Complexity
Volume 2018 (2018), Article ID 3497640, 15 pages
https://doi.org/10.1155/2018/3497640

Research Article

Analysis and Implementation of a New Switching Memristor Scroll Hyperchaotic System and Application in Secure Communication

Ping Liu,¹ Rui Xi,¹ Pengbo Ren,¹ Jialin Hou,¹ and Xiang Li²

¹College of Mechanical and Electrical Engineering, Shandong Agricultural University, Taian 271018, China
²State Key Laboratory of Crop Biology, College of Life Sciences, Shandong Agricultural University, Taian 271018, China

Correspondence should be addressed to Xiang Li; lixiang@sdau.edu.cn

Received 27 February 2018; Revised 22 April 2018; Accepted 27 May 2018; Published 2 July 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Ping Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proposed a novel switching scroll hyperchaotic system based on a memristor device and explored its application to secure communication. The new system could be switched between the double-scroll chaotic system and multiscroll one by switch and switch . We gave the construction process of the novel system, its numerical simulations, and dynamical properties, firstly. Moreover, the memristive circuit implementation of the new switching system was presented and the results were also in agreement with those of numerical simulation. Finally, the new switching memristive system was applied to secure communication by means of the drive-response synchronization with chaotic masking. When the voice signal is a rising waveform, it is encrypted by the double-scroll memristive system. When the voice signal is a falling waveform, the multiscroll memristive system works. The voice signal is completely submerged in the chaotic signal and could not be distinguished at all. Security analyses show that it is a successful application to secure communication.

1. Introduction

Chaotic systems based on memristor have widely attracted attention recently. It has many applications such as in secure communication [1–4], neural network [5–9], and chemical route [10]. A lot of researches on memristive chaotic (hyperchaotic) system have been reported, for example, global synchronization [11], state estimation [12], time delay [13], and adaptive synchronization [14] of the memristive chaotic system. On the other hand, scroll chaotic systems have been explored in many papers extensively. For instance, Ma et al. introduced simulation and circuit implementation of 12-scroll chaotic system in [15]. Chen et al. investigated the generation of grid multiscroll chaotic attractors in [16]. In [17], Chen et al. studied the fractional-order multiscroll chaotic system. And in [18-19], García-Martínez et al. and Liu et al. discussed the multiscroll hyperchaotic system and its application to secure communication, but the memristor has not been introduced to chaotic system to produce multiscroll.

With respect to the chaotic system based on memrister, its application to secure communication has been studied in many works, such as in [1–4], since memristors are nonlinear elements with memory function, which are different from resistors, capacitors, and inductors. So the applications to secure communication based on chaotic system with memristor have become a hot topic. Meanwhile, researches on application of the scroll chaotic system have also existed. Whether the memristive chaotic (hyperchaotic) system or the scroll system, application to secure communication has been very common. Till now, secure communication based on the switching memristor scroll hyperchaotic system has not been explored. Thus, it has a great significance to investigate the switching memristive scroll hyperchaotic system and its application to secure communication.

The rest of the paper was organized as follows. Section 2 introduced the construction, numerical simulations, and dynamic analysis of the switching scroll hyperchaotic system based on a memristor. The circuit implementation was given in Section 3. In Section 4, application to voice encryption was discussed using the drive-response synchronization with chaotic masking and some conclusions are given in the last.

2. System Construction and Dynamic Analysis

2.1. System Construction

The Chua’s circuit is composed of a linear resistor, a linear inductor, two linear capacitors, and a nonlinear Chua’s diode as shown in Figure 1.

Figure 1: Chua’s circuit diagram.

Based on Figure 1, the differential equation [20] could be obtained as

where when taking system parameters and with initial values , we can get the projection on the phase plane of the double-scroll attractors shown in Figure 2.

Figure 2: A projection of the double-scroll attractor onto the plane.

In Figure 1, the resistor and the Chua’s diode are replaced by a magnetically controlled memristor and a linear function, respectively. So, we can get a new hyperchaotic memristive circuit demonstrated by Figure 3. The memristor is the fourth kind of basic passive circuit element which is proposed by Professor Chua. There are two mathematical definitions of memristor, which are charge-controlled memristor and flux-controlled memristor, respectively [21]. The expression of a charge-controlled memristor is

Figure 3: The new hyperchaotic memristive circuit.

The flux-controlled memristor could be expressed as

And basic model of the memristor can be found in [22]. The definition of memristor was extended in [23],

where and are the input signal and output signal of the memristor, respectively. is the system state variable. The input of the magnetically controlled memristor is voltage , the output is the current following through the memristor, and flux is the state variable.

According to the definition of memristor, we obtain a novel expression as

where is called memory conductance, let . According to the Kirchhoff’s law and component parameter constraints, circuit dynamic equation could be obtained as

where is the time constant. Dimensionless time could be obtained by the scaling time and state space. The dimensional state variable , , , , , , , , and , so we can obtain

where , , and , , , and are system parameters. The simulation results are given in Figure 4 by taking initial values , , and parameters , , , and . Then, based on (7), the N-scroll is realized in direction by using the step function.

(a)

(b)

Figure 4: Projections of the double-scroll hyperchaotic attractors onto the and plane. (a) plane. (b) plane.

There are two forms of step function, when

scrolls could be generated in direction.

When

scrolls could be generated in direction.

N-scroll could be obtained by a reasonable set of step function parameters. For example, when , ,

The 2 × 3 multiscroll hyperchaotic system based on a memristor device is obtained as

When system parameters are , , , and with initial values and , simulation results are presented in Figure 5.

(a)

(b)

Figure 5: Projections of the 2 × 3 multiscroll hyperchaotic attractors onto the and plane. (a) plane. (b) plane.

Similarly, 2 × 5 and 2 × 6 multiscroll hyperchaotic system could be also obtained by taking , , and , , and

respectively.

When selecting system parameters , , , and with initial values and , the numerical simulations of 2 × 5 and 2 × 6 multiscroll hyperchaotic system are shown in Figures 6 and 7.

(a)

(b)

Figure 6: Projections of the 2 × 5 multiscroll hyperchaotic attractors onto the and plane. (a) plane. (b) plane.

(a)

(b)

Figure 7: Projections of the 2 × 6 multiscroll hyperchaotic attractors onto the and plane. (a) plane. (b) plane.

2.2. Dynamic Analysis

2.2.1. The Double-Scroll Memristive System Is Taken as an Example to Analyse the Dynamic Properties

(1) Lyapunov Exponents. When selecting system parameters , , , and with initial values and , we can get the Lyapunov exponents diagram as shown in Figure 8.

Figure 8: Lyapunov exponents diagram.

From Figure 8, we can see that the chaotic system has two positive Lyapunov exponents. So, the system is a hyperchaotic system and has two scrolls.

(2) Bifurcation Diagram. Lyapunov exponents and bifurcation diagram of the system due to the variation of parameter is displayed in Figure 9. We could see that when and , the system shows chaotic behavior.

(a)

(b)

Figure 9: Lyapunov exponents and bifurcation diagram with respect to parameter . (a) Lyapunov exponents diagram. (b) Bifurcation diagram.

(3) Dissipation. Generation of chaotic behavior is decided by whether the system has a dissipative structure or not [24]. The dissipative formula of the system is

so the system is dissipative, converging at an exponential rate until it becomes 0. When , every volume element which contains trajectories of the system shrinks to zero with the rate of exponential convergent.

(4) Equilibrium Points and Stability. The Jacobian matrix of the system is

The equilibrium points of the system could be calculated by making (7) as . We could get three equilibrium points as

The Jacobian matrix at point becomes

Eigenvalues at point are , , , and . The Jacobian matrix at point and becomes

The Equilibria and could have eigenvalues , , , and , which are called saddle points of index 2 since the two complex conjugate eigenvalues have positive real parts[25, 26]. It is clear that is the first type saddle point since the real eigenvalue is positive [25, 26]. It is noticed that the scrolls are generated only around the equilibria of saddle points of index 2 [25, 26]. Moreover, equilibria and correspond to the two saturated plateaus, which are responsible for generating the two scrolls in the double-scroll attractor. However, the equilibrium point corresponds to the saturated slope and is responsible for connecting these two symmetrical scrolls.

2.2.2. The Dynamic Analysis of 2 × 3 Multiscroll Hyperchaotic System

The dynamic analysis of 2 × 3 multiscroll hyperchaotic system is introduced briefly as follows. Let

Equation of equilibrium points could be obtained as

From (19), we could get the coordinate of system equilibrium points in the y-axis as , that is, , ±12, ±24, . The figures of equilibrium points and chaotic attractors of the multiscroll hyperchaotic system are demonstrated in Figure 10. Therefore, the number of equilibrium points in the y-axis direction is 5 × 2. The solid points represent the second type saddle point, which correspond to the six saturated plateaus and are responsible for generating the six scrolls in the 2 × 3-scroll attractors. The hollow points represent the first type saddle point, which correspond to the saturated slope and are responsible for connecting these six symmetrical scrolls.

(a)

(b)

Figure 10: Equilibrium points and attractors of the 2 × 3 multiscroll hyperchaotic system. (a) Equilibrium points. (b) 2 × 3 multiscroll attractors.

3. The Circuit Implementation of the Switching Scroll Hyperchaotic System

3.1. The Circuit Implementation of the Double-Scroll Hyperchaotic System

Circuit diagram of the double-scroll hyperchaotic system is displayed in Figure 3. With respect to the memristor shown in Figure 11(a), its internal circuit structure could be obtained in [27], shown in Figure 11(b). Circuit implementation results are shown in Figure 12, which are in agreement with numerical simulation results in Figure 4.

(a)

(b)

Figure 11: The memristor and its internal circuit. (a) The memristor. (b) Its internal circuit.

(a)

(b)

Figure 12: Circuit implementation of the double-scroll hyperchaotic system. (a) plane. (b) plane.

Symbolic function circuit is designed by the saturated output voltage of the operational amplifier to achieve step function. The saturated voltage of the operational amplifier in Figure 13 is .

Figure 13: Step function circuit.

The step function circuit designed by the method is easy to expand and achieve, that is, more complex function could be realized by increasing the number of corresponding comparison circuit. In the following part, we choose the 2 × 3 scroll system as an example to discuss the multiscroll system. Based on Figure 13, step function is connected between the capacitor and the ground to get the circuit diagram of the multiscroll hyperchaotic system shown in Figure 14, and its circuit simulation results are displayed in Figure 15 which are in accordance with the numerical simulation results.

Figure 14: Circuit diagram of the multiscroll hyperchaotic system.

(a)

(b)

Figure 15: Circuit implementation of the multiscroll hyperchaotic system. (a) plane. (b) plane.

4. Applications to Secure Communication

4.1. Drive-Response Synchronization of the Switching Systems

The synchronization of the switching systems is briefly stated as follows. The drive system from (11) is

where , , , and are system parameters, , , , and , and initial values and .

And the response system is

where , , , , , and are system parameters, , , , and , and initial values and .

The state error is defined as

where is diagonal matrix, and . is scaling function matrix (if , , ( is a constant), and , respectively, then the synchronization is called as complete synchronization, antisynchronization, projective synchronization, and modified projective synchronization). In this paper, we choose , that is to say, the synchronization type is antisynchronization.

The synchronization controller of the response system is designed as

According to the systems (21), (22), (23), and controller (24), we could obtain the final expression of the error system as

Let , , , and , it is obvious that all roots of the error system have negative real parts. When , the error system (25) converge to 0 and therefore the synchronization between the drive system and the response system is realized.

Next, we illustrate the validity of the proposed controller by MATLAB 2010a. Selecting initial values , , , , , , , and , we could get those of the error system as , , , and . Keeping the system parameters, the initial values, the scale factors, and synchronization controllers unchanged, simulation results are illustrated in Figure 16. One can see that the error dynamic system is stable asymptotically by using the designed controllers, which implies that the drive system and the response system could achieve synchronization well.

(a)

(b)

(c)

(d)

Figure 16: The state trajectories of the error system. (a) State trajectories of . (b) State trajectories of . (c) State trajectories of . (d) State trajectories of .

4.2. Drive-Response Circuit Synchronization of the Switching Systems

Drive-response synchronous circuit of the multiscroll hyperchaotic system is demonstrated in Figure 17.

Figure 17: Drive-response synchronous circuit of the multiscroll hyperchaotic system.

The and waveforms of the drive system and the and waveforms of the response system are, respectively, shown in Figures 18(a) and 18(b). The waveform at the time of synchronization is shown in Figure 19. From the simulation results, we could see that voltage signal waveforms of the drive and response system are exactly the same, and the synchronous phase diagram of the corresponding state is a straight line through the origin point with 45 degrees, indicating that the corresponding state variables in the two circuits achieve a good synchronization. When it is applied to the secure communication, the synchronization method is simple and the circuit implementation is convenient.

(a)

(b)

Figure 18: The and waveforms and the and waveforms.

Figure 19: The waveform at the time of synchronization.

The drive-response synchronous circuit of the double-scroll hyperchaotic system is similar to that of the multiscroll one and will be omitted here.

4.3. Application to Secure Communication

In the circuit implementation, the double-scroll and multiscroll circuits are encapsulated, respectively, so as to simplify the circuit structure.

4.3.1. Description of Encryption and Decryption Algorithm

In this subsection, the synchronization circuit above is applied to secure communications. A voice signal carrying the message to be transmitted could be masked by the chaotic sequence , which is a key sequence from the double-scroll and multiscroll hyperchaotic system. And

The chaotic synchronous circuit given above could be applied to extract the message at the receiver. Some strategies could be used to make the actual transmitted signal as broadband as possible, that is, to make its detection through spectral techniques difficult. In general, three strategies are proposed in chaotic secure communications [28]. One is signal masking, where ; the second is modulation, ; the third is a combination of masking and modulation, . Here, chaotic masking is used to encryption. The transmitted signal is and injected into the transmitter and, simultaneously, transmitted to the receiver. By the above synchronous circuit, a chaotic receiver is then derived to recover the voice signal at the receiving end, that is,

The switching encryption circuit is shown in Figure 20. System 1 and system 2 represent the double-scroll and multiscroll hyperchaotic circuit, respectively. When the voice signal is a rising waveform, switch connects to , and switch connects to . At this time, is used as the key sequence in the encryption, that is to say, the double-scroll hyperchaotic system works. When the voice signal is a falling waveform, switch connects to , and switch connects to . At this point, is used to encrypt the voice signal, and the multiscroll hyperchaotic system does. With the voice waveform changes, the connection mode of the switch and switch is different. Finally, the switching between the double-scroll and the multiscroll hyperchaotic system is realized. In the decryption process, the chaotic signal generated by the response system is used to decrypt the signal. The schematic diagram is shown in Figure 21. The key sequence generated by the drive system is used to cover the voice signal and that generated by the response system works in removing the cover of the encrypted signal.

Figure 20: The switching encryption circuit.

Figure 21: Encryption and decryption flow chart.

Results of encryption and decryption are displayed in Figure 22. We can know that the voice signal is completely covered by the chaotic signal, and the original shape could not be seen at all after the voice signal and the chaotic signal generated by the driving system is superimposed. The cracker could not get any information of the voice signal from the channel [29-30]. It has proved that the switching encryption method has pretty good privacy.

(a)

(b)

(c)

Figure 22: Encrypted and decrypted waveforms. (a) Original voice signal. (b) Encrypted voice signal. (c) Decrypted voice signal.

Moreover, the decrypted voice signal and the original voice signal are identical, indicating that the original signal could be recovered well by the chaotic signal generated by the response system. This proves the effectiveness and reliability of the encryption method.

4.3.2. Security Analyses

(1) Key Space Analysis. The size of key space is the total number of different keys used in the encryption process. The key space should be large enough to resist attacks. In the encryption scheme, thirteen key parameters are used, which are , , , and from the double-scroll hyperchaotic system, , , , and from the multiscroll hyperchaotic system, scaling function matrix , synchronization controller , and system parameters , , , and . If the precision is chosen as 10⁻¹⁴, the total key space is (10¹⁴)¹⁴, which is obviously larger than that in [31].

(2) Key Sensitivity Analysis. In order to test the sensitivity of the encryption algorithm, key parameter is changed to and others unchanged in the decryption, the result is shown in Figure 23. We could see that the encryption scheme in the paper is so sensitive to key parameters that a small change can lead to a completely different result.

Figure 23: Wrong decryption result.

(3) Encryption Speed. For a good encryption algorithm, running speed is an important reference aspect. MATLAB 2014a is used to run the program that realizes the proposed algorithm in a personal computer with a Pentium 4 CPU 3.0 GHz, 4.0 GB RAM, 500 GB hard disk, and Microsoft Windows 7 operating system. The encryption algorithm in the paper is compared with that in [32] and [33]; results between different algorithms are shown in Table 1. We could see that the speed of our encryption algorithm is much faster.

Table 1: Comparison of encryption speed between different algorithms.

Algorithms
Encryption speed (MB/s)

Algorithm of [32]
4.80

Algorithm of [33]
6.21

The proposed algorithm
7.91

5. Conclusions

In this paper, a new switching scroll hyperchaotic system was proposed, which consisted of the double-scroll and multiscroll hyperchaotic system. Based on a memristor device, the double-scroll hyperchaotic system was taken as an example to analyse the dynamic properties, such as Lyapunov exponents, bifurcation diagram, dissipation and equilibrium points, and stability. Furthermore, circuit implementation of the system was introduced in detail and the results are consistent with those of numerical simulations. At last, the switching scroll hyperchaotic system was successfully applied to voice encryption and showed its effectiveness.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (no. 31700644), Postdoctoral Science Foundation of China (nos. 2015M582122 and 2016T90644), Key Research and Development Project of Shandong Province (nos. 2016ZDJS02A07 and 2017GNC12105), the Agricultural Machinery Research and Development Project of Shandong Province (2018YF004).

References

X. Shi, S. Duan, L. Wang, T. Huang, and C. Li, “A novel memristive electronic synapse-based Hermite chaotic neural network with application in cryptography,” Neurocomputing, vol. 166, pp. 487–495, 2015.

J. Yang, L. Wang, Y. Wang, and T. Guo, “A novel memristive hopfield neural Network with application in associative memory,” Neurocomputing, vol. 227, pp. 142–148, 2017.

L. Wang, X. Wang, S. Duan, and H. Li, “A spintronic memristor bridge synapse circuit and the application in memrisitive cellular automata,” Neurocomputing, vol. 167, pp. 346–351, 2015.

S. Zhu, L. Wang, and S. Duan, “Memristive pulse coupled neural network with applications in medical image processing,” Neurocomputing, vol. 227, pp. 149–157, 2017.

J. Gao, P. Zhu, A. Alsaedi, F. E. Alsaadi, and T. Hayat, “A new switching control for finite-time synchronization of memristor-based recurrent neural networks,” Neural Networks, vol. 86, pp. 1–9, 2017.

A. Abdurahman, H. Jiang, and Z. Teng, “Exponential lag synchronization for memristor-based neural networks with mixed time delays via hybrid switching control,” Journal of the Franklin Institute, vol. 353, no. 13, pp. 2859–2880, 2016.

A. Abdurahman and H. Jiang, “New results on exponential synchronization of memristor-based neural networks with discontinuous neuron activations,” Neural Networks, vol. 84, pp. 161–171, 2016.

J. Gao, P. Zhu, W. Xiong, J. Cao, and L. Zhang, “Asymptotic synchronization for stochastic memristor-based neural networks with noise disturbance,” Journal of the Franklin Institute, vol. 353, no. 13, pp. 3271–3289, 2016.

P. Jiang, Z. Zeng, and J. Chen, “On the periodic dynamics of memristor-based neural networks with leakage and time-varying delays,” Neurocomputing, vol. 219, pp. 163–173, 2017.

T. D. Dongale, K. V. Khot, S. S. Mali et al., “Development of Ag/ZnO/FTO thin film memristor using aqueous chemical route,” Materials Science in Semiconductor Processing, vol. 40, pp. 523–526, 2015.

Z. Guo, S. Yang, and J. Wang, “Global synchronization of memristive neural networks subject to random disturbances via distributed pinning control,” Neural Networks, vol. 84, pp. 67–79, 2016.

S. Ding, Z. Wang, J. Wang, and H. Zhang, “ state estimation for memristive neural networks with time-varying delays: the discrete-time case,” Neural Networks, vol. 84, pp. 47–56, 2016.

V.-T. Pham, S. Vaidyanathan, C. K. Volos, S. Jafari, N. V. Kuznetsov, and T. M. Hoang, “A novel memristive time–delay chaotic system without equilibrium points,” The European Physical Journal Special Topics, vol. 225, pp. 127–136, 2016.

S. Song, X. Song, and I. T. Balsera, “Adaptive projective synchronization for fractional-order T-S fuzzy neural networks with time-delay and uncertain parameters,” Optik, vol. 129, pp. 140–152, 2017.

Y. Ma, Y. Li, and X. Jiang, “Simulation and circuit implementation of 12-scroll chaotic system,” Chaos, Solitons and Fractals, vol. 75, pp. 127–133, 2015.

Z. Chen, G. Wen, H. Zhou, and J. Chen, “Generation of grid multi-scroll chaotic attractors via hyperbolic tangent function series,” Optik, vol. 130, pp. 594–600, 2017.

L. Chen, W. Pan, R. Wu, K. Wang, and Y. He, “Generation and circuit implementation of fractional-order multi-scroll attractors,” Chaos, Solitons and Fractals, vol. 85, pp. 22–31, 2016.

M. García-Martínez, L. J. Ontañón-García, E. Campos-Cantón, and S. Čelikovský, “Hyperchaotic encryption based on multi-scroll piecewise linear systems,” Applied Mathematics and Computation, vol. 270, pp. 413–424, 2015.

H. Liu, A. Kadir, and Y. Li, “Audio encryption scheme by confusion and diffusion based on multi-scroll chaotic system and one-time keys,” Optik, vol. 127, no. 19, pp. 7431–7438, 2016.

B. S. T. Alkahtani, “Chua’s circuit model with Atangana–Baleanu derivative with fractional order,” Chaos, Solitons and Fractals, vol. 89, pp. 547–551, 2016.

L. Chua, “Memristor-the missing circuit element,” IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971.

D. B. Strukov, G. S. Snider, D. R. Stewart, and R. Stanley Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, 2008.

S. Wen, Z. Zeng, and T. Huang, “Adaptive synchronization of memristor-based Chua’s circuits,” Physics Letters A, vol. 376, no. 44, pp. 2775–2780, 2012.

P. Liu and S. Liu, “Anti-synchronization between different chaotic complex systems,” Physica Scripta, vol. 83, no. 6, article 065066, 2011.

D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring,” International Journal of Bifurcation and Chaos, vol. 13, no. 10, pp. 2889–2903, 2003.

L. Chua, M. Komuro, and T. Matsumoto, “The double scroll family,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1118, 1986.

Y. Wang and X. Liao, “Stability analysis of multimode oscillations in three coupled memristor-based circuits,” AEU-International Journal of Electronics and Communications, vol. 70, no. 12, pp. 1569–1579, 2016.

B. Mensour and A. Longtin, “Synchronization of delay-differential equations with application to private communication,” Physics Letters A, vol. 244, no. 1-3, pp. 59–70, 1998.

L. Wang, H. Song, and P. Liu, “A novel hybrid color image encryption algorithm using two complex chaotic systems,” Optics and Lasers in Engineering, vol. 77, pp. 118–125, 2016.

X. Li, L. Wang, Y. Yan, and P. Liu, “An improvement color image encryption algorithm based on DNA operations and real and complex chaotic systems,” Optik, vol. 127, no. 5, pp. 2558–2565, 2016.

Z. Wang, X. Huang, Y. X. Li, and X. N. Song, “A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,” Chinese Physics B, vol. 22, no. 1, article 010504, 2013.

X. Wu, Y. Li, and J. Kurths, “A new color image encryption scheme using CML and a fractional-order chaotic system,” PLoS One, vol. 10, no. 3, article e0119660, 2015.

M. Ahmad, U. Shamsi, and I. R. Khan, “An enhanced image encryption algorithm using fractional chaotic systems,” Procedia Computer Science, vol. 57, pp. 852–859, 2015.