# Chaos and Symbol Complexity in a Conformable Fractional-Order Memcapacitor System

Authors: Shaobo He; Santo Banerjee; Bo Yan
Journal: Complexity
 (2018)

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## Abstract

Application of conformable fractional calculus in nonlinear dynamics is a new topic, and it has received increasing interests in recent years. In this paper, numerical solution of a conformable fractional nonlinear system is obtained based on the conformable differential transform method. Dynamics of a conformable fractional memcapacitor (CFM) system is analyzed by means of bifurcation diagram and Lyapunov characteristic exponents (LCEs). Rich dynamics is found, and coexisting attractors and transient state are observed. Symbol complexity of the CFM system is estimated by employing the symbolic entropy (SybEn) algorithm, symbolic spectral entropy (SybSEn) algorithm, and symbolic C0 (SybC0) algorithm. It shows that pseudorandom sequences generated by the system have high complexity and pass the rigorous NIST test. Results demonstrate that the conformable memcapacitor nonlinear system can also be a good model for real applications.


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## Body

## 1. Introduction

In 1971, Chua postulated the concept of memristor that describes a relationship between flux and charge [1]. In 2008, researchers in Hewlett-Packard announced that a solid-state implementation of memristor has been successfully fabricated [2]. Since then, designing memory circuits has received significant attention of researchers, and many different kinds of memristor-based circuits have been designed [3–5]. In 2009, Ventra et al. [6] reported memcapacitors and meminductors. Compared with memristors, memcapacitors and meminductors have received much less attention. Currently, memcapacitor and meminductor can be designed based on the memristor. For example, Biolek and Biolkova [7] designed a memcapacitor model based on memristor by means of off-the-shelf circuits. As a matter of fact, memory electronic elements are usually designed nonlinearly. Thus, chaos can be easily found in those memory electronic element-based circuits [8–16]. Bao et al. [8–11] presented many valuable works on chaotic memristor circuits. For instance, their most recent work reported quasiperiodic behavior and chaotic busting in a third-order autonomous memristive oscillator [11]. Moreover, Mou et al. [12] designed a memory circuit with two memcapacitors that exhibited complex phenomena of state transition and transient chaos accompanied with time evolution and coexisting states. Fractional calculus has been studied for about 300 years, and there are a large number of literatures reporting chaos in the fractional-order nonlinear systems [17–20]. Moreover, fractional-order memory electronic element-based systems increasingly attracted attention of scholars [21, 22]. Since not much research exists about the fractional-order memcapacitor system, a fractional-order system with two memcapacitors is considered in this paper.All of the abovementioned systems are integer-order systems or fractional-order chaotic systems under Caputo definition or Riemann-Liouville (R-L) definition [23]. In 2014, Khalil et al. [24] proposed a new fractional derivative, and it is called the conformable fractional (CF) derivative. Since the CF definition is prominently compatible with the integer-order derivative, it has been widely studied in different research fields [25–28]. For example, İskender Eroğlu et al. [26] proposed an optimal boundary temperature control for a time-conformable fractional heat conduction equation. However, to our best knowledge, there are only two literatures reporting numerical analysis of CF chaotic systems. He et al. [29] firstly solved the nonlinear CF equations by the conformable Adomian decomposition method (CADM) and found chaos in the CF simplified Lorenz system. Later, Ruan et al. [30] investigated dynamics of a CF memristor system based on CADM, and rich dynamical behaviors were found. It shows that the CF nonlinear systems also generate chaos, and it is an interesting topic to explore complexity in these systems. Recently, Ünal and Gökdoğan [31] modified the differential transform method (DTM) and applied this method to solve CF nonlinear equations. But it has not been used to solve CF chaotic systems. Thus, in this paper, we will use conformable DTM to solve the CF memcapacitor system and analyze this system numerically.Meanwhile, measuring complexity is also an important method to analyze dynamics of chaotic systems. It reflects the security of the system to some extent. When a system has higher complexity, it means that the time series generated by the system is more random. Currently, there are several methods to measure complexity of time series, such as the permutation entropy (PE) [32], sample entropy (SampEn) [33], spectral entropy (SE) [34], and C0 algorithms [35]. It should be noted out that complexity of chaotic systems is mainly estimated based on the original time series, and complexity analysis of nonlinear symbol sequence has aroused interests of researchers [36, 37]. Meanwhile, there are many kinds of pseudorandom sequence generation algorithms. How complexity and dynamics of a chaotic system are determined by the pseudorandom quantization algorithms should be investigated. And whether the CFM system can be actually used in real applications should be verified.The rest of the article is organized as follows. In Section2, definitions of conformable fractional derivative and a numerical solution algorithm are proposed. Solution of the CFM system is obtained. In Section 3, dynamics of the CFM system is analyzed by means of Lyapunov characteristic exponents (LCEs), bifurcation diagram, and phase portraits. In Section 4, three different symbol complexity measuring algorithms are designed and the complexity of the CFM system is analyzed. Meanwhile, the NIST test is carried out. Finally, we summarize the results in Section 5.


## 2. Definitions and Numerical Solution Algorithm

In this section, the system model and definitions about conformable fractional derivative are presented. A numerical solution algorithm for conformable fractional nonlinear systems is designed based on the differential transform method.


### 2.1. The Conformable Fractional Memcapacitor System

Mou et al. [12] proposed a circuit with memcapacitor, and it is denoted by
(1)ẋ=cf1z,ẏ=d−ef2y+ef1z,ż=ef2y−f1z−x,where c, d, and e are the system parameters, x, y, and z are the state variables, and f1z=a1z+b1z3 and f2y=a2y+b2y3 represent the two memcapacitors in the circuit in which a1, b1, a2, and b2 are the intrinsic parameters of the two memcapacitors. In [12], c=8.96, d=4, and e is the bifurcation parameter. Moreover, there are three different sets of intrinsic parameters for different types of attractors. The three sets of intrinsic parameters are shown in Table 1.Table 1
Parameters for the three kinds of chaotic attractors.
Type
(a1, b1)
(a2, b2)
I
(−0.17, 10)
(0.25, 0.6)
II
(0.15, 10)
(−0.2, 0.6)
III
(−0.17, 10)
(−0.2, 0.6)By introducing the conformable fractional derivative to the system, the conformable fractional memcapacitor (CFM) system is defined as(2)Tt0q1x=cf1z,Tt0q2y=d−ef2y+ef1z,Tt0q3z=ef2y−f1z−x,where Tt0q is the conformable fractional derivative and 0<q1,q2,q3≤1. Definitions and characteristics of the conformable fractional derivative are given as follows.Definition 1 [24].
For a given functionf:0,∞→R, its conformable fractional derivative of order α is defined by
(3)Tt0qft=limε→0ft+εt1−q−ftε,where t>t0≥0 and q∈0,1.
Letq∈0,1 and f,g be q-differentiable at a point t>t0≥0. Then,
(1)
Tt0qaf+bg=aTt0qf+bTt0qg, for all a,b∈ℝ;
(2)
Tt0qtp=ptp−q, for all p∈ℝ;
(3)
Tt0qλ=0, for all constant functions ft=λ;
(4)
Tt0qfg=fTt0qg+gTt0qf;
(5)
Tt0qf/g=gTt0qf−fTt0qg/g2;
(6)
Tt0qft=t1−qdf/dtt.Definition 2 [24].
The conformable fractional integral of functionf:q,∞→ℝ is defined by
(4)It0qft=∫t0tfxx−t01−qdx,where t>t0≥0, q∈0,1, and f is q-differentiable at t0,t.


### 2.2. Conformable Fractional Differential Transform Method

The differential transform method is one of the most effective methods for semianalytic analysis of differential equations. Here, the conformable fractional differential transform method (CFDTM) is introduced to solve the conformable fractional chaotic system.Assume thatf is an infinitely q-differentiable function, for 0<q≤1 at a neighborhood of a point t0. Then, f has the fractional power series expansion [31].
(5)ft=∑k=0∞Tt0qft0t−t0qkqkk!,where t0<t<t0+R1/q, R>0, and Tt0qfkt0 denotes the conformable fractional derivative for k times. Define the conformable fractional differential transform of ft as
(6)Fqk=1qkk!Tt0qfktt=t0,where Tt0qfkt denotes the application of the fractional derivative for k times. Thus, the inverse conformable fractional differential transform of Fk is defined as
(7)ft=∑k=0∞Fqkt−t0qk.Lemma 1 [31].
Ifft=ut±vt, then Fqk=Uqk±Vqk.Lemma 2 [31].
Ifft=αut, then Fqk=αUqk.Lemma 3 [31].
Ifft=ut3=ututut, then Fqk=∑i=0k∑j=0i∑l=0jUqlUqj−lUqk−j.The multistep CFDTM method is proposed to solve the CFM system. Divide the time interval0,T into small subintervals tn,tn+1, where n=0,1,2,…,N, tN=T, and tn+1−tn=h. According to (7), the solution of the system over interval tn,tn+1 is given by
(8)xtn+1≈∑k=0DXkhkq1,ytn+1≈∑k=0DYkhkq2,ztn+1≈∑k=0DZkhkq3,where
(9)Xk+1=1q1k+1cF1k,Yk+1=1q2k+1d−eF2k+eF1k,Zk+1=1q3k+1eF2k−F1k−Xk,and
(10)F1k=a1Zk+b1∑i=0k∑j=0i∑l=0jZlZj−lZk−j,F2k=a2Yk+b2∑i=0k∑j=0i∑l=0jYlYj−lYk−j.It should be pointed out that(11)X0=xtn,Y0=ytn,Z0=ztn.It means thatxtn+1, ytn+1, and ztn+1 can be obtained based on xtn, ytn, and ztn, and the solution can be given as xtn+1=Fxtn. As for the contribution of this section, there are two aspects that could be specified. Firstly, it is the first time that the solution of the conformable fractional-order chaotic (memcapacitor) system is obtained by employing DTM. Secondly, we modified the method as the multistep CFDTM method by dividing the solution into subintervals tn,tn+1, and the obtained solution can be represented as xtn+1=Fxtn. Thus, the numerical solution of the conformable fractional-order chaotic system can be obtained in MATLAB.In addition, we chooseD=3 for the approximation of the system. Let e=6.9; phase diagrams with different derivative orders are shown in Figure 1. As shown in Figure 1, type I and type III attractors do not change much with the decrease of derivative orders while type II attractor is changed from chaos to periodical circle. Obviously, these three types of attractors are different. According to [12], there is no steady state in the type I system and type III system, but the type II system is steady when 0<e<4.Figure 1
Different types of chaotic attractors in the CFM system with different fractional derivative orders: (a) type I attractor withq1=q2=q3=0.95; (b) type I attractor with q1=q2=q3=0.9; (c) type I attractor with q1=q2=q3=0.8; (d) type II attractor with q1=q2=q3=0.95; (e) type II attractor with q1=q2=q3=0.9; (f) type II attractor with q1=q2=q3=0.8; (g) type III attractor with q1=q2=q3=0.95; (h) type III attractor with q1=q2=q3=0.9; (i) type III attractor with q1=q2=q3=0.8.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 2.1. The Conformable Fractional Memcapacitor System

Mou et al. [12] proposed a circuit with memcapacitor, and it is denoted by
(1)ẋ=cf1z,ẏ=d−ef2y+ef1z,ż=ef2y−f1z−x,where c, d, and e are the system parameters, x, y, and z are the state variables, and f1z=a1z+b1z3 and f2y=a2y+b2y3 represent the two memcapacitors in the circuit in which a1, b1, a2, and b2 are the intrinsic parameters of the two memcapacitors. In [12], c=8.96, d=4, and e is the bifurcation parameter. Moreover, there are three different sets of intrinsic parameters for different types of attractors. The three sets of intrinsic parameters are shown in Table 1.Table 1
Parameters for the three kinds of chaotic attractors.
Type
(a1, b1)
(a2, b2)
I
(−0.17, 10)
(0.25, 0.6)
II
(0.15, 10)
(−0.2, 0.6)
III
(−0.17, 10)
(−0.2, 0.6)By introducing the conformable fractional derivative to the system, the conformable fractional memcapacitor (CFM) system is defined as(2)Tt0q1x=cf1z,Tt0q2y=d−ef2y+ef1z,Tt0q3z=ef2y−f1z−x,where Tt0q is the conformable fractional derivative and 0<q1,q2,q3≤1. Definitions and characteristics of the conformable fractional derivative are given as follows.Definition 1 [24].
For a given functionf:0,∞→R, its conformable fractional derivative of order α is defined by
(3)Tt0qft=limε→0ft+εt1−q−ftε,where t>t0≥0 and q∈0,1.
Letq∈0,1 and f,g be q-differentiable at a point t>t0≥0. Then,
(1)
Tt0qaf+bg=aTt0qf+bTt0qg, for all a,b∈ℝ;
(2)
Tt0qtp=ptp−q, for all p∈ℝ;
(3)
Tt0qλ=0, for all constant functions ft=λ;
(4)
Tt0qfg=fTt0qg+gTt0qf;
(5)
Tt0qf/g=gTt0qf−fTt0qg/g2;
(6)
Tt0qft=t1−qdf/dtt.Definition 2 [24].
The conformable fractional integral of functionf:q,∞→ℝ is defined by
(4)It0qft=∫t0tfxx−t01−qdx,where t>t0≥0, q∈0,1, and f is q-differentiable at t0,t.


## 2.2. Conformable Fractional Differential Transform Method

The differential transform method is one of the most effective methods for semianalytic analysis of differential equations. Here, the conformable fractional differential transform method (CFDTM) is introduced to solve the conformable fractional chaotic system.Assume thatf is an infinitely q-differentiable function, for 0<q≤1 at a neighborhood of a point t0. Then, f has the fractional power series expansion [31].
(5)ft=∑k=0∞Tt0qft0t−t0qkqkk!,where t0<t<t0+R1/q, R>0, and Tt0qfkt0 denotes the conformable fractional derivative for k times. Define the conformable fractional differential transform of ft as
(6)Fqk=1qkk!Tt0qfktt=t0,where Tt0qfkt denotes the application of the fractional derivative for k times. Thus, the inverse conformable fractional differential transform of Fk is defined as
(7)ft=∑k=0∞Fqkt−t0qk.Lemma 1 [31].
Ifft=ut±vt, then Fqk=Uqk±Vqk.Lemma 2 [31].
Ifft=αut, then Fqk=αUqk.Lemma 3 [31].
Ifft=ut3=ututut, then Fqk=∑i=0k∑j=0i∑l=0jUqlUqj−lUqk−j.The multistep CFDTM method is proposed to solve the CFM system. Divide the time interval0,T into small subintervals tn,tn+1, where n=0,1,2,…,N, tN=T, and tn+1−tn=h. According to (7), the solution of the system over interval tn,tn+1 is given by
(8)xtn+1≈∑k=0DXkhkq1,ytn+1≈∑k=0DYkhkq2,ztn+1≈∑k=0DZkhkq3,where
(9)Xk+1=1q1k+1cF1k,Yk+1=1q2k+1d−eF2k+eF1k,Zk+1=1q3k+1eF2k−F1k−Xk,and
(10)F1k=a1Zk+b1∑i=0k∑j=0i∑l=0jZlZj−lZk−j,F2k=a2Yk+b2∑i=0k∑j=0i∑l=0jYlYj−lYk−j.It should be pointed out that(11)X0=xtn,Y0=ytn,Z0=ztn.It means thatxtn+1, ytn+1, and ztn+1 can be obtained based on xtn, ytn, and ztn, and the solution can be given as xtn+1=Fxtn. As for the contribution of this section, there are two aspects that could be specified. Firstly, it is the first time that the solution of the conformable fractional-order chaotic (memcapacitor) system is obtained by employing DTM. Secondly, we modified the method as the multistep CFDTM method by dividing the solution into subintervals tn,tn+1, and the obtained solution can be represented as xtn+1=Fxtn. Thus, the numerical solution of the conformable fractional-order chaotic system can be obtained in MATLAB.In addition, we chooseD=3 for the approximation of the system. Let e=6.9; phase diagrams with different derivative orders are shown in Figure 1. As shown in Figure 1, type I and type III attractors do not change much with the decrease of derivative orders while type II attractor is changed from chaos to periodical circle. Obviously, these three types of attractors are different. According to [12], there is no steady state in the type I system and type III system, but the type II system is steady when 0<e<4.Figure 1
Different types of chaotic attractors in the CFM system with different fractional derivative orders: (a) type I attractor withq1=q2=q3=0.95; (b) type I attractor with q1=q2=q3=0.9; (c) type I attractor with q1=q2=q3=0.8; (d) type II attractor with q1=q2=q3=0.95; (e) type II attractor with q1=q2=q3=0.9; (f) type II attractor with q1=q2=q3=0.8; (g) type III attractor with q1=q2=q3=0.95; (h) type III attractor with q1=q2=q3=0.9; (i) type III attractor with q1=q2=q3=0.8.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 3. Dynamical Analysis of the CFM System

### 3.1. LCE Calculation Algorithm

As mentioned above, the solution of the CFM system can be written asxtn+1=Fxtn; thus, it is actually shown as a given map xn+1=Fxn. Here, the QR decomposition method is employed to calculate the LCEs of the system. The computational process is denoted as
(12)qrJnJn−1⋯J1=qrJnJn−1⋯J2J1Q0=QnRn⋯R2R1,where qr· represents the QR decomposition function, J is the Jacobian matrix of the given map, Q is an orthogonal matrix, and R is an upper triangular matrix. All LCEs are calculated according to the upper triangular matrix, and they are given by [38]
(13)λη=1Mh∑i=1MlnRiη,η,where η=1,2,3 (dimension of the system) and M is the maximum iteration number. Here, the Jacobian matrix is obtained by the MATLAB symbolic operation function J=Jacobian⋅. LCEs of attractors in Figure 1 are calculated, and the results are shown in Table 2.Table 2
LCEs of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.0987, 0, −3.4849
0.1290, 0, −4.7279
0.2287, 0, −8.4289
II
0.1099, 0, −6.422
0.0855, 0, −7.1556
0, 0, −16.1767
III
0.0531, 0, −9.0147
0.0789, 0, −12.4373
0.2831, 0, −28.5537


### 3.2. Bifurcation and Chaos

As with [12], we also treat parameter e as a bifurcation parameter. Meanwhile, dynamics with the variation of fractional derivative orders q1, q2, and q3 is also analyzed. Since type II attractor changes more with the decrease of derivative orders, it is chosen as the representation of the three kinds of chaotic attractors for further analysis. When plotting bifurcation diagrams, the initial condition for the blue dots is xt0,yt0,zt0=0.1,0,0, while the initial condition for the red dots is xt0,yt0,zt0=−0.1,0,0.Case 1.
Fixq1=q2=q3=0.95 and vary parameter e from 6.7 to 7.05 with a step size of 0.0007. The bifurcation diagram and LCEs with parameter e varying are shown in Figure 2. It shows that dynamical behaviors of the CFM system change with the variation of parameter e. The system is chaotic when q∈6.7,0.7561U0.7617,6.9974, while the system is periodical when q∈0.7561,0.7617U6.9974,7.05.Case 2.
Lete=6.9 and q1=q2=q3=q. Vary q from 0.72 to 1 where the variation step size is 0.002. As shown in Figure 3(a), the system is divergent when q<0.75. When q∈0.7568,0.802U0.817,0.8496, the system is periodical while the chaotic interval is q∈0.75,0.7568U0.802,0.817U0.8496,1. LCE curves agree well with the analysis results. It shows that rich dynamics is found with the decrease of q.Case 3.
Lete=6.9 and q2=q3=1, and vary q1 from 0.995 to 1 with a step size of 0.00056. The bifurcation diagram and its corresponding LCEs are shown in Figure 4. The system is chaotic when q∈0.9958,0.9975U0.9984,1, while the system is nonchaotic for the rest values of q1.Case 4.
Lete=6.9 and q1=q3=1, and vary q2 from 0.995 to 1 with a step size of 0.00001. The bifurcation diagram and its corresponding LCEs are shown in Figure 5. When q2<0.9959, the transient state is found. The system is chaotic at the beginning and then becomes divergent finally. The system is periodical when q∈0.9967,0.9982, while for the rest range, the system is chaotic.Case 5.
Lete=6.9 and q1=q2=1, and vary q3 from 0.995 to 1 with a step size of 0.00001. The bifurcation diagram and its corresponding LCEs with derivative order q3 are shown in Figure 6. The system is chaotic when q∈0.942,0.95U0.961,1, and the system is periodical when q∈0.85,0.942U0.95,0.961.As shown above, after introducing the conformable fractional derivative, the system still has rich dynamical behaviors like parametere and derivative orders q1, q2, and q3. Moreover, when varying one derivative order and the other two to be equal to 1, the chaotic region shrinks much, compared with that when all derivative orders are varied simultaneously. The minimum order for chaos in the CFM system is 2.25, when the system is solved by CFDTM. Meanwhile, according to Figures 3–6, the system is chaotic when q1=q2=q3=1. However, when the derivative orders become smaller, the periodical state can be observed. It means that derivative orders can change the dynamics of the system distinctly. Thus, the conformable derivative orders q1, q2, and q3 are also the bifurcation parameters. Chaotic pseudorandom sequence (CPRS) has been widely used in real applications. In the next section, complexity of the CPRS generated by the CFM system is measured and the potential application values of the system are discussed.Figure 2
Dynamics of the CFM system with parametere varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 3
Dynamics of the CFM system with derivative orderq1=q2=q3=q varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 4
Dynamics of the CFM system with derivative orderq2=q3=1 and q1 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 5
Dynamics of the CFM system with derivative orderq1=q3=1 and q2 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 6
Dynamics of the CFM system with derivative orderq2=q2=1 and q3 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)


### 3.3. Coexisting Attractors and Transient State

As shown in the above bifurcation diagrams, when the initial conditions are different, the red and blue dots show two different routes to chaos. Moreover, the coexistence of multiple attractors is produced mainly due to the reason that symmetry and invariance exist in the system. For the CFM system, it is symmetric and invariant under the transformationx,y,z→−x,−y,−z for all values of parameter e. Thus, the coexistence of multiple attractors should be observed. As mentioned above, two different sets of initial conditions are chosen which are xt0,yt0,zt0=0.1,0,0 and xt0,yt0,zt0=−0.1,0,0. Coexisting attractors under different orders are shown in Figure 7. Obviously, the phase portraits of the CFM system with two symmetric initial values are symmetric, and the system can generate coexisting periodical cycles, chaotic attractors.Figure 7
Coexisting attractors of the CFM system wheree=6.9. The blue line is obtained with the initial condition xt0,yt0,zt0=0.1,0,0, and the red line is obtained with the initial condition xt0,yt0,zt0=−0.1,0,0: (a) q1=q2=q3=0.79; (b) q2=q3=1, q1=0.95; (c) q1=q3=1, q2=0.997; (d) q1=q2=1, q3=0.948.
(a)
(b)
(c)
(d)In some cases, there is no steady state in the system, and dynamical behaviors of the system are different under certain parameters when starting from a different initial state. As mentioned above, whene=6.9, q1=q3=1, and q2<0.9959, the system has a transient state. Here, phase diagrams with q2=0.9955 and 0.9958 are shown in Figure 8, where initial conditions are the same as those mentioned above. The coexisting state and transient state are observed as shown in Figure 8. The system becomes divergent through different directions with different initial conditions, and it is chaotic at the beginning and becomes divergent finally.Figure 8
Transient state in the CFM system where the blue line is obtained with the initial conditionxt0,yt0,zt0=0.1,0,0 and the red line is obtained with the initial condition xt0,yt0,zt0=−0.1,0,0: (a) q2=0.9955; (b) q2=0.9958.
(a)
(b)


## 3.1. LCE Calculation Algorithm

As mentioned above, the solution of the CFM system can be written asxtn+1=Fxtn; thus, it is actually shown as a given map xn+1=Fxn. Here, the QR decomposition method is employed to calculate the LCEs of the system. The computational process is denoted as
(12)qrJnJn−1⋯J1=qrJnJn−1⋯J2J1Q0=QnRn⋯R2R1,where qr· represents the QR decomposition function, J is the Jacobian matrix of the given map, Q is an orthogonal matrix, and R is an upper triangular matrix. All LCEs are calculated according to the upper triangular matrix, and they are given by [38]
(13)λη=1Mh∑i=1MlnRiη,η,where η=1,2,3 (dimension of the system) and M is the maximum iteration number. Here, the Jacobian matrix is obtained by the MATLAB symbolic operation function J=Jacobian⋅. LCEs of attractors in Figure 1 are calculated, and the results are shown in Table 2.Table 2
LCEs of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.0987, 0, −3.4849
0.1290, 0, −4.7279
0.2287, 0, −8.4289
II
0.1099, 0, −6.422
0.0855, 0, −7.1556
0, 0, −16.1767
III
0.0531, 0, −9.0147
0.0789, 0, −12.4373
0.2831, 0, −28.5537


## 3.2. Bifurcation and Chaos

As with [12], we also treat parameter e as a bifurcation parameter. Meanwhile, dynamics with the variation of fractional derivative orders q1, q2, and q3 is also analyzed. Since type II attractor changes more with the decrease of derivative orders, it is chosen as the representation of the three kinds of chaotic attractors for further analysis. When plotting bifurcation diagrams, the initial condition for the blue dots is xt0,yt0,zt0=0.1,0,0, while the initial condition for the red dots is xt0,yt0,zt0=−0.1,0,0.Case 1.
Fixq1=q2=q3=0.95 and vary parameter e from 6.7 to 7.05 with a step size of 0.0007. The bifurcation diagram and LCEs with parameter e varying are shown in Figure 2. It shows that dynamical behaviors of the CFM system change with the variation of parameter e. The system is chaotic when q∈6.7,0.7561U0.7617,6.9974, while the system is periodical when q∈0.7561,0.7617U6.9974,7.05.Case 2.
Lete=6.9 and q1=q2=q3=q. Vary q from 0.72 to 1 where the variation step size is 0.002. As shown in Figure 3(a), the system is divergent when q<0.75. When q∈0.7568,0.802U0.817,0.8496, the system is periodical while the chaotic interval is q∈0.75,0.7568U0.802,0.817U0.8496,1. LCE curves agree well with the analysis results. It shows that rich dynamics is found with the decrease of q.Case 3.
Lete=6.9 and q2=q3=1, and vary q1 from 0.995 to 1 with a step size of 0.00056. The bifurcation diagram and its corresponding LCEs are shown in Figure 4. The system is chaotic when q∈0.9958,0.9975U0.9984,1, while the system is nonchaotic for the rest values of q1.Case 4.
Lete=6.9 and q1=q3=1, and vary q2 from 0.995 to 1 with a step size of 0.00001. The bifurcation diagram and its corresponding LCEs are shown in Figure 5. When q2<0.9959, the transient state is found. The system is chaotic at the beginning and then becomes divergent finally. The system is periodical when q∈0.9967,0.9982, while for the rest range, the system is chaotic.Case 5.
Lete=6.9 and q1=q2=1, and vary q3 from 0.995 to 1 with a step size of 0.00001. The bifurcation diagram and its corresponding LCEs with derivative order q3 are shown in Figure 6. The system is chaotic when q∈0.942,0.95U0.961,1, and the system is periodical when q∈0.85,0.942U0.95,0.961.As shown above, after introducing the conformable fractional derivative, the system still has rich dynamical behaviors like parametere and derivative orders q1, q2, and q3. Moreover, when varying one derivative order and the other two to be equal to 1, the chaotic region shrinks much, compared with that when all derivative orders are varied simultaneously. The minimum order for chaos in the CFM system is 2.25, when the system is solved by CFDTM. Meanwhile, according to Figures 3–6, the system is chaotic when q1=q2=q3=1. However, when the derivative orders become smaller, the periodical state can be observed. It means that derivative orders can change the dynamics of the system distinctly. Thus, the conformable derivative orders q1, q2, and q3 are also the bifurcation parameters. Chaotic pseudorandom sequence (CPRS) has been widely used in real applications. In the next section, complexity of the CPRS generated by the CFM system is measured and the potential application values of the system are discussed.Figure 2
Dynamics of the CFM system with parametere varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 3
Dynamics of the CFM system with derivative orderq1=q2=q3=q varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 4
Dynamics of the CFM system with derivative orderq2=q3=1 and q1 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 5
Dynamics of the CFM system with derivative orderq1=q3=1 and q2 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)Figure 6
Dynamics of the CFM system with derivative orderq2=q2=1 and q3 varying: (a) bifurcation diagram; (b) LCEs.
(a)
(b)


## 3.3. Coexisting Attractors and Transient State

As shown in the above bifurcation diagrams, when the initial conditions are different, the red and blue dots show two different routes to chaos. Moreover, the coexistence of multiple attractors is produced mainly due to the reason that symmetry and invariance exist in the system. For the CFM system, it is symmetric and invariant under the transformationx,y,z→−x,−y,−z for all values of parameter e. Thus, the coexistence of multiple attractors should be observed. As mentioned above, two different sets of initial conditions are chosen which are xt0,yt0,zt0=0.1,0,0 and xt0,yt0,zt0=−0.1,0,0. Coexisting attractors under different orders are shown in Figure 7. Obviously, the phase portraits of the CFM system with two symmetric initial values are symmetric, and the system can generate coexisting periodical cycles, chaotic attractors.Figure 7
Coexisting attractors of the CFM system wheree=6.9. The blue line is obtained with the initial condition xt0,yt0,zt0=0.1,0,0, and the red line is obtained with the initial condition xt0,yt0,zt0=−0.1,0,0: (a) q1=q2=q3=0.79; (b) q2=q3=1, q1=0.95; (c) q1=q3=1, q2=0.997; (d) q1=q2=1, q3=0.948.
(a)
(b)
(c)
(d)In some cases, there is no steady state in the system, and dynamical behaviors of the system are different under certain parameters when starting from a different initial state. As mentioned above, whene=6.9, q1=q3=1, and q2<0.9959, the system has a transient state. Here, phase diagrams with q2=0.9955 and 0.9958 are shown in Figure 8, where initial conditions are the same as those mentioned above. The coexisting state and transient state are observed as shown in Figure 8. The system becomes divergent through different directions with different initial conditions, and it is chaotic at the beginning and becomes divergent finally.Figure 8
Transient state in the CFM system where the blue line is obtained with the initial conditionxt0,yt0,zt0=0.1,0,0 and the red line is obtained with the initial condition xt0,yt0,zt0=−0.1,0,0: (a) q2=0.9955; (b) q2=0.9958.
(a)
(b)


## 4. Complexity Analysis of the CFM System

In this section, complexity of the CFM system is analyzed by means of symbolic complexity measures. Generally, if the system has higher complexity, it means that the system is securer in practical applications. Besides, it is more convenient to calculate complexity since it just needs a segment of time series. Thus, complexity analysis result provides a basis for parameter choice of chaotic systems.


### 4.1. Symbolic Complexity Measures

In real application, a chaotic time series should be discretized as a pseudorandom sequence or symbolic sequence, which usually varies between 0 and 1 (0-1 time series) or varies from 0 to 255 (8-bit numbers). Here, complexity of chaotic symbolic time series is analyzed using different algorithms including the symbolic entropy (SybEn) algorithm, symbolic spectral entropy (SybSEn) algorithm, and symbolic C0 (SybC0) algorithm. These three complexity algorithms are defined as follows.Step 1 (pseudorandom process).
Here, suppose that there is a chaotic series defined asxn,n=0,1,2,…,N−1; it is discretized as sn,n=0,1,2,…,N−1 by employing the following method:
(14)sn=0,ifminx≤xn<Δx,1,ifΔx≤xn<2Δx,⋮255,if255Δx≤xn≤maxx,where Δx=maxx−minx/256. Thus, sn,n=0,1,2,…,N−1 is an 8-bit number time series.Step 2 (calculating SybEn).
Count the number of each symbol (from 0 to 255), and obtain the probability of each symbol as(15)Pi=#i:sn=i,n=0,1,…,N−1N,where # stands for number and i=0, 1,…,255. Thus, SybEn is defined by
(16)SybEn=−∑i=0255PilogPi.Step 3 (discrete Fourier transformation (DFT)).
Before the DFT, the following normalization is employed to the symbol time series, and it is denoted as(17)Sn=sn−meansstds,where means and stds are the mean value and the standard deviation of the time series s, respectively, and n=0,1,…,N−1. Thus, the mean value of the new time is zero and there is no direct current signal. DFT is carried out on time series S, and it is given by
(18)Xk=∑n=0N−1Sne−j2πnk/N,where k=0,1,…,N−1 and j is the imaginary unit.Step 4 (calculating SybSEn).
If the power of a discrete power spectrum with thekth frequency is Xk2, then the “probability” of this frequency is defined as
(19)ρk=Xk2∑k=0N/2−1Xk2.
When the DFT is employed, the summation runs fromk=0 to k=N/2−1. The normalization of SybSEn is denoted by [34]
(20)SybSEn=−1lnN/2∑k=0N/2−1ρklnρk.
lnN/2 is the entropy of the completely random signal.Step 5 (inverse DFT).
Define the mean square value ofXk as
(21)GN=1N∑k=0N−1Xk2.
Let(22)X~k=Xk,ifXk2>rGN,0,ifXk2≤rGN,where rr>0 is the control parameter. The inverse DFT of X~k is
(23)S~n=1N∑k=0N−1X~kej2πnk/N,where n=0,1,…,N−1. S~n reflects the regular part of the time series with detailed information removed.Step 6 (calculating SybC0).
SybC0 complexity is defined as [35]
(24)SybC0=∑n=0N−1Sn−S~n2Ω.where Ω=∑n=0N−1Sn2.The three complexity measuring algorithms estimate complexity of an 8-bit symbol time series from different aspects. Firstly, SybEn analyzes complexity in the time domain while SybSEn and SybC0 are defined in the frequency domain. Secondly, SybEn and SybSEn are defined based on the definition of the Shannon entropy while SybC0 reflects the ratio of an irregular part in the time series. Let e=6.9 and q1=q2=q3=0.95; we obtain a type II chaotic time series x. The time series is shown in Figure 9(a), and the symbol time series is illustrated in Figure 9(b). By employing (15) and (19), plots of the two different “probabilities” are shown in Figures 9(c) and 9(d), respectively. According to (22) and (24), we illustrate the irregular part of the time series with r=10, where the red line represents Sn2 and the green line represents Sn−S~n2 which shows the “difference” or “irregular part.”Figure 9
Analysis of complexity algorithms: (a) original time seriesx; (b) pseudorandom sequence s; (c) probability of each symbol Pi; (d) probability of the frequency ρi; (e) irregular part in the SybC0 algorithm.
(a)
(b)
(c)
(d)
(e)As shown in Figure9, the principles of different algorithms are different. For SybEn and SybSEn, if the probability density is more uniform, values of entropy are larger, while if the proportion of the green part is larger, values of SybC0 are larger. As a result, larger values of SybEn, SybSEn, and SybC0 mean that the system has higher complexity. In this paper, to analyze the complexity of the CMF system, the time series x with a length of 55000 is sampled with τ=10; thus, x=x1:10:55000. Finally, the sampled time series is changed into a symbol time series. For SybC0 complexity, we choose r=10.


### 4.2. Complexity Analysis

Complexity of the CFM system is analyzed by means of SybEn, SybSEn, and SybC0. Firstly, SybEn, SybSEn, and SybC0 of different types of CFM systems with different derivative orders are analyzed, and the results are shown in Tables 3–5, respectively. As with Figure 1 and Table 2, values of derivative orders are set as q1=q2=q3=0.95,0.90,and0.80 and e=6.9. It shows that complexity analysis results agree well with the LCE analysis results. When the system is chaotic, higher complexity can be found. Moreover, the system has relative higher complexity when the derivative orders are smaller. It should be pointed out that SybEn cannot distinguish the chaotic state and periodical state well similar to SybSEn and SybC0. Actually, it is also one of the reasons why we furtherly design complexity analysis methods in the frequency domain.Table 3
SybEn complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
5.4834
5.4646
5.4868
II
4.7810
4.7422
4.6255
III
4.8680
4.8711
4.9467Table 4
SybSEn complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.4497
0.5131
0.5589
II
0.5635
0.6135
0.3190
III
0.3580
0.3776
0.5386Table 5
SybC0 complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.1263
0.1756
0.2885
II
0.1913
0.2424
0.0776
III
0.1795
0.1963
0.3128Complexity of the CFM system with parametere varying is analyzed, and results are shown in Figure 10. Here, q1=q2=q3=q equals to 0.95, and parameter e varies from 6.7 to 7.05 with a step size of 0.0007. As shown in Figure 10, SybSEn and SybC0 agree with the maximum LCEs better than SybEn, and they identify more periodical windows which show relative lower complexity.Figure 10
Complexity analysis results of the CFM system with parametere varying: (a) SybEn; (b) SybSEn; (c) SybC0.
(a)
(b)
(c)Complexity of the type II CFM system with parametere=6.9 and derivative orders q1, q2, and q3 varying is analyzed, and the analysis results are shown in Figure 11. Let q1=q2=q3=q and let it vary from 0.75 to 1. Complexity analysis results are shown in Figures 11(a)–11(c). As shown in Figures 11(b) and 11(c), complexity of the system increases with the decrease of order q, which means that the system has higher complexity with smaller values of q. Thus, the system has a good application value in the engineering field. Fix q2=q3=1 and vary q1 from 0.995 to 1, and the complexity results are shown in Figures 11(d)–11(f). Fix q1=q2=1 and vary q2 from 0.996 to 1, and the complexity results are displayed in Figures 11(g)–11(i). Moreover, let q1=q2=1 and vary q3 from 0.85 to 1, and the complexity results are illustrated in Figures 11(j)–11(l). As shown in those figures, complexity of the CFM system does not increase with the decrease of derivative order q1, q2, or q3. When q1=q2=q3=1, the integer-order system is chaotic with high complexity, but the low-complexity region can be found when the fractional derivative orders decrease. In real application, one should choose those orders with which the system generates high-complexity time series.Figure 11
Complexity analysis results of the CFM system with derivative orders varying: (a) SybEn withq1=q2=q3=q varying; (b) SybSEn with q1=q2=q3=q varying; (c) SybC0 with q1=q2=q3=q varying; (d) SybEn with q2=q3=1 and q1 varying; (e) SybSEn with q2=q3=1 and q1 varying; (f) SybC0 with q2=q3=1 and q1 varying; (g) SybEn with q1=q3=1 and q2 varying; (h) SybSEn with q1=q3=1 and q2 varying; (i) SybC0 with q1=q3=1 and q2 varying; (j) SybEn with q1=q2=1 and q3 varying; (k) SybSEn with q1=q2=1 and q3 varying; (l) SybC0 with q1=q2=1 and q3 varying.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)According to Figures11 and 12, SybEn, SybSEn, and SybC0 complexity analysis results are consistent with the corresponding maximum LCEs to a certain degree. Overall, SybSEn and SybC0 analysis results agree better with the corresponding maximum LCE results than with those of SybEn. Compared with calculating LCEs, calculating SybEn, SybSEn, and SybC0 needs much less time and it is more convenient in real application since results can be obtained with a time series. On the one hand, it shows that the system has a potential application value in practice. On the other hand, it provides a basis for parameter choice of the CFM system in real applications.Figure 12
Complexity analysis results of the CFM system in thee-q parameter plane: (a) SybSEn; (b) SybC0.
(a)
(b)SybSEn and SybC0 complexity in the q-e parameter plane is calculated, and the results are shown in Figure 12. Here, the parameter varies from 6.7 to 7.05 similar to that mentioned above, while derivative order q varies from 6.5 to 1, simultaneously. As shown in Figure 12, the minimum order for chaos is about 0.65 when e=6.7. Meanwhile, it shows that the system has higher complexity.The pseudorandom time seriessn by (14) fluctuates with the original system variables. As we all know that a chaotic system is a good source for entropy. There are many different kinds of chaotic systems that can be used for designing a pseudorandom sequence generator (PRSG), such as the Lorenz system [39], logistic map, sine map, and 2D-SIMM [40]. And these PRSGs are widely used in real application fields such as image encryption [41], speech encryption [42], and chaotic watermark [43]. The quantization algorithms used in this applications are more complex than the method given in (14). One of the most commonly used methods is designed by expressing the original number or its converted number as a 64-bit binary number as DB63-DB0; then, one can obtain a new 8-bit number by choosing DB7-DB0. The details of this method are given as follows.The original number is converted as(25)φn=roundxn×10w,where n=0,1,2,…,N−1 and w is a control number. Here, in this paper, w=10. Thus, φn is an integer number and it can be expressed as
(26)φn=DB63DB62⋯DB1DB0.By choosing the first 8-bit number, a new symbol time seriessn is given as
(27)sn=DB7⋯DB1DB0.Obviously,sn can be expressed as a decimal integer number varying from 0 to 255, as mentioned above. By using this quantization algorithm, complexity of pseudorandom sequences generated by different chaotic systems is calculated and the results are shown in Table 6. The length of each segment symbol time series is 5000, and complexity of such 100 time series is calculated. The results are given as mean ± std of complexity values in these windows. As shown in Table 6, entropy or complexity of different chaotic pseudorandom sequences is at about the same high level. It shows that as with other different chaotic systems, the CFM system is also a good system for high entropy.Table 6
Complexity measuring results of different chaotic pseudorandom sequences with the second quantization method.
Systems
Equations
SybEn
SybSEn
SybC0
CFM system
Tt0q1x=cf1zTt0q2y=d−ef2y+ef1zTt0q3z=ef2y−f1z−x
5.4825 ± 0.0036
0.9466 ± 0.0014
0.9995 ± 0.0014
Lorenz system
ẋ=10y−xẏ=28x−xz−yż=xy−8z/3
5.5206 ± 0.0012
0.9583 ± 0.0003
0.9995 ± 4.4 × 10−4
Logistic map
xn+1=4xn1−xn
5.4962 ± 0.0053
0.9461 ± 0.0012
0.9994 ± 0.0015
Sine map
xn+1=4sinπxn
5.4966 ± 0.0043
0.9459 ± 0.0013
0.9992 ± 0.0019
2D-SIMM
xn+1=10sinπynsin10/xnyn+1=10sinπxn+1sin10/yn
5.4966 ± 0.0046
0.9461 ± 0.0014
0.9996 ± 0.0013According to Table6, complexity of the CFM system is high as other systems. Thus, it is necessary to check whether the pseudorandom sequence generated by this system passes the test suite of NIST. The package used for the NIST test is sts-2.1.2 which can be downloaded from the website. Two indicators, which are p values and the proportion of passing sequences, are used to show whether the sequence passes the test or not. The minimum value for p value is 0.0001. It means that when the p value is larger than 0.0001 and the confidence interval satisfies
(28)1−η−31−ηηM,1−η+31−ηηM,where M is the sample size and η is the given significance level, then the pseudorandom bit generator passes the test successfully. In our test, the length of the pseudorandom sequence is 106, and η=0.01 with a confidence interval given by 96.015%,1. The test result is illustrated in Table 7. For those items including C. Sums (2 times), N.O. Temp (148 times), R. Excur. (8 times), R. Excur. V. (18 times), and Serial (2 times), we only illustrate the lowest values of p value and proportion. It is shown in Table 7 that all p values are larger than 0.0001 and the computed proportion for each test lies inside the confidence interval. Hence, the tested binary sequences generated by the proposed pseudorandom bit generator are random.Table 7
NIST test result of binary sequences generated by the CFM system.
Tests
p value
Proportion
Success
Frequency (1)
0.3345
97/100
✓
B. Frequency (1)
0.6163
100/100
✓
C. Sums (2)
0.9558
99/100
✓
Runs (1)
0.1816
99/100
✓
Longest Run (1)
0.2133
99/100
✓
Rank (1)
0.3191
99/100
✓
FFT (1)
0.1968
99/100
✓
N.O. Temp (148)
0.0288
97/100
✓
O. Temp (1)
0.7197
97/100
✓
Universal (1)
0.5341
99/100
✓
App. Entropy (1)
0.8165
57/58
✓
R. Excur. (8)
0.010
57/58
✓
R. Excur. V. (18)
0.0179
58/58
✓
Serial (2)
0.5544
99/100
✓
L. Complexity (1)
0.1025
99/100
✓In this section, two different pseudorandom quantization algorithms are designed. As shown in Figures9(a) and 9(b), fluctuation of pseudorandom sequence obtained by the first method given (14) agrees well with the original time series. Thus, it is the reason why complexity analysis results match well with the corresponding maximum LCEs. It provides a good symbol sequence for complexity analysis of chaotic systems when analyzed by complexity measuring methods. However, according to Table 6, pseudorandom sequences generated by different systems by employing the second method have the same level of high complexity. Meanwhile, the pseudorandom sequence passes all the NIST tests. It means that the obtained sequence is random. As with other chaotic systems, the CFM system is also a good secret key generator for real applications including information encryption, secure communication, and chaotic digital watermark.Finally, it is necessary to explain why we choose 8-bit or 256-quantization-level symbols to form the numbers and how complexity analysis result is determined by the quantization levels. Lete=6.9 and q1=q2=q3=0.95. A segment of type II chaotic time series is used to calculate the complexity. The relation between the number of quantization levels and its complexity measuring results is shown in Figure 13, where SybEn is normalized via dividing lnm. Here, m is the number of bits or 2m is the quantization levels. It shows that the values of the two frequency domain methods, namely, SybC0 and SybSEn, do not change with the quantization levels. However, values of SybEn (the time-domain method) increase but tend to become stable with the quantization levels. Meanwhile, the 8-bit number is widely used in real applications. Thus, 8-bit or 256-quantization-level symbol sequences are employed for complexity analysis.Figure 13
Relation between the number of quantization levels and its complexity: (a) quantization method one; (b) quantization method two.
(a)
(b)


## 4.1. Symbolic Complexity Measures

In real application, a chaotic time series should be discretized as a pseudorandom sequence or symbolic sequence, which usually varies between 0 and 1 (0-1 time series) or varies from 0 to 255 (8-bit numbers). Here, complexity of chaotic symbolic time series is analyzed using different algorithms including the symbolic entropy (SybEn) algorithm, symbolic spectral entropy (SybSEn) algorithm, and symbolic C0 (SybC0) algorithm. These three complexity algorithms are defined as follows.Step 1 (pseudorandom process).
Here, suppose that there is a chaotic series defined asxn,n=0,1,2,…,N−1; it is discretized as sn,n=0,1,2,…,N−1 by employing the following method:
(14)sn=0,ifminx≤xn<Δx,1,ifΔx≤xn<2Δx,⋮255,if255Δx≤xn≤maxx,where Δx=maxx−minx/256. Thus, sn,n=0,1,2,…,N−1 is an 8-bit number time series.Step 2 (calculating SybEn).
Count the number of each symbol (from 0 to 255), and obtain the probability of each symbol as(15)Pi=#i:sn=i,n=0,1,…,N−1N,where # stands for number and i=0, 1,…,255. Thus, SybEn is defined by
(16)SybEn=−∑i=0255PilogPi.Step 3 (discrete Fourier transformation (DFT)).
Before the DFT, the following normalization is employed to the symbol time series, and it is denoted as(17)Sn=sn−meansstds,where means and stds are the mean value and the standard deviation of the time series s, respectively, and n=0,1,…,N−1. Thus, the mean value of the new time is zero and there is no direct current signal. DFT is carried out on time series S, and it is given by
(18)Xk=∑n=0N−1Sne−j2πnk/N,where k=0,1,…,N−1 and j is the imaginary unit.Step 4 (calculating SybSEn).
If the power of a discrete power spectrum with thekth frequency is Xk2, then the “probability” of this frequency is defined as
(19)ρk=Xk2∑k=0N/2−1Xk2.
When the DFT is employed, the summation runs fromk=0 to k=N/2−1. The normalization of SybSEn is denoted by [34]
(20)SybSEn=−1lnN/2∑k=0N/2−1ρklnρk.
lnN/2 is the entropy of the completely random signal.Step 5 (inverse DFT).
Define the mean square value ofXk as
(21)GN=1N∑k=0N−1Xk2.
Let(22)X~k=Xk,ifXk2>rGN,0,ifXk2≤rGN,where rr>0 is the control parameter. The inverse DFT of X~k is
(23)S~n=1N∑k=0N−1X~kej2πnk/N,where n=0,1,…,N−1. S~n reflects the regular part of the time series with detailed information removed.Step 6 (calculating SybC0).
SybC0 complexity is defined as [35]
(24)SybC0=∑n=0N−1Sn−S~n2Ω.where Ω=∑n=0N−1Sn2.The three complexity measuring algorithms estimate complexity of an 8-bit symbol time series from different aspects. Firstly, SybEn analyzes complexity in the time domain while SybSEn and SybC0 are defined in the frequency domain. Secondly, SybEn and SybSEn are defined based on the definition of the Shannon entropy while SybC0 reflects the ratio of an irregular part in the time series. Let e=6.9 and q1=q2=q3=0.95; we obtain a type II chaotic time series x. The time series is shown in Figure 9(a), and the symbol time series is illustrated in Figure 9(b). By employing (15) and (19), plots of the two different “probabilities” are shown in Figures 9(c) and 9(d), respectively. According to (22) and (24), we illustrate the irregular part of the time series with r=10, where the red line represents Sn2 and the green line represents Sn−S~n2 which shows the “difference” or “irregular part.”Figure 9
Analysis of complexity algorithms: (a) original time seriesx; (b) pseudorandom sequence s; (c) probability of each symbol Pi; (d) probability of the frequency ρi; (e) irregular part in the SybC0 algorithm.
(a)
(b)
(c)
(d)
(e)As shown in Figure9, the principles of different algorithms are different. For SybEn and SybSEn, if the probability density is more uniform, values of entropy are larger, while if the proportion of the green part is larger, values of SybC0 are larger. As a result, larger values of SybEn, SybSEn, and SybC0 mean that the system has higher complexity. In this paper, to analyze the complexity of the CMF system, the time series x with a length of 55000 is sampled with τ=10; thus, x=x1:10:55000. Finally, the sampled time series is changed into a symbol time series. For SybC0 complexity, we choose r=10.


## 4.2. Complexity Analysis

Complexity of the CFM system is analyzed by means of SybEn, SybSEn, and SybC0. Firstly, SybEn, SybSEn, and SybC0 of different types of CFM systems with different derivative orders are analyzed, and the results are shown in Tables 3–5, respectively. As with Figure 1 and Table 2, values of derivative orders are set as q1=q2=q3=0.95,0.90,and0.80 and e=6.9. It shows that complexity analysis results agree well with the LCE analysis results. When the system is chaotic, higher complexity can be found. Moreover, the system has relative higher complexity when the derivative orders are smaller. It should be pointed out that SybEn cannot distinguish the chaotic state and periodical state well similar to SybSEn and SybC0. Actually, it is also one of the reasons why we furtherly design complexity analysis methods in the frequency domain.Table 3
SybEn complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
5.4834
5.4646
5.4868
II
4.7810
4.7422
4.6255
III
4.8680
4.8711
4.9467Table 4
SybSEn complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.4497
0.5131
0.5589
II
0.5635
0.6135
0.3190
III
0.3580
0.3776
0.5386Table 5
SybC0 complexity of the CFM system with different derivative orders.
Type
q1=q2=q3=0.95
q1=q2=q3=0.90
q1=q2=q3=0.80
I
0.1263
0.1756
0.2885
II
0.1913
0.2424
0.0776
III
0.1795
0.1963
0.3128Complexity of the CFM system with parametere varying is analyzed, and results are shown in Figure 10. Here, q1=q2=q3=q equals to 0.95, and parameter e varies from 6.7 to 7.05 with a step size of 0.0007. As shown in Figure 10, SybSEn and SybC0 agree with the maximum LCEs better than SybEn, and they identify more periodical windows which show relative lower complexity.Figure 10
Complexity analysis results of the CFM system with parametere varying: (a) SybEn; (b) SybSEn; (c) SybC0.
(a)
(b)
(c)Complexity of the type II CFM system with parametere=6.9 and derivative orders q1, q2, and q3 varying is analyzed, and the analysis results are shown in Figure 11. Let q1=q2=q3=q and let it vary from 0.75 to 1. Complexity analysis results are shown in Figures 11(a)–11(c). As shown in Figures 11(b) and 11(c), complexity of the system increases with the decrease of order q, which means that the system has higher complexity with smaller values of q. Thus, the system has a good application value in the engineering field. Fix q2=q3=1 and vary q1 from 0.995 to 1, and the complexity results are shown in Figures 11(d)–11(f). Fix q1=q2=1 and vary q2 from 0.996 to 1, and the complexity results are displayed in Figures 11(g)–11(i). Moreover, let q1=q2=1 and vary q3 from 0.85 to 1, and the complexity results are illustrated in Figures 11(j)–11(l). As shown in those figures, complexity of the CFM system does not increase with the decrease of derivative order q1, q2, or q3. When q1=q2=q3=1, the integer-order system is chaotic with high complexity, but the low-complexity region can be found when the fractional derivative orders decrease. In real application, one should choose those orders with which the system generates high-complexity time series.Figure 11
Complexity analysis results of the CFM system with derivative orders varying: (a) SybEn withq1=q2=q3=q varying; (b) SybSEn with q1=q2=q3=q varying; (c) SybC0 with q1=q2=q3=q varying; (d) SybEn with q2=q3=1 and q1 varying; (e) SybSEn with q2=q3=1 and q1 varying; (f) SybC0 with q2=q3=1 and q1 varying; (g) SybEn with q1=q3=1 and q2 varying; (h) SybSEn with q1=q3=1 and q2 varying; (i) SybC0 with q1=q3=1 and q2 varying; (j) SybEn with q1=q2=1 and q3 varying; (k) SybSEn with q1=q2=1 and q3 varying; (l) SybC0 with q1=q2=1 and q3 varying.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)According to Figures11 and 12, SybEn, SybSEn, and SybC0 complexity analysis results are consistent with the corresponding maximum LCEs to a certain degree. Overall, SybSEn and SybC0 analysis results agree better with the corresponding maximum LCE results than with those of SybEn. Compared with calculating LCEs, calculating SybEn, SybSEn, and SybC0 needs much less time and it is more convenient in real application since results can be obtained with a time series. On the one hand, it shows that the system has a potential application value in practice. On the other hand, it provides a basis for parameter choice of the CFM system in real applications.Figure 12
Complexity analysis results of the CFM system in thee-q parameter plane: (a) SybSEn; (b) SybC0.
(a)
(b)SybSEn and SybC0 complexity in the q-e parameter plane is calculated, and the results are shown in Figure 12. Here, the parameter varies from 6.7 to 7.05 similar to that mentioned above, while derivative order q varies from 6.5 to 1, simultaneously. As shown in Figure 12, the minimum order for chaos is about 0.65 when e=6.7. Meanwhile, it shows that the system has higher complexity.The pseudorandom time seriessn by (14) fluctuates with the original system variables. As we all know that a chaotic system is a good source for entropy. There are many different kinds of chaotic systems that can be used for designing a pseudorandom sequence generator (PRSG), such as the Lorenz system [39], logistic map, sine map, and 2D-SIMM [40]. And these PRSGs are widely used in real application fields such as image encryption [41], speech encryption [42], and chaotic watermark [43]. The quantization algorithms used in this applications are more complex than the method given in (14). One of the most commonly used methods is designed by expressing the original number or its converted number as a 64-bit binary number as DB63-DB0; then, one can obtain a new 8-bit number by choosing DB7-DB0. The details of this method are given as follows.The original number is converted as(25)φn=roundxn×10w,where n=0,1,2,…,N−1 and w is a control number. Here, in this paper, w=10. Thus, φn is an integer number and it can be expressed as
(26)φn=DB63DB62⋯DB1DB0.By choosing the first 8-bit number, a new symbol time seriessn is given as
(27)sn=DB7⋯DB1DB0.Obviously,sn can be expressed as a decimal integer number varying from 0 to 255, as mentioned above. By using this quantization algorithm, complexity of pseudorandom sequences generated by different chaotic systems is calculated and the results are shown in Table 6. The length of each segment symbol time series is 5000, and complexity of such 100 time series is calculated. The results are given as mean ± std of complexity values in these windows. As shown in Table 6, entropy or complexity of different chaotic pseudorandom sequences is at about the same high level. It shows that as with other different chaotic systems, the CFM system is also a good system for high entropy.Table 6
Complexity measuring results of different chaotic pseudorandom sequences with the second quantization method.
Systems
Equations
SybEn
SybSEn
SybC0
CFM system
Tt0q1x=cf1zTt0q2y=d−ef2y+ef1zTt0q3z=ef2y−f1z−x
5.4825 ± 0.0036
0.9466 ± 0.0014
0.9995 ± 0.0014
Lorenz system
ẋ=10y−xẏ=28x−xz−yż=xy−8z/3
5.5206 ± 0.0012
0.9583 ± 0.0003
0.9995 ± 4.4 × 10−4
Logistic map
xn+1=4xn1−xn
5.4962 ± 0.0053
0.9461 ± 0.0012
0.9994 ± 0.0015
Sine map
xn+1=4sinπxn
5.4966 ± 0.0043
0.9459 ± 0.0013
0.9992 ± 0.0019
2D-SIMM
xn+1=10sinπynsin10/xnyn+1=10sinπxn+1sin10/yn
5.4966 ± 0.0046
0.9461 ± 0.0014
0.9996 ± 0.0013According to Table6, complexity of the CFM system is high as other systems. Thus, it is necessary to check whether the pseudorandom sequence generated by this system passes the test suite of NIST. The package used for the NIST test is sts-2.1.2 which can be downloaded from the website. Two indicators, which are p values and the proportion of passing sequences, are used to show whether the sequence passes the test or not. The minimum value for p value is 0.0001. It means that when the p value is larger than 0.0001 and the confidence interval satisfies
(28)1−η−31−ηηM,1−η+31−ηηM,where M is the sample size and η is the given significance level, then the pseudorandom bit generator passes the test successfully. In our test, the length of the pseudorandom sequence is 106, and η=0.01 with a confidence interval given by 96.015%,1. The test result is illustrated in Table 7. For those items including C. Sums (2 times), N.O. Temp (148 times), R. Excur. (8 times), R. Excur. V. (18 times), and Serial (2 times), we only illustrate the lowest values of p value and proportion. It is shown in Table 7 that all p values are larger than 0.0001 and the computed proportion for each test lies inside the confidence interval. Hence, the tested binary sequences generated by the proposed pseudorandom bit generator are random.Table 7
NIST test result of binary sequences generated by the CFM system.
Tests
p value
Proportion
Success
Frequency (1)
0.3345
97/100
✓
B. Frequency (1)
0.6163
100/100
✓
C. Sums (2)
0.9558
99/100
✓
Runs (1)
0.1816
99/100
✓
Longest Run (1)
0.2133
99/100
✓
Rank (1)
0.3191
99/100
✓
FFT (1)
0.1968
99/100
✓
N.O. Temp (148)
0.0288
97/100
✓
O. Temp (1)
0.7197
97/100
✓
Universal (1)
0.5341
99/100
✓
App. Entropy (1)
0.8165
57/58
✓
R. Excur. (8)
0.010
57/58
✓
R. Excur. V. (18)
0.0179
58/58
✓
Serial (2)
0.5544
99/100
✓
L. Complexity (1)
0.1025
99/100
✓In this section, two different pseudorandom quantization algorithms are designed. As shown in Figures9(a) and 9(b), fluctuation of pseudorandom sequence obtained by the first method given (14) agrees well with the original time series. Thus, it is the reason why complexity analysis results match well with the corresponding maximum LCEs. It provides a good symbol sequence for complexity analysis of chaotic systems when analyzed by complexity measuring methods. However, according to Table 6, pseudorandom sequences generated by different systems by employing the second method have the same level of high complexity. Meanwhile, the pseudorandom sequence passes all the NIST tests. It means that the obtained sequence is random. As with other chaotic systems, the CFM system is also a good secret key generator for real applications including information encryption, secure communication, and chaotic digital watermark.Finally, it is necessary to explain why we choose 8-bit or 256-quantization-level symbols to form the numbers and how complexity analysis result is determined by the quantization levels. Lete=6.9 and q1=q2=q3=0.95. A segment of type II chaotic time series is used to calculate the complexity. The relation between the number of quantization levels and its complexity measuring results is shown in Figure 13, where SybEn is normalized via dividing lnm. Here, m is the number of bits or 2m is the quantization levels. It shows that the values of the two frequency domain methods, namely, SybC0 and SybSEn, do not change with the quantization levels. However, values of SybEn (the time-domain method) increase but tend to become stable with the quantization levels. Meanwhile, the 8-bit number is widely used in real applications. Thus, 8-bit or 256-quantization-level symbol sequences are employed for complexity analysis.Figure 13
Relation between the number of quantization levels and its complexity: (a) quantization method one; (b) quantization method two.
(a)
(b)


## 5. Conclusions

In this work, we introduced the conformable fractional calculus in a nonlinear system with two memcapacitors (CFM system). The conformable fractional differential transform method is employed to solve the nonlinear conformable differential system for the first time. The ALCE calculation algorithm is designed based on the obtained solution, and bifurcation and chaos in the CFM system are explored. It shows that the CFM system has rich dynamics with the variation of system parameter and derivative orders. Meanwhile, coexisting attractors and transient state were observed under some specific parameters. Three symbol complexity measuring algorithms are designed, namely, symbolic entropy (SybEn) algorithm, symbolic spectral entropy (SybSEn) algorithm, and symbolic C0 (SybC0) algorithm. Complexity analysis results match well with the corresponding maximum LCE analysis results. Finally, by using a proper quantization algorithm, the obtained pseudorandom sequence has high complexity as other common chaotic systems and is randomly verified by the NIST test. It shows that the memory electronic element-based systems have a potential engineering application value.



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