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

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 order q. 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.


Introduction
Some studies have shown that fractional-order chaotic systems have more complex dynamic properties than the corresponding integer-order systems.The differential order  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][5][6].The properties and characteristics of its attractor depend on the control parameters a, b, and .Some researchers investigated the chaotic dynamics of fractional-order Rössler systems [7][8][9].In [7], Li and Chen analyzed the dynamics of fractional-order Rössler systems by employing frequencydomain method [10] based on the approximation of the transition function 1/s  and discussed only cases of q=0.7, 0.8, 0.9.For this approximation, in general,  ∈ [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  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][15][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  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 fractionalorder chaotic systems in [1,2] are implemented on DSP platform.He et 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 2 Complexity 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][22][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][26][27][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 Section 2, 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.
The fundamental properties of the integral operator    0 are described by [31]    * Here, r≥0,  >-1, and  is a real constant.According to (4), the following equation is obtained by applying the operator    0 to both sides of (1) [32]: Based on ADM, the nonlinear terms in (8) are expressed by Adomian polynomials [32] Here, x i is the No. i (i = 0, 1, . .., ∞) of Adomian polynomials, and    is obtained according to where j = 1, 2, . .., n (the system dimension).Then, the solution of (1) is expressed as x = ∑ ∞ =0 x  , where x i is derived from . . .

Numerical Solution of Fractional-Order Rössler System.
The fractional-order Rössler system is [7] 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). [ where Here ℎ is iteration step size.In this paper, we set h=0.01.

Lyapunov Exponent Spectra.
According to ( 13)-( 18), the chaotic sequences of the fractional-order Rössler system are obtained with initial values [ 0 ,  0 ,  0 ], q, and appropriate parameters a, 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] Here, qr [.] denotes the QR-factorization process.J is the Jacobin matrix of the iteration (13).Q is the orthogonal matrix, and R are the diagonal elements.m is the iteration number.Then, the Lyapunov exponent spectra are calculated by where the system dimension k = 1, 2, . .., n.Dynamics of system ( 12) are investigated with the variation of derivative  and system parameter a.

Numerical Analysis of the Fractional-Order Rössler System
3.1.Dynamics with Variation of Derivative q.For a=0.55, b=2, and c=4, the bifurcation diagrams versus q∈ [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  increases, the system is periodic at q=0.268, and it gradually enters chaotic state by period-doubling bifurcations.The first pitchfork bifurcation occurs at q=0.346.The lowest order at which the chaos exists is about 3 (the dimension of the system) ×0.346=1.119when q=0.373, and the corresponding phase portrait is shown in Figure 2(a).However, in [7], the lowest order obtained is about q=0.7-0.8.For q∈[0.346, 1], the system maintains the chaotic state in most regions except for some periodic windows.The phase portrait when q=0.564 is plotted as shown in Figure 2(b).Figure 1 indicates that the differential order  is another bifurcation parameter except for the system parameters a, b, and  in the fractional-order Rössler system.Compared with q=1, the system has different dynamic characteristics when  is fractional, including chaotic, periodic, period-doubling, and tangent bifurcations.In Figure 1, the step size of  is 0.001 when the fractional-order chaotic system is analyzed.In theory, the step size of  will be smaller, and it is not limited according to the iteration.Their bifurcation structures are similar.However, the first pitchfork bifurcations occur at different positions, located at a=0.330, a=0.339, and a=0.366 for q=1, q=0.8, and q=0.6, respectively.The maximum  at which the chaos exists are also different, and they are at a=0.556, a=0.560, and a=0.572 for q=1, q=0.8, and q=0.6, respectively.The chaotic regions of the system move to the right along with the decreasing . the smaller the differential order  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  affects the dynamic characteristics of the fractional-order Rössler system.Figure 4 shows phase portraits on x-y plane of attractor obtained by simulation for different  and .It illustrates that the other conditions are the same, but  is different, and the phase portraits of the system are different.When a=0.504, the system is periodic for q=1, and it is chaotic for q=0.8 and q=0.6.However, when a=0.441, the system is periodic for q=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 by q.

Dynamics with
3.3.Implementation on DSP.The realization of fractionalorder 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).
For software design, the operational procedure is shown in Figure 6.It is based on the discrete iterative equations ( 13)- (18).The initial values, including h, q, [ 0 ,  0 ,  0 ], parameters and iteration number are set after initializing DSP.To improve the calculation speed, all Γ(⋅) and h nq are calculated before iterative computation.According to ( 13)- (18), some sequences are negative, and input data of DAC8552 must be integers in [0, 2 16 −1].Thus, to display the attractors of the fractional-order chaotic system, the data processing includes three steps.Firstly, a positive integer  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 .Finally, the above results are rounded up.A and  are different for the sequences with different parameters and .In addition, the iterative computation is not affected by data processing by employing the operation of pushing and popping [2].
We set h = 0.01, initial values [ 0 ,  0 ,  0 ] = [2, -3, 2], and the parameters a, b, c are the same as those in Section 3.3; A = 15 and B = 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.

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 where u=[ 1 ,  2 ,  3 ] T are the controllers.By designing proper controllers, variable states   ,   , and   in system (21) track the signals  1 ,  2 , and  3 , respectively.Here we define the error between variable states and target signals as follows: Definition 2. For the tracking system (21), if there is a controller u making   () → 0, ( = 1, 2, 3), then variable states   ,   , and   of the system are synchronized with the signals  1 ,  2 , and  3 , respectively.
For a fractional-order linear system given by *    0   () = −  (), where  is a real constant, if k>0, then x(t) convergences to zero with the increase of time .In this paper, the obtained error system *    0   () = −    () (i=1, 2, 3) will satisfy this property.

Tracking the Constant Signals.
Let variable states   ,   , and   in system (21) track the constants  1 ,  2 , and  3 , respectively.It means that the system will be controlled to a fixed point.Suppose that we have the following controllers: By subtracting (11) from ( 21), we have * Linearizing error system (24) at its equilibrium point  1 = 2 = 3 =0 yields the Jacobin matrix J as shown in (25).The eigenvalues of J are  1 =  2 =  3 = −1, and they satisfy |arg(  )| > /2.So, the error system ( 24) is asymptotically stable at its equilibrium point.
The error system (24) with controller is solved by adopting ADM, too.The obtained approximate solution is shown in ( 26)- (31).
In (36), let a=0.55, b=2, c=4, q=0.95, and [ 1 ,  2 ,  3 ]=[1, 2, 3].The simulation results are shown in Figure 9. Figure 9(a) demonstrates that synchronization between two fractionalorder 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.

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 parameters a, b, and c.
(2) When the fractional-order Rössler system is solved by employing ADM, the lowest order at which the chaos exists is about 1.119 when q=0.373 with a=0.55, b=2, c=4, and h=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.

and ℎ nq 6 :
Flow chart for DSP implementation of a fractional-order chaotic system.