.2,1] and the corresponding Lyapunov exponent spectra are shown in Figure 1. The bifurcation diagram is consistent with the Lyapunov exponent spectra. As q increases, the system is periodic atq=0.268, and it gradually enters chaotic state by period-doubling bifurcations. The first pitchfork bifurcation occurs atq=0.346. The lowest order at which the chaos exists is about 3 (the dimension of the system) ×0.346=1.119 whenq=0.373, and the corresponding phase portrait is shown in Figure 2(a). However, in [7], the lowest order obtained is aboutq=0.7-0.8. Forq∈[0.346, 1], the system maintains the chaotic state in most regions except for some periodic windows. The phase portrait whenq=0.564 is plotted as shown in Figure 2(b). Figure 1 indicates that the differential order q is another bifurcation parameter except for the system parametersa,b, and c in the fractional-order Rössler system.Figure 1
Bifurcation diagram and Lyapunov exponent spectra for fractional-order Rössler system witha=0.55,b=2, andc=4. (a) Bifurcation diagram. (b) Lyapunov exponent spectra.
(a) (b)Figure 2
Phase portraits onx-y plane of attractor obtained by simulation. (a)q=0.373. (b)q=0.564.
(a) (b)Compared withq=1, the system has different dynamic characteristics when q is fractional, including chaotic, periodic, period-doubling, and tangent bifurcations. In Figure 1, the step size of q is 0.001 when the fractional-order chaotic system is analyzed. In theory, the step size of q will be smaller, and it is not limited according to the iteration.


### 3.2. Dynamics with Variation of Parametera

Letb=2 andc=4; the dynamics are investigated versusa∈[0.2,0.6] with the step sizes of 0.001. Figures 3(a), 3(b), and 3(c) are the bifurcation diagrams forq=1,q=0.8, andq=0.6, respectively. Their bifurcation structures are similar. However, the first pitchfork bifurcations occur at different positions, located ata=0.330,a=0.339, anda=0.366 forq=1,q=0.8, andq=0.6, respectively. The maximum a at which the chaos exists are also different, and they are ata=0.556,a=0.560, anda=0.572 forq=1,q=0.8, andq=0.6, respectively. The chaotic regions of the system move to the right along with the decreasing q.Figure 3(d) shows the maximum Lyapunov exponent spectra versus a with different q. Obviously, the smaller q is, the larger the maximum Lyapunov exponent is. It illustrates that the smaller the differential order q is, the more complex the fractional-order chaotic system is. This result indicates that the fractional-order Rössler system has broader applications than its integer-order counterpart. Figure 3 verifies that the differential order q affects the dynamic characteristics of the fractional-order Rössler system. Figure 4 shows phase portraits onx-y plane of attractor obtained by simulation for different q and a. It illustrates that the other conditions are the same, but q is different, and the phase portraits of the system are different. Whena=0.504, the system is periodic forq=1, and it is chaotic forq=0.8 andq=0.6. However, whena=0.441, the system is periodic forq=0.6, and in the other two cases, the system is chaotic. Figure 4 also verifies that the dynamic characteristics of the fractional-order Rössler system are affected byq.Figure 3
Bifurcation diagrams and Lyapunov exponent spectra for fractional-order Rössler system withb=2 andc=4. (a) Bifurcation diagram withq=1. (b) Bifurcation diagram withq=0.8. (c) Bifurcation diagram withq=0.6. (d) Maximum Lyapunov exponent spectra.
(a) (b) (c) (d)Figure 4
Phase portraits onx-y plane of attractor obtained by simulation. (a)q=1,a=0.504. (b)q=0.8,a=0.504. (c)q=0.6,a=0.504. (d)q=1,a=0.441. (e)q=0.8,a=0.441. (f)q=0.6,a=0.441.
(a) (b) (c) (d) (e) (f)


### 3.3. Implementation on DSP

The realization of fractional-order chaotic system is an important part of its application. We implement the fractional-order Rössler system on DSP platform. In [2], we realized the fractional-order simplified Lorenz system in the DSP board based on Adomian decomposition method, and the same technique is employed in this paper. For hardware design, the block diagram of the working principle is shown in Figure 5. In the experiments, the Texas Instrument DSP device TMS320F2812 is employed. TMS320F2812 is a 32-bit DSP running at 150MHz with fixed point operation. Such a high-speed clock rate is considered to be sufficient for our experiments. It can easily interface with a 16-bit dual channels digital-to-analog converter DAC8552 by SPI (serial peripheral interface). Phase portraits of the system are captured randomly by oscilloscope (Tektronix MSO 4102B-L).Figure 5
Simplified block diagram for DSP implementation of a fractional-order chaotic system.For software design, the operational procedure is shown in Figure6. It is based on the discrete iterative equations (13)-(18). The initial values, includingh,q, [x0, y0, z0], parameters and iteration number are set after initializing DSP. To improve the calculation speed, all Γ(·) andhnq are calculated before iterative computation. According to (13)-(18), some sequences are negative, and input data of DAC8552 must be integers in [0, 216−1]. Thus, to display the attractors of the fractional-order chaotic system, the data processing includes three steps. Firstly, a positive integer A is added to all sequences. It makes sure that each dataset is positive. Secondly, a scaling process is carried out by multiplying a positive integer B. Finally, the above results are rounded up.A and B are different for the sequences with different parameters and q. In addition, the iterative computation is not affected by data processing by employing the operation of pushing and popping [2].Figure 6
Flow chart for DSP implementation of a fractional-order chaotic system.We seth = 0.01, initial values [x0, y0, z0] = [2, -3, 2], and the parametersa,b,c are the same as those in Section 3.3;A = 15 andB = 1400. Phase portraits of the system are captured by the oscilloscope as shown in Figure 7. The experimental results qualify the simulation analysis, as shown in Figure 2. It indicates that the fractional-order Rössler system is realized successfully on DSP platform. For the DSP implementation of the fractional-order Rössler system, the number of variations is not changed in the calculation. So, it is not worrisome that the computer resources will be exhausted, and ADM is suitable for engineering applications.Figure 7
Phase portraits onx-y plane of attractor displayed on oscilloscope. (a)q=0.373. (b)q=0.564.
(a) (b)


## 3.1. Dynamics with Variation of Derivativeq

Fora=0.55,b=2, andc=4, the bifurcation diagrams versusq∈[0.2,1] and the corresponding Lyapunov exponent spectra are shown in Figure 1. The bifurcation diagram is consistent with the Lyapunov exponent spectra. As q increases, the system is periodic atq=0.268, and it gradually enters chaotic state by period-doubling bifurcations. The first pitchfork bifurcation occurs atq=0.346. The lowest order at which the chaos exists is about 3 (the dimension of the system) ×0.346=1.119 whenq=0.373, and the corresponding phase portrait is shown in Figure 2(a). However, in [7], the lowest order obtained is aboutq=0.7-0.8. Forq∈[0.346, 1], the system maintains the chaotic state in most regions except for some periodic windows. The phase portrait whenq=0.564 is plotted as shown in Figure 2(b). Figure 1 indicates that the differential order q is another bifurcation parameter except for the system parametersa,b, and c in the fractional-order Rössler system.Figure 1
Bifurcation diagram and Lyapunov exponent spectra for fractional-order Rössler system witha=0.55,b=2, andc=4. (a) Bifurcation diagram. (b) Lyapunov exponent spectra.
(a) (b)Figure 2
Phase portraits onx-y plane of attractor obtained by simulation. (a)q=0.373. (b)q=0.564.
(a) (b)Compared withq=1, the system has different dynamic characteristics when q is fractional, including chaotic, periodic, period-doubling, and tangent bifurcations. In Figure 1, the step size of q is 0.001 when the fractional-order chaotic system is analyzed. In theory, the step size of q will be smaller, and it is not limited according to the iteration.


## 3.2. Dynamics with Variation of Parametera

Letb=2 andc=4; the dynamics are investigated versusa∈[0.2,0.6] with the step sizes of 0.001. Figures 3(a), 3(b), and 3(c) are the bifurcation diagrams forq=1,q=0.8, andq=0.6, respectively. Their bifurcation structures are similar. However, the first pitchfork bifurcations occur at different positions, located ata=0.330,a=0.339, anda=0.366 forq=1,q=0.8, andq=0.6, respectively. The maximum a at which the chaos exists are also different, and they are ata=0.556,a=0.560, anda=0.572 forq=1,q=0.8, andq=0.6, respectively. The chaotic regions of the system move to the right along with the decreasing q.Figure 3(d) shows the maximum Lyapunov exponent spectra versus a with different q. Obviously, the smaller q is, the larger the maximum Lyapunov exponent is. It illustrates that the smaller the differential order q is, the more complex the fractional-order chaotic system is. This result indicates that the fractional-order Rössler system has broader applications than its integer-order counterpart. Figure 3 verifies that the differential order q affects the dynamic characteristics of the fractional-order Rössler system. Figure 4 shows phase portraits onx-y plane of attractor obtained by simulation for different q and a. It illustrates that the other conditions are the same, but q is different, and the phase portraits of the system are different. Whena=0.504, the system is periodic forq=1, and it is chaotic forq=0.8 andq=0.6. However, whena=0.441, the system is periodic forq=0.6, and in the other two cases, the system is chaotic. Figure 4 also verifies that the dynamic characteristics of the fractional-order Rössler system are affected byq.Figure 3
Bifurcation diagrams and Lyapunov exponent spectra for fractional-order Rössler system withb=2 andc=4. (a) Bifurcation diagram withq=1. (b) Bifurcation diagram withq=0.8. (c) Bifurcation diagram withq=0.6. (d) Maximum Lyapunov exponent spectra.
(a) (b) (c) (d)Figure 4
Phase portraits onx-y plane of attractor obtained by simulation. (a)q=1,a=0.504. (b)q=0.8,a=0.504. (c)q=0.6,a=0.504. (d)q=1,a=0.441. (e)q=0.8,a=0.441. (f)q=0.6,a=0.441.
(a) (b) (c) (d) (e) (f)


## 3.3. Implementation on DSP

The realization of fractional-order chaotic system is an important part of its application. We implement the fractional-order Rössler system on DSP platform. In [2], we realized the fractional-order simplified Lorenz system in the DSP board based on Adomian decomposition method, and the same technique is employed in this paper. For hardware design, the block diagram of the working principle is shown in Figure 5. In the experiments, the Texas Instrument DSP device TMS320F2812 is employed. TMS320F2812 is a 32-bit DSP running at 150MHz with fixed point operation. Such a high-speed clock rate is considered to be sufficient for our experiments. It can easily interface with a 16-bit dual channels digital-to-analog converter DAC8552 by SPI (serial peripheral interface). Phase portraits of the system are captured randomly by oscilloscope (Tektronix MSO 4102B-L).Figure 5
Simplified block diagram for DSP implementation of a fractional-order chaotic system.For software design, the operational procedure is shown in Figure6. It is based on the discrete iterative equations (13)-(18). The initial values, includingh,q, [x0, y0, z0], parameters and iteration number are set after initializing DSP. To improve the calculation speed, all Γ(·) andhnq are calculated before iterative computation. According to (13)-(18), some sequences are negative, and input data of DAC8552 must be integers in [0, 216−1]. Thus, to display the attractors of the fractional-order chaotic system, the data processing includes three steps. Firstly, a positive integer A is added to all sequences. It makes sure that each dataset is positive. Secondly, a scaling process is carried out by multiplying a positive integer B. Finally, the above results are rounded up.A and B are different for the sequences with different parameters and q. In addition, the iterative computation is not affected by data processing by employing the operation of pushing and popping [2].Figure 6
Flow chart for DSP implementation of a fractional-order chaotic system.We seth = 0.01, initial values [x0, y0, z0] = [2, -3, 2], and the parametersa,b,c are the same as those in Section 3.3;A = 15 andB = 1400. Phase portraits of the system are captured by the oscilloscope as shown in Figure 7. The experimental results qualify the simulation analysis, as shown in Figure 2. It indicates that the fractional-order Rössler system is realized successfully on DSP platform. For the DSP implementation of the fractional-order Rössler system, the number of variations is not changed in the calculation. So, it is not worrisome that the computer resources will be exhausted, and ADM is suitable for engineering applications.Figure 7
Phase portraits onx-y plane of attractor displayed on oscilloscope. (a)q=0.373. (b)q=0.564.
(a) (b)


## 4. Tracking Synchronization

Synchronization control of fractional-order chaotic system is one of the main aspects of its applications. In practical applications, we need to eliminate the chaos or transform it into some useful signals. Therefore, tracking control, which transforms the chaos signal into desired bounded signal, is significant in practice. We investigate the tracking synchronization of fractional-order Rössler systems based on ADM.The tracking system with controllers is defined as(21)​∗Dt0qx′=-y′-z′+u1​∗Dt0qy′=x′+ay′+u2​∗Dt0qz′=b-cz′+x′z′+u3,whereu=[u1, u2, u3]T are the controllers. By designing proper controllers, variable states x′, y′, and z′ in system (21) track the signals φ1, φ2, and φ3, respectively. Here we define the error between variable states and target signals as follows: (22)e1t=x′t-φ1te2t=y′t-φ2te3t=z′t-φ3t.Definition 2.
For the tracking system (21), if there is a controlleru making eit→0, i=1,2,3, then variable states x′, y′, and z′ of the system are synchronized with the signals φ1, φ2, and φ3, respectively.
For a fractional-order linear system given by​∗Dt0qx′t=-kx′t, where k is a real constant, ifk>0, thenx(t) convergences to zero with the increase of time t. In this paper, the obtained error system ​∗Dt0qeit=-kieit (i=1, 2, 3) will satisfy this property.


### 4.1. Tracking the Constant Signals

Let variable statesx′, y′, and z′ in system (21) track the constants φ1, φ2, and φ3, respectively. It means that the system will be controlled to a fixed point. Suppose that we have the following controllers:(23)u1=-k1x′-φ1+y′+z′u2=-k2y′-φ2-x′-ay′u3=-k3z′-φ3-b+cz′-x′z′.By subtracting (11) from (21), we have(24)​∗Dt0qe1=​∗Dt0qx′=-k1x′-φ1=-k1e1​∗Dt0qe2=​∗Dt0qy′=-k2y′-φ2=-k2e2​∗Dt0qe3=​∗Dt0qz′=-k3z′-φ3=-k3e3.Linearizing error system (24) at its equilibrium point e1=e2=e3=0 yields the Jacobin matrixJ as shown in (25). The eigenvalues ofJ are λ1=λ2=λ3=-1, and they satisfy arg(λi)>qπ/2. So, the error system (24) is asymptotically stable at its equilibrium point. (25)J=-k1000-k2000-k3.The error system (24) with controller is solved by adopting ADM, too. The obtained approximate solution is shown in (26)-(31).(26)x′m+1y′m+1z′m+1=x′10+x′11+x′12+x′13+x′14y′10+y′11+y′12+y′13+y′14z′10+z′11+z′12+z′13+z′14,where(27)C10=x′mC20=y′mC30=z′mC100=k1φ1C200=k2φ2C300=k3φ3x′10=C10+C100hqΓq+1y′10=C20+C200hqΓq+1z′10=C30+C300hqΓq+1,(28)C11=-k1C10C110=-k1C100C21=-k2C20C210=-k2C200C31=-k3C30C310=-k3C300x′11=C11hqΓq+1+C110h2qΓ2q+1y′11=C21hqΓq+1+C210h2qΓ2q+1z′11=C31hqΓq+1+C310h2qΓ2q+1,(29)C12=-k1C11C120=-k1C110C22=-k2C21C220=-k2C210C32=-k3C31C320=-k3C310x′12=C12h2qΓ2q+1+C120h3qΓ3q+1y′12=C22h2qΓ2q+1+C220h3qΓ3q+1z′12=C32h2qΓ2q+1+C320h3qΓ3q+1,(30)C13=-k1C12C130=-k1C120C23=-k2C22C230=-k2C220C33=-k3C32C330=-k3C320x′13=C13h3qΓ3q+1+C130h4qΓ4q+1y′13=C23h3qΓ3q+1+C230h4qΓ4q+1z′13=C33h3qΓ3q+1+C330h4qΓ4q+1,(31)C14=-k1C13C140=-k1C130C24=-k2C23C240=-k2C230C34=-k3C33C340=-k3C330x′14=C14h4qΓ4q+1+C140h5qΓ5q+1y′14=C24h4qΓ4q+1+C240h5qΓ5q+1z′14=C340h4qΓ4q+1+C341h5qΓ5q+1Settingq=0.98, [k1, k2, k3]=[2, 3, 4], and [φ1, φ2, φ3]=[1, 2, 3] in (24), respectively, the synchronizations of fractional-order Rössler systems are simulated. The error curves of simulation results are displayed in Figure 8(a). It shows that the evolution of the synchronization errors asymptotically converges to zero. Eventually [x′, y′z′]= [φ1, φ2, φ3]=[1, 2, 3] as shown in Figure 8(b). The results verify that the variable states x′, y′, and z′ in system (21) track the constants φ1, φ2, and φ3, successfully.Figure 8
Tracking constant signals. (a) Error curves of synchronization system. (b) Tracked variable states.
(a) (b)


### 4.2. Tracking the Fractional-Order Chaotic Signals

In this case,φ1, φ2, and φ3 are fractionalq-order differentiable function signals. Suppose that φ1t=xt, φ2t=yt, and φ3t=zt, from the driving system given by (32)​∗Dt0qx=-y-z​∗Dt0qy=x+ay​∗Dt0qz=b-cz+xz.The controller’s equations are as given as follows:(33)u1=-k1x′-φ1+y′+z′+​∗Dt0qφ1tu2=-k2y′-φ2-x′-ay′+​∗Dt0qφ2tu3=-k3z′-φ3-b+cz′-x′z′+​∗Dt0qφ3t.Now system (21) is denoted as (34)​∗Dt0qx′=-k1x′-φ1+​∗Dt0qφ1t​∗Dt0qy′=-k2y′-φ2+​∗Dt0qφ2t​∗Dt0qz′=-k3z′-φ3+​∗Dt0qφ3t,Obviously, (35)​∗Dt0qx′-​∗Dt0qφ1t=​∗Dt0qe1=-k1x′-φ1=-k1e1​∗Dt0qy′-​∗Dt0qφ2t=​∗Dt0qe2=-k2y′-φ2=-k2e2​∗Dt0qz′-​∗Dt0qφ3t=​∗Dt0qe3=-k3z′-φ3=-k3e3,and system (21) can be synchronized with the driving signals. Moreover, since φ1t=xt, φ2t=yt, and φ3t=zt, system (21) is given by (36)​∗Dt0qx′=-k1x′-x+​∗Dt0qxt=-k1x′-x-y-z​∗Dt0qy′=-k2y′-y+​∗Dt0qyt=-k2y′-y+x+ay​∗Dt0qz′=-k3z′-z+​∗Dt0qzt=-k3z′-z+b-cz+xz,System (36) is solved by adopting ADM, and the solution is as follows:(37)x′m+1y′m+1z′m+1=x′10+x′11+x′12+x′13+x′14y′10+y′11+y′12+y′13+y′14z′10+z′11+z′12+z′13+z′14,where(38)x′10=x′my′10=y′mz′10=z′m+bhqΓq+1G10=x′10G20=y′10G30=z′0,(39)G11=-k1G10-C10+C11G110=C110x′11=G11hqΓq+1+G110h2qΓ2q+1G21=-k2G20-C20+C21y′11=G21hqΓq+1G31=-k3G30-C30+C31G310=C310-k3bz11=G31hqΓq+1+G310h2qΓ2q+1,(40)G12=-k1G11-C11+C12G120=-k1G110-C110+C120x′12=G12h2qΓ2q+1+G120h3qΓ3q+1G22=-k2G21-C21+C22G220=C220y′12=G22h2qΓ2q+1+G220h3qΓ3q+1G32=-k3G31-C31+C32G320=-k2G310-C320+C320G321=C321z′12=G32h2qΓ2q+1+G320h3qΓ3q+1+G321h4qΓ4q+1,(41)G13=-k1G12-C12+C13G130=-k1G120-C120+C130G131=C131x′13=G13h3qΓ3q+1+G130h4qΓ4q+1+G131h5qΓ5q+1G23=-k2G22-C22+C23G230=-k2G220-C220+C230y′13=G23h3qΓ3q+1+G230h4qΓ4q+1G33=-k3G32-C32+C33G330=-k3G320-C320+C330G331=C331z′13=G33h3qΓ3q+1+G330h4qΓ4q+1+G331h5qΓ5q+1,(42)G14=-k1G13-C13+C14G140=-k1G130-C130+C140G141=-k1G131-C131+C141x′14=G14h4qΓ4q+1+G140h5qΓ5q+1+G141h6qΓ6q+1G24=-k2G23-C23+C24G240=-k2G230-C230+C240G241=C131y′14=G24h4qΓ4q+1+G240h5qΓ5q+1+G241h6qΓ6q+1G340=-k3G33-C33+C340G341=-k3G330-C330+C341G342=-k3G331-C331+C342G343=C343z′14=G340h4qΓ4q+1+G341h5qΓ5q+1+G342h6qΓ6q+1+G343h7qΓ7q+1In (37)-(42),Cis the same as that in the solution (see (13)-(18)) of the driving system.In (36), leta=0.55,b=2,c=4,q=0.95, and [k1, k2, k3]=[1, 2, 3]. The simulation results are shown in Figure 9. Figure 9(a) demonstrates that synchronization between two fractional-order Rössler systems with the designed controller is achieved by employing ADM. Figures 9(b), 9(c), and 9(d) illustrate the changes of the tracked variable states, and they show that the system can track the chaotic signals from master system successfully.Figure 9
Tracking chaotic signal from the master system. (a) Error curves of synchronization system. (b) Tracked variable state (x and x′). (c) Tracked variable state (y and y′). (d) Tracked variable state (z and z′).
(a) (b) (c) (d)


## 4.1. Tracking the Constant Signals

Let variable statesx′, y′, and z′ in system (21) track the constants φ1, φ2, and φ3, respectively. It means that the system will be controlled to a fixed point. Suppose that we have the following controllers:(23)u1=-k1x′-φ1+y′+z′u2=-k2y′-φ2-x′-ay′u3=-k3z′-φ3-b+cz′-x′z′.By subtracting (11) from (21), we have(24)​∗Dt0qe1=​∗Dt0qx′=-k1x′-φ1=-k1e1​∗Dt0qe2=​∗Dt0qy′=-k2y′-φ2=-k2e2​∗Dt0qe3=​∗Dt0qz′=-k3z′-φ3=-k3e3.Linearizing error system (24) at its equilibrium point e1=e2=e3=0 yields the Jacobin matrixJ as shown in (25). The eigenvalues ofJ are λ1=λ2=λ3=-1, and they satisfy arg(λi)>qπ/2. So, the error system (24) is asymptotically stable at its equilibrium point. (25)J=-k1000-k2000-k3.The error system (24) with controller is solved by adopting ADM, too. The obtained approximate solution is shown in (26)-(31).(26)x′m+1y′m+1z′m+1=x′10+x′11+x′12+x′13+x′14y′10+y′11+y′12+y′13+y′14z′10+z′11+z′12+z′13+z′14,where(27)C10=x′mC20=y′mC30=z′mC100=k1φ1C200=k2φ2C300=k3φ3x′10=C10+C100hqΓq+1y′10=C20+C200hqΓq+1z′10=C30+C300hqΓq+1,(28)C11=-k1C10C110=-k1C100C21=-k2C20C210=-k2C200C31=-k3C30C310=-k3C300x′11=C11hqΓq+1+C110h2qΓ2q+1y′11=C21hqΓq+1+C210h2qΓ2q+1z′11=C31hqΓq+1+C310h2qΓ2q+1,(29)C12=-k1C11C120=-k1C110C22=-k2C21C220=-k2C210C32=-k3C31C320=-k3C310x′12=C12h2qΓ2q+1+C120h3qΓ3q+1y′12=C22h2qΓ2q+1+C220h3qΓ3q+1z′12=C32h2qΓ2q+1+C320h3qΓ3q+1,(30)C13=-k1C12C130=-k1C120C23=-k2C22C230=-k2C220C33=-k3C32C330=-k3C320x′13=C13h3qΓ3q+1+C130h4qΓ4q+1y′13=C23h3qΓ3q+1+C230h4qΓ4q+1z′13=C33h3qΓ3q+1+C330h4qΓ4q+1,(31)C14=-k1C13C140=-k1C130C24=-k2C23C240=-k2C230C34=-k3C33C340=-k3C330x′14=C14h4qΓ4q+1+C140h5qΓ5q+1y′14=C24h4qΓ4q+1+C240h5qΓ5q+1z′14=C340h4qΓ4q+1+C341h5qΓ5q+1Settingq=0.98, [k1, k2, k3]=[2, 3, 4], and [φ1, φ2, φ3]=[1, 2, 3] in (24), respectively, the synchronizations of fractional-order Rössler systems are simulated. The error curves of simulation results are displayed in Figure 8(a). It shows that the evolution of the synchronization errors asymptotically converges to zero. Eventually [x′, y′z′]= [φ1, φ2, φ3]=[1, 2, 3] as shown in Figure 8(b). The results verify that the variable states x′, y′, and z′ in system (21) track the constants φ1, φ2, and φ3, successfully.Figure 8
Tracking constant signals. (a) Error curves of synchronization system. (b) Tracked variable states.
(a) (b)


## 4.2. Tracking the Fractional-Order Chaotic Signals

In this case,φ1, φ2, and φ3 are fractionalq-order differentiable function signals. Suppose that φ1t=xt, φ2t=yt, and φ3t=zt, from the driving system given by (32)​∗Dt0qx=-y-z​∗Dt0qy=x+ay​∗Dt0qz=b-cz+xz.The controller’s equations are as given as follows:(33)u1=-k1x′-φ1+y′+z′+​∗Dt0qφ1tu2=-k2y′-φ2-x′-ay′+​∗Dt0qφ2tu3=-k3z′-φ3-b+cz′-x′z′+​∗Dt0qφ3t.Now system (21) is denoted as (34)​∗Dt0qx′=-k1x′-φ1+​∗Dt0qφ1t​∗Dt0qy′=-k2y′-φ2+​∗Dt0qφ2t​∗Dt0qz′=-k3z′-φ3+​∗Dt0qφ3t,Obviously, (35)​∗Dt0qx′-​∗Dt0qφ1t=​∗Dt0qe1=-k1x′-φ1=-k1e1​∗Dt0qy′-​∗Dt0qφ2t=​∗Dt0qe2=-k2y′-φ2=-k2e2​∗Dt0qz′-​∗Dt0qφ3t=​∗Dt0qe3=-k3z′-φ3=-k3e3,and system (21) can be synchronized with the driving signals. Moreover, since φ1t=xt, φ2t=yt, and φ3t=zt, system (21) is given by (36)​∗Dt0qx′=-k1x′-x+​∗Dt0qxt=-k1x′-x-y-z​∗Dt0qy′=-k2y′-y+​∗Dt0qyt=-k2y′-y+x+ay​∗Dt0qz′=-k3z′-z+​∗Dt0qzt=-k3z′-z+b-cz+xz,System (36) is solved by adopting ADM, and the solution is as follows:(37)x′m+1y′m+1z′m+1=x′10+x′11+x′12+x′13+x′14y′10+y′11+y′12+y′13+y′14z′10+z′11+z′12+z′13+z′14,where(38)x′10=x′my′10=y′mz′10=z′m+bhqΓq+1G10=x′10G20=y′10G30=z′0,(39)G11=-k1G10-C10+C11G110=C110x′11=G11hqΓq+1+G110h2qΓ2q+1G21=-k2G20-C20+C21y′11=G21hqΓq+1G31=-k3G30-C30+C31G310=C310-k3bz11=G31hqΓq+1+G310h2qΓ2q+1,(40)G12=-k1G11-C11+C12G120=-k1G110-C110+C120x′12=G12h2qΓ2q+1+G120h3qΓ3q+1G22=-k2G21-C21+C22G220=C220y′12=G22h2qΓ2q+1+G220h3qΓ3q+1G32=-k3G31-C31+C32G320=-k2G310-C320+C320G321=C321z′12=G32h2qΓ2q+1+G320h3qΓ3q+1+G321h4qΓ4q+1,(41)G13=-k1G12-C12+C13G130=-k1G120-C120+C130G131=C131x′13=G13h3qΓ3q+1+G130h4qΓ4q+1+G131h5qΓ5q+1G23=-k2G22-C22+C23G230=-k2G220-C220+C230y′13=G23h3qΓ3q+1+G230h4qΓ4q+1G33=-k3G32-C32+C33G330=-k3G320-C320+C330G331=C331z′13=G33h3qΓ3q+1+G330h4qΓ4q+1+G331h5qΓ5q+1,(42)G14=-k1G13-C13+C14G140=-k1G130-C130+C140G141=-k1G131-C131+C141x′14=G14h4qΓ4q+1+G140h5qΓ5q+1+G141h6qΓ6q+1G24=-k2G23-C23+C24G240=-k2G230-C230+C240G241=C131y′14=G24h4qΓ4q+1+G240h5qΓ5q+1+G241h6qΓ6q+1G340=-k3G33-C33+C340G341=-k3G330-C330+C341G342=-k3G331-C331+C342G343=C343z′14=G340h4qΓ4q+1+G341h5qΓ5q+1+G342h6qΓ6q+1+G343h7qΓ7q+1In (37)-(42),Cis the same as that in the solution (see (13)-(18)) of the driving system.In (36), leta=0.55,b=2,c=4,q=0.95, and [k1, k2, k3]=[1, 2, 3]. The simulation results are shown in Figure 9. Figure 9(a) demonstrates that synchronization between two fractional-order Rössler systems with the designed controller is achieved by employing ADM. Figures 9(b), 9(c), and 9(d) illustrate the changes of the tracked variable states, and they show that the system can track the chaotic signals from master system successfully.Figure 9
Tracking chaotic signal from the master system. (a) Error curves of synchronization system. (b) Tracked variable state (x and x′). (c) Tracked variable state (y and y′). (d) Tracked variable state (z and z′).
(a) (b) (c) (d)


## 5. Conclusions

The fractional-order Rössler system is solved by adopting Adomian decomposition method, and the numerical solution is presented. Based on the given iteration, dynamics and synchronizations of the fractional-order Rössler system are investigated.( 1 ) Compared with its integer-order counterpart, the fractional-order Rössler system shows different dynamics.q affects the dynamic characteristics of the fractional-order chaotic system, and it is another bifurcation parameter except for the system parametersa,b, andc.( 2 ) When the fractional-order Rössler system is solved by employing ADM, the lowest order at which the chaos exists is about 1.119 whenq=0.373 witha=0.55,b=2,c=4, andh=0.01. The results are more accurate than that by adopting frequency-domain method.( 3 ) Based on ADM, the tracking synchronization of fractional-order Rössler systems is realized. It lays the foundation for the extensive applications research of the fractional-order chaotic system.



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*Source: 4019749-2018-12-02.xml*