Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System

The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, bifurcation diagrams, phase portraits, complexity algorithms. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing fractional order q in this system. It also shows that the system is more complex when q takes smaller values. SE and C0 complexity algorithms provide a parameter choice criteria for practice applications of fractional-order chaotic systems. The fractional-order system is implemented by digital signal processor (DSP), and a pseudo-random bit generator is designed based on the implemented system, which passes the NIST test successfully.


Introduction
In recent years, dynamics of fractional-order chaotic systems have become a hot topic [1][2][3].Secure communication and information encryption based on these fractional-order systems have also aroused people's interests [4][5][6].Now, three approaches are mainly derived to solve fractional-order chaotic systems: frequency-domain method [7], Adomian Decomposition Method (ADM) [8] and Adams-Bashforth-Moulton (ABM) algorithm [9].However, Tavazoei et al. [10] reported that the frequency-domain method is not always reliable in detecting chaos behavior in nonlinear systems.On the other hand, ABM and ADM are more accurate and convenient to analyze dynamical behaviors of a nonlinear system.Compared with the ABM, ADM yields more accurate results and needs less computing resources as well as memory resources [11].So the numerical simulations in this paper are done by applying ADM scheme.
Lyapunov Characteristic Exponents (LCEs) are necessary and more convenient for detecting hyperchaos in fractional-order hyperchaotic system.A definition of LCEs for fractional differential systems was given in Reference [12] based on frequency-domain approximations, but the limitations of frequency-domain approximations are highlighted in Reference [10].Time series based LCEs calculation methods like Wolf algorithm [13], Jacobian method [14] and neural network algorithm [15] are associated with difficulties in choosing the embedding dimension and the delay parameter inherent of the phase-space reconstruction.Recently, LCEs of fractional-order chaotic systems are calculated based on ADM by applying QR decomposition method [16].However, there are few references discussing LCEs of fractional-order chaotic system versus parameters and fractional order q.
In addition, compared with analog circuit implementation, digital circuit realization is a more reliable and accurate way for the application of fractional-order chaotic systems.We will also focus on the digital circuit realization of the fractional-order chaotic system.
The rest of the paper is organized as follows.In Section 2, the fractional-order Lorenz hyperchaotic system is solved numerically.In Section 3, dynamics and complexity of the system are analyzed and some interesting results are illustrated.In Section 4, the system is implemented by DSP, then a pseudo-random bit generator is designed based on the implemented system.Finally, we summarize the results.

Adomian Decomposition Method
For a given fractional-order chaotic system with the form of * D q t 0 x(t) = f (x(t)), where x(t) = [x(t), y(t), z(t), u(t)] T are the state variables.* D q t 0 is the Caputo fractional derivative with the order (0 < q ≤ 1), and it is defined as [26] (1) This system can be separated into two parts as following form Here, m = ceil(q), b k is a specified constant relating to the initial values, and Lx(t) and Nx(t) are the linear and nonlinear terms of the fractional differential equations respectively.By applying the R-L fractional integral operator to both sides of Equation (2), the following equation is obtained [26,27] where the definition of R-L fractional-order integral operator According to [27], the nonlinear terms can be evaluated by where i = 0, 1, ..., ∞.Then the nonlinear terms are expressed as According to Reference [8], the solution of Equation ( 2) is derived by The analytical solution of the fractional-order system is presented by Because ADM converges fast [11,16], the first 6 terms are used to get the solution of system (2).In the real cases, it is impossible to estimate the high accuracy value of x when t takes large value.So, it is necessary to design a time discretization method.That is to say, for a time interval [t 0 , t], we divide the interval into subintervals [t n , t n+1 ].Then we get the value of x(t n+1 ) based on x(t n ) by applying function F(•).Finally, the numerical solution of the fractional-order chaotic system are denoted as a discrete map x(n + 1) = F(x(n)).

Fractional-Order Lorenz Hyperchaotic System
By introducing a nonlinear quadratic controller u to the second equation of Lorenz system, a four dimensional dynamic system is obtained [28] where k(0 < k ≤ 1) is a parameter.When k ∈ (0, 0.152), the system is a hyperchaotic system.When k ∈ [0.152, 0.21)∪[0.34,0.49), the system ( 9) is chaotic, and the system undergoes periodic orbits for the rest region of k [28].By introducing fractional derivative to this system, the fractional-order Lorenz hyperchaotic is obtained Here, the fractional-order system is solved by ADM.By introducing some intermediate variables, a rapid iteration scheme of the fractional-order hyperchaotic Lorenz system is obtained.It is shown as The intermediate variables K j i (i = 1, ..., 4, j = 0, 1, ..., 6) are shown as follows.
where h = t m+1 − t m , and Γ(•) is the Gamma function.Based on the discrete map Equation (11), it is easy to get the time series by the way of computer programming.Also it provides the necessary discrete iterative equations for the fractional-order hyperchaotic Lorenz system implemented on DSP platform.In this paper, we set h = 0.01.Dynamics under two parameters q and k are analyzed in the following sections.

Bifurcation Analysis
For the discrete map Equation ( 11), LCEs can be calculated by employing QR decomposition method [16].The computational process is shown as follows.
where qr[•] represents QR decomposition function, and J is the Jacobian matrix of the given map Equation (11).All LCEs can be calculated by where k = 1, 2, 3, 4, and M is the maximum iteration number.Let M = 20, 000 for accurate and stable outputs.In this study, we just show the first three λ 1 , λ 2 and λ 3 for better observation as well as for legibility.LCEs and the corresponding bifurcation diagram of the fractional-order Lorenz hyperchaotic system with q varying and k varying are analyzed.
(1) Fix k = 0.26 and vary q from 0.60 to 1.00 with the step of ∆q = 0.001.The bifurcation diagram and Lyapunov characteristic exponents are shown in Figure 1.It illustrates that the states of the system are different as q increases.The period windows are observed at q ∈ [0.6340, 0.6700] ∪ (0.906, 1].When q ∈ (0.670, 0.860], the system is a hyperchaotic system.When q ∈ (0.860, 0.906], this system is chaotic.It is worth pointing out that the periodic state of the integer order system becomes chaotic or hyperchaotic when order q changes to fractional order.

Observation of Chaotic Attractors
To further observe the dynamical behavior, some typical chaotic attractors of system (10) are recorded according to the bifurcation analysis results above.These phase portraits are shown in Figures 3 and 4 with k = 0.26, q varying and q = 0.96, k varying, respectively.
Results with k = 0.26 and q varying are summarized as follows.
(1) The system is periodic when q = 0.65 as shown in Figure 3a; (2) The system is hyperchaotic when q = 0.72 as shown in Figure 3b, and the attractor has two wings; (3) The system is chaotic when q = 0.89 as shown in Figure 3c, which is different from the above two attractors; (4) The system is periodic when q = 1.00 as shown in Figure 3d.It is interesting that this fractional-order system generates chaos and hyperchaos, while its integer-order counterpart is non-chaotic.Results with q = 0.96 and k varying are summarized as follows.(1) When k = 0.0.05, the hyperchaos strange attractor is shown in Figure 4a, the attractor is similar with the hyperchaos strange attractor in Figure 3a; (2) When k = 0.20, the chaos attractor is shown in Figure 4b, a typical chaotic attractor of this system; (3) When k = 0.25, the periodic orbits is shown in Figure 4c; (4) When k = 0.30, the periodic orbits is shown in Figure 4d; (5) When k = 0.40, the chaos attractor is shown in Figure 4e; (6) When k = 0.50, a quasi-periodic orbits is shown in Figure 4f; (7) When k = 0.80, the periodic orbits is shown in Figure 4g; (8) When k = 1.00, the periodic orbits is shown in Figure 4h, it is different from those in the above situations.It can be seen from the simulation results that the system undergoes hyperchaos, chaos, and some different periodic orbits when the parameter k and fractional-order q vary.Fractional order q should be treated as a bifurcation parameter as it also determines dynamics of the system.By applying ADM, the fractional-order Lorenz hyperchaotic system can be used in fields like secure communication, image encryption and digital watermark.

Spectral Entropy Complexity Analysis
SE reflects the disorder in the Fourier transformation domain.A flatter spectrum has a larger value of SE, which shows a higher complexity of the time series.SE is described as follows.Given a time series {x(n), n = 0, 1, 2,..., N − 1} with a length of N, let x(n) = x(n) − x , where x is the mean value of time series.Its corresponding Discrete Fourier Transformation (DFT) is defined by where k = 0, 1..., N − 1 and j is the imaginary unit.If the power of a discrete power spectrum with the k th frequency is |X(k)| 2 , then the "probability" of this frequency is defined as When the DFT is employed, the summation runs from k = 0 to k = N/2 − 1.The normalization entropy is denoted by [21] where ln(N/2) is the entropy of completely random signal.Complexity of x series of the fractional-order system (10) is calculated and illustrated in Figure 5.The length N for SE is 4 × 10 4 after removing the first 10 4 points of data.It shows in Figure 5 that the SE complexity agrees well with bifurcation results in Figures 1 and 2. Thus, it is a faster and more convenient way for dynamics analysis of fractional-order chaotic system.Compared with complexity versus a parameter, complexity in the parameter plane can show us more information of the system.SE complexity in the q − k parameter plane is calculated and shown in Figure 5c.It shows that complexity of the fractional-order chaotic system decreases as fractional order q increases when k is smaller than 0.2.When k is larger than 0.9, there is no chaos or hyperchaos generated for this system.Overall, high complexity region takes up about 40 percent of total parameter plane.As the SE algorithm estimates complexity very fast and illustrates the dynamics of the fractional-order chaotic system, complexity analysis is a practical basis for parameter choice in real applications.

C 0 Complexity Analysis
C 0 complexity is also based on the DFT, but it reflects the ratio of irregular in the series.The corresponding DFT process of the time series {x(n), n = 0, 1, 2, ..., N − 1} is shown as formula (21).Define the mean square value of X(k) as where r (r > 0) is the control parameter.The inverse Fourier Transformation of X(k) is where n = 0, 1, ..., N − 1.Finally, the C 0 complexity [22] is defined as In this paper, we set r = 15 to calculate the C 0 complexity.C 0 complexity of x series of the fractional-order system (10) is calculated as shown in Figure 6.The data applied for complexity calculation is the same as above.According to Figure 6, C 0 complexity is consistent with SE complexity as shown in Figure 5.As C 0 algorithm is also fast at complexity estimation, thus it can be employed to analyze dynamics and complexity of fractional-order Lorenz hyperchaotic system.It shows that C 0 algorithm provides a parameter selection method for a fractional-order Lorenz hyperchaotic system in practical application.

DSP Implementation
In this section, the digital circuit of the fractional-order Lorenz hyperchaotic system is implemented based on DSP technology.A hardware block diagram of the digital circuit is shown in Figure 7, and the DSP board used to perform digital implementation is shown in Figure 8.The DSP board is chosen for its powerful computing ability, and the key chip in this board is the fixed-point DSP TMS320F2812.A 16-bit dual-channel DA converter DAC8552 is used to convert time series generated by DSP.It is controlled by the DSP development board via a SPI interface.Then the converted data is sent to a oscilloscope (tektronix MDO3104) which is used to record phase portraits of the system.The flow diagram for DSP implementation of the fractional-order system is shown in Figure 9, and Equation (11) provides a necessary iterative algorithm for the fractional-order Lorenz hyperchaotic system.C language is used to realized the discrete iterative equation and then the program is downloaded to the DSP board.To improve the iteration speed, we use a function to calculate some items like h q , Γ(q + 1), h q /Γ(q + 1), h 2q , Γ(2q + 1), h 2q /Γ(2q + 1) and so on in Equations ( 15)-( 18) before iteration.In the step of data processing, all data is converted to analog signals.To reduce the impact of the data processing on the iterative computation, the operations of pushing and popping are introduced.This means that values of the state variables are placed and protected at a location pointed to a stack pointer.In a real world application, proper parameter q and k can be chosen based on the above complexity results or bifurcation results.To compare computer simulation results with digital circuit simulation results, we set the simulation parameter k = 0.26 and vary q (q = 0.65, 0.72, 0.89, and 1.00) as previously mentioned, then x − z phase diagrams are shown in Figure 10.We also set q = 0.96 and vary k (k = 0.05, 0.20, 0.50, and 1.00), and x − z phase diagrams are shown in Figure 11.It shows in Figures 10 and 11 that phase diagrams of fractional-order Lorenz hyperchaotic system by DSP consist with the computer simulation results.In conclusion, the fractional-order hyperchaotic Lorenz system is implemented in the digital circuit.Currently, based on DSP/FPGA technology, chaos-based applications are widely used in the engineering fields, especially in image encryption [31] and secret communication [32].In these applications, the pseudo-random bit generator usually plays an important role.Moreover, it shows in this paper that the fractional-order chaotic systems can be used in the engineering fields.

Conclusions
The dynamics of the fractional-order Lorenz hyperchaotic system is investigated in this paper.It is solved as a discrete map by applying ADM.The algorithm used to calculate LCEs of the fractional-order chaotic system is presented based on the discrete map and QR decomposition algorithm.Dynamics of the fractional-order Lorenz hyperchaotic system are analyzed by means of LCEs, bifurcation diagram, phase diagram, SE and C 0 complexity algorithms.Finally, the system is implemented in the digital circuit.The conclusions are drawn as follows.
(1) This fractional-order Lorenz hyperchaotic system contains rich dynamical behaviors.It is interesting that the system with integer-order is periodic, but chaos and hyperchaos are observed with the decrease of the fractional-order q.
(2) SE and C 0 complexities agree well with LCEs and bifurcation diagram results.It shows that complexity analysis is a more convenient method to choose parameters of fractional-order chaotic system in the real applications.We also find that the complexity of this system decreases when q increases.
(3) The fractional-order Lorenz hyperchaotic system is implemented on DSP digital circuit.According to the implemented fractional-order system, a DSP-based pseudo-random bit generator is designed, which passes the NIST test successfully.Our further work will focus on the information encryption application of the fractional-order Lorenz hyperchaotic system.

Figure 8 .
Figure 8. Digital signal processor (DSP) board used to perform digital implementation.

Figure 9 .
Figure 9. Flow diagram for DSP implementation of the fractional-order system.