Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization

Liu, Jiaxun; Wang, Zuoxun; Shu, Minglei; Zhang, Fangfang; Leng, Sen; Sun, Xiaohui

doi:https://doi.org/10.1155/2019/7242791

Complexity

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 7242791 | https://doi.org/10.1155/2019/7242791

Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization

Jiaxun Liu,¹Zuoxun Wang,¹Minglei Shu,²Fangfang Zhang,¹Sen Leng,¹and Xiaohui Sun³

Academic Editor: Cornelio Posadas-Castillo

Received10 Apr 2019

Accepted10 Jul 2019

Published18 Aug 2019

Abstract

Fractional complex chaotic systems have attracted great interest recently. However, most of scholars adopted integer real chaotic system and fractional real and integer complex chaotic systems to improve the security of communication. In this paper, the advantages of fractional complex chaotic synchronization (FCCS) in secure communication are firstly demonstrated. To begin with, we propose the definition of fractional difference function synchronization (FDFS) according to difference function synchronization (DFS) of integer complex chaotic systems. FDFS makes communication secure based on FCCS possible. Then we design corresponding controller and present a general communication scheme based on FDFS. Finally, we respectively accomplish simulations which transmit analog signal, digital signal, voice signal, and image signal. Especially for image signal, we give a novel image cryptosystem based on FDFS. The results demonstrate the superiority and good performances of FDFS in secure communication.

1. Introduction

With the booming of multimedia and wireless networks, secure communication of information is more and more significant in real applications. In particular, encryption of voice, images, and other real information has got a lot of attentions, such as recording of business meetings, military and architectural images, computer code, and so on. Many of the theoretical results make improvements in communication complexity as justification and a measure of success, and this seems logical as the communication complexity is the lower bound of the computation complexity [1].

In the last three decades, chaotic synchronization of real variables has been a hotspot in nonlinear science since Caroll and Pecora [2] firstly achieved chaotic synchronization in electronic circuit. Chaos can exhibit a noise-like behavior such as randomicity, ergodicity, unpredictability, high sensitivity for the initial condition, and broadband nature. It is the great advantage in chaos communication and allows chaos to elegantly cover up the communication information. Moreover, chaos communication is usually easy to be realized by numerous simple circuits, which is more convenient than traditional cryptosystem.

Since Fowler et al. [3] came up with the concept of complex chaos in 1982, many studies have been proposed in the field of new complex chaotic systems (CCS) and their applications [4–12]. Some actual physical models also were discovered as CCS, such as amplitudes of electromagnetic and detuned laser systems [13–17]. The number of the variables in CCS is twice as many as real chaotic systems (RCS), which is the major difference between RCS and CCS. In some sense, it means that CCS have nature superiority in increasing transformation and channels. Moreover, complex variables are simpler to be accomplished by RLC circuit in actual applications than real variables. Therefore, the complicated dynamics and easy implementations of CCS are born to secure communication. Some methods of integer complex chaotic synchronization (ICCS) used in secure communication were discussed in literatures. Reference [18] firstly put complex function projective synchronization (CFPS) into secure communication and got the superb result. However, when the signal of the master system is close to zero in CFPS, it affects the encryption since the denominator is close to zero. Therefore, [9] gave the definition of difference function synchronization (DFS) and solved this problem. DFS is proposed to the point of difference in two state variables and breaks the previous concept that scholars only study synchronization from the proportional relation between state variables. It is somewhat great innovation. DFS extends the difference between two state variables from zero to any desired functions. Therefore, it is the synchronization with much broader context. Complete synchronization and phase synchronization are its special cases. E.Mahmoud and Abo-Dahab studied another chaotic complex nonlinear framework and achieved communication of analog signal based on complex antisynchronization [19]. As far as we know, all above literatures of secure communication were just on the basis of ICCS.

Fractional calculus has been recommended over three hundred years, but it had not received too much attention because of the unclear physical background of fractional calculus. Until the last decade, some scholars found that it is more accurate than integer order calculus in describing some actual physical models, for instance, viscoelastic system [20], viscoelastic material [21], finance system [22], nuclear spin generator system [23], industrial system [24], human immunodeficiency virus model [25], and other interdisciplinary fields. Therefore, fractional chaos also caused abundant interests. Besides, because of complicated geometric interpretation of nonlocal effects of fractional derivatives in time and space [26], fractional chaos exhibits more unpredictable and complex nonlinear dynamic behaviors than integer chaotic systems. These merits were caught hold of by a few researchers who focused on chaos communication. Reference [27] studied the modified generalized projective synchronization of fractional real hyperchaotic systems and applied it to secure communication. Sarah et al. [28] proposed a novel secure image transmission method based on fractional real discrete-time chaotic systems. Muthukumar et al. [29] presented a fractional sliding mode controller and studied its application for a cryptosystem. Li and Wu accomplished secure communication of fractional chaotic systems with teaching-learning-feedback based optimization [30]. Mohammadzadeh and Ghaemi researched a secure communication with uncertain fractional hyperchaotic synchronization [31]. These proposed literatures make significant contributions in secure communication of fractional chaotic systems. However, we find most papers were based on fractional real chaotic synchronization.

Enlightened by the above discussions, we will put the FCCS with DFS into secure communication in this paper and gain higher security and better reliability than traditional methods and other chaos communication schemes. It will combine the advantages of DFS in complex chaotic synchronization and fractional chaotic systems. Compared with secure communication based on integer real chaos synchronization, FCCS not only increase the number of potential channels, but also complicate the types of encryption keys. When it comes to secure communication based on fractional real chaos synchronization, FCCS have nature superiority in transmitting complex signal and choosing variable signal channels. As for secure communication based on integer complex chaos synchronization, FCCS could generate more unpredictable secret keys by means of different fractional orders. Due to the existence of the double variables and fractional derivative factors, it is difficult for the unauthorized third party to extract the useful information because of its complicated dynamic behavior.

The main contributions of this paper are as follows: (1) Firstly, we extend the DFS to FDFS and investigate the general controller. (2) Four types of transmitted signals including analog signal, digital signal, voice signal, and image signal are accomplished to verify the high security and good performance of the communication scheme based on FDFS. (3) As for the image signal, we propose a novel cryptosystem with FDFS.

The structure of the remaining paper is organized as follows: some basic mathematical theorems are given in Section 2. Section 3 introduces the FDFS and general controllers. The application of FCCS is finished by four types of signals and the method of image cryptosystem is proposed in Section 4. Finally, the conclusions are drawn in Section 5.

2. Mathematical Background

There are three main types of definition of fractional derivative, such as Caputo definition, Riemann Liouville definition, and Grunwald Letnikobv definition. In this paper, we use the Caputo definition as it includes the conventional initial conditions and Caputo derivative of the constant is zero.

Definition 1 (see [32]). The Caputo derivative definition is as follows:where is integer part of and is the gamma function. and are the upper and lower bounds, and is called the order Caputo differential operator. The gamma function is

Lemma 2 (see [33]). There is an autonomous fractional systemwhere and . The system is asymptotically stable if and only ifAnd the component of the state decays toward like .

3. Fractional Difference Function Synchronization

3.1. The Definition of FDFS

Recently, a new type of chaotic synchronization was proposed, which is called DFS in [9]. It is the expanding form of complete synchronization (CS) and phase synchronization (PHS). Particularly, for some signals near zero, it is effective for DFS in secure communication. In this part, we expand the DFS into FDFS and give the general synchronization controller, aiming to lay the foundation for secure communication in the following part.

Delighted by the concept of DFS, we firstly present the FDFS as follows:

Definition 3. For two -dimensional and -order general form of fractional chaotic systems and , we call the difference between and as the fractional difference function vector,where , , , and .

Definition 4. For two arbitrary fractional chaotic state vector and difference vector in (5), they are said to be FDFS if there existswhere is the matrix norm.

Remark 5. When , the FDFS would be DFS.

Remark 6. CS indicates the difference vectors with different or same initial value converge to zero when and . The CS is a special case of the FDFS.

Remark 7. When , the difference function vector will be a constant value, so FDFS is also the extension of PHS.

3.2. The Control Laws for FDFS

In order to increase generality, we consider a general form of coupled fractional chaotic system as follows:where is the fractional operator and , are nonlinear continuous vector functions, and are the Jacobian matrices of systems . is the controller part of the coupled system and is the combination between nonlinear and linear function.

As for the state vectors , one has the following.

Case 1. When they are real vectors, then , and , , where .

Case 2. When they are complex vectors, , where are the real parts and are the imaginary parts. is the imaginary unit and . , , , , , and .
We assume the difference function is , then where . According to (7) and the definition of FDFS, we have the error system as follows:where .

According to the fractional stable theory and Lemma 2, we will get the following theorem.

Theorem 8. For the coupled fractional chaotic system (7) with the difference function and the initial value , it could accomplish FDFS with the following controller:where parameter matrix satisfies .

Proof. In order to verify the effect of the control law, we put (9) into (8) and get According to Lemma 2, since in the coupled fractional chaotic system, the system could arrive FDFS asymptotically with the controller (9) and the error function could tend to zero.

4. Secure Communication of FCCS

In this section, we focus on secure communication of FDFS with complex variables. Due to the broadband characters of chaotic systems, we could effectively cover up the signal by the chaos carrier. The block diagram of information transmission based on FDFS is illustrated in Figure 1. From Figure 1, we can know that the overall structure is composed of two parts approximately. One part S is the sending end, which provides the carrier of chaotic masking. Message modulation between chaotic driver system and information signal occurs here. The other main part R is receiving end, which aims at information demodulation. Response system and received signal have demodulated with the controller in part R. is the information signal and is the potential disturbance signal in process of transmitting. is the transmission signal, where . are the other alternative transmission channels. is used for filter interference and we can get accurate recovered signals.

In order to verify the security of the proposed cryptosystem, we will simulate the transmission of four kinds of information signals in this part. In 2013, the coupled complex fractional Lorenz system was presented in [8], which is described as follows:andwhere and the overbar of means the complex conjugate of . are correlation controllers.

We choose (9) as the controller in secure communication and the controller iswhere are the scale parameters of controller and are the difference factors of FDFS which also represent information signal () in secure communication.

For the disturbance signal, we choose the stochastic Gaussian noise as the , which is described as follows:where the mean value and variance and is the scale parameter of stochastic Gaussian noise.

4.1. Communication of Analog Signals

In this section, we put the analog signal into transmission system. In order to get the clear result of simulations, we firstly choose the information signal ; then we havewhere is the -order derivative of . are the real and imaginary parts of , respectively.

As for the choice of communication channels, we select as the sending end to transmit the real and imaginary part of analog signal and as the receiving end to compute the real and imaginary part of encrypted analog signal. To further enhance the confidentiality of the transmission system and simulate the potential disturbance signal, we add stochastic Gaussian noise into transmitting process. Set in (14). In the receiving end, we could use some easy filters in real engineering and it allows assuring the accuracy of recovered information.

Figure 2 shows the encrypted transmission signal, which is complicated and irregular. It is hard for unauthorized third party to find the information signal. The recovered signal, original signal, and their error are shown in Figure 3. With the controller (13) in the receiving end, information signal has been recovered correctly in Figure 3. Figure 4 shows the error of receiving end and sending end. According to the definition of FDFS, as the choice of communication channel is complex variable and , their error is the information signal. Other errors of complex variables and and real variables and are zero asymptotically.

4.2. Communication of Digital Signal

Firstly, the original digital signal produced by the random function is shown in Figure 5. In order to achieve a fast transmission without increasing the system complexity, the binary bits are transformed into one corresponding fractional difference functions by -ary. In this part, we set and transform digital signal to the fractional difference function by -ary. The signal duration is iterations. We choose the of the drive system as the sending terminal and as the receiving terminal. Figure 5 shows the transmitted process of digital signals without noise, where the error between original digital signal and the recovered signal is zero and the transmitted signal is an analog signal which completely covers up the binary digital sequence.

Considering that there is potential disturbance in transmitting process of digital signals, we get the transmitting process shown in Figure 6. The transmitted signal is almost noise-like and someone like espionage hardly extracts the information signal without authorization. In order to reduce the effect of noise, we firstly compute the average value of the fractional difference function during each signal duration and then round off to the nearest integer that is bounded between and . Therefore, this scheme guarantees the accuracy and low bit error rate in process of recovery.

4.3. Communication of Voice Signal

In this section, a novel audio cryptosystem is presented for transmitting voice signal. In the sending end, we choose the mellifluous song “traveling light” as the information signal, which is shown in Figure 7.

There are five alternative communication channels to transmit the voice signal. We choose to encrypt and decrypt the information signal.

Because the voice signal has a great number of samples, we extract the sample from to , which is enough to get an excellent simulation. From Figure 8, the error between the original voice and recovered voice approaches to zero quickly and the transmitted signal of encryption voice completely covers the information signal. Moreover, due to the existence of five alternative communication channels, there is less possibility for the eavesdropper to extract the original voice.

4.4. Communication Image Signal

In this section, a new key cryptosystem is presented for sharing image messages to increase the security and anticrack ability of secure communication. Generally speaking, the purpose of key cryptography is to allow two different organizations to communicate the confidential message, even though they have never met and communicated with each other or they are supervised by an adversary. The proposed cryptosystem consists of three parts: key generation, encryption, and decryption.

Key generation

Sending end and receiving end both agree on this:

(1) The fractional order .

(2) Complex variable and the real variable of couple system.

(3) Initial value of fractional order complex chaotic system.

(4) The parameters of couple system.

(5) Scaling parameters .

Encryption

(1) The pixel matrix of original picture is .

(2) There are also five zeros matrices, . As the coupled chaotic system has five alternative encryption sending terminals, , we could generate five arrays with the initial value of coupled system, where the number of every array is more than and are the matrix size parameters.

(3) Let and ,

and ,

; if has remainder, we could adjust the number of correspondingly.

(4) .

(5) The sending end sent to the receiving end.

Decryption

(1) On condition that we received the and agree on the keys, we must form corresponding five decryption matrices , , , , and in the decryption receiving terminal, with the controller (9). The method of generation of these five decryption matrices is similar to encryption.

(2)

The recovered image pixel matrix .

(3) We can get the recovered image by the pixel matrix

Figures 9 and 10 are the diagrams of original information image and recovered image, respectively. The picture of encrypted matrix is shown in Figure 11, where the encryption image completely covers the information picture. Due to the complexity of the secret key and the process of encryption, the unauthorized organization cannot recover the original image absolutely without the whole information of the secret key.

5. Conclusions

In the last two decades, chaos communication has been a hotspot and got astonishing progress. In this paper, we propose a novel secure communication scheme of fractional complex chaotic systems based on FDFS. We firstly extend the DFS from integer complex chaotic systems to fractional complex chaotic systems and design corresponding controller. FDFS is one of FCCS in essence. In order to verify the effectiveness and advantages of FCCS, we present novel secure communication schemes based on FDFS and transmit analog signal, digital signal, voice signal, and image. Moreover, we design an image cryptosystem with high security. The numerical simulations demonstrate the great effect of encryption, transmission, and decryption. Particularly, the results exhibit the advantages of FDFS integrating with fractional complex chaotic system.

Secure communication of FCCS is a completely new field. We hope that more and more researchers will extend some traditional synchronization to FCCS and increase the diversity of FCCS in secure communication, which will deeply develop chaos communication.

Data Availability

The Matlab programs used to support the findings of this study are currently under embargo while the research findings are not published. Requests for data, 6 months after publication of this article, will be considered by the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is partially supported by Young doctorate Cooperation Fund Project of Qilu University of Technology (Shandong Academy of Sciences) (no. 2018BSHZ001), International Collaborative Research Project of Qilu University of Technology (no. QLUTGJHZ2018020), National Nature Science Foundation of China (nos. 61603203 and 61773010), and Nature Science Foundation of Shandong Province (no. ZR2017MF064).

References

F. Kerschbaum, D. Dahlmeier, A. Schröpfer, and D. Biswas, “On the practical importance of communication complexity for secure multi-party computation protocols,” in Proceedings of the the 2009 ACM symposium, pp. 2008–2015, Honolulu, Hawaii, 2009.
View at: Publisher Site | Google Scholar
T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 38, no. 4, pp. 453–456, 1991.
View at: Publisher Site | Google Scholar
A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The complex Lorenz equations,” Physica D: Nonlinear Phenomena, vol. 4, no. 2, pp. 139–163, 1981/82.
View at: Publisher Site | Google Scholar | MathSciNet
C. Jiang, F. Zhang, H. Qin, and T. Li, “Anti-synchronization of fractional-order chaotic complex systems with unknown parameters via adaptive control,” The Journal of Nonlinear Science and Applications, vol. 10, no. 11, pp. 5608–5621, 2017.
View at: Publisher Site | Google Scholar
F. Zhang, M. Li, S. Leng, and J. Liu, “Linear correlation of complex vector space and its application on complex parameter identification,” Journal of Qilu University of Technology, vol. 1, pp. 70–73, 2019.
View at: Google Scholar
F. Zhang, K. Sun, Y. Chen, H. Zhang, and C. Jiang, “Parameters identification and adaptive tracking control of uncertain complex-variable chaotic systems with complex parameters,” Nonlinear Dynamics, pp. 1–16, 2019.
View at: Google Scholar
Z. Wang, J. Liu, F. Zhang, and S. Leng, “Hidden Chaotic Attractors and Synchronization for a New Fractional-Order Chaotic System,” Journal of Computational and Nonlinear Dynamics, vol. 14, no. 8, p. 081010, 2019.
View at: Publisher Site | Google Scholar
C. Luo and X. Wang, “Chaos in the fractional-order complex Lorenz system and its synchronization,” Nonlinear Dynamics, vol. 71, no. 1-2, pp. 241–257, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Chen, H. Zhang, and F. Zhang, “Difference function projective synchronization for secure communication based on complex chaotic systems,” in Proceedings of the 5th IEEE International Conference on Cyber Security and Cloud Computing and 4th IEEE International Conference on Edge Computing and Scalable Cloud, CSCloud/EdgeCom 2018, pp. 52–57, China, June 2018.
View at: Google Scholar
H. S. Nik, S. Effati, and J. Saberi-Nadjafi, “Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization,” Complexity, vol. 20, pp. 30–44, 2014.
View at: Publisher Site | Google Scholar
S. Zheng, “Further results on the impulsive synchronization of uncertain complex-variable chaotic delayed systems,” Complexity, vol. 21, no. 5, pp. 131–142, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
B. Sun, M. Li, F. Zhang, H. Wang, and J. Liu, “The characteristics and self-time-delay synchronization of two-time-delay complex Lorenz system,” Journal of The Franklin Institute, vol. 356, no. 1, pp. 334–350, 2019.
View at: Publisher Site | Google Scholar
C.-Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 41, no. 7, pp. 3826–3837, 1990.
View at: Publisher Site | Google Scholar
V. Y. Toronov and V. L. Derbov, “Boundedness of attractors in the complex Lorenz model,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 55, no. 3, pp. 3689–3692, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
A. D. Mengue and B. Z. Essimbi, “Secure communication using chaotic synchronization in mutually coupled semiconductor lasers,” Nonlinear Dynamics, vol. 70, no. 2, pp. 1241–1253, 2012.
View at: Publisher Site | Google Scholar
L. F. Abdulameer, J. D. Jignesh, U. Sripati, and M. Kulkarni, “BER performance enhancement for secure wireless optical communication systems based on chaotic MIMO techniques,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 7–16, 2014.
View at: Publisher Site | Google Scholar
Y. Fu, M. Cheng, X. Jiang et al., “Wavelength division multiplexing secure communication scheme based on an optically coupled phase chaos system and PM-to-IM conversion mechanism,” Nonlinear Dynamics, vol. 94, no. 3, pp. 1949–1959, 2018.
View at: Publisher Site | Google Scholar
S. Liu and F. Zhang, “Complex function projective synchronization of complex chaotic system and its applications in secure communication,” Nonlinear Dynamics, vol. 76, no. 2, pp. 1087–1097, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
E. E. Mahmoud and S. M. Abo-Dahab, “Dynamical properties and complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model,” Chaos, Solitons & Fractals, vol. 106, pp. 273–284, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
R. L. Bagley and R. A. Calico, “Fractional order state equations for the control of viscoelastically damped structures,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 2, pp. 304–311, 1991.
View at: Publisher Site | Google Scholar
G. T. Oumbé Tékam, C. A. Kitio Kwuimy, and P. Woafo, “Analysis of tristable energy harvesting system having fractional order viscoelastic material,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 25, no. 1, 013112, 10 pages, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
N. Laskin, “Fractional market dynamics,” Physica A: Statistical Mechanics and its Applications, vol. 287, no. 3-4, pp. 482–492, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
S. Hassan Hosseinnia, R. L. Magin, and B. M. Vinagre, “Chaos in fractional and integer order NSG systems,” Signal Processing, vol. 107, pp. 302–311, 2015.
View at: Publisher Site | Google Scholar
M. Ö. Efe, “Fractional order systems in industrial automation—a survey,” IEEE Transactions on Industrial Informatics, vol. 7, pp. 582–591, 2011.
View at: Google Scholar
Y. Ding, Z. Wang, and H. Ye, “Optimal control of a fractional-order HIV-immune system with memory,” IEEE Transactions on Control Systems Technology, vol. 20, no. 3, pp. 763–769, 2012.
View at: Publisher Site | Google Scholar
I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
View at: Google Scholar | MathSciNet
X. Wu, H. Wang, and H. Lu, “Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1441–1450, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Kassim, H. Hamiche, S. Djennoune, and M. Bettayeb, “A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems,” Nonlinear Dynamics, vol. 88, no. 4, pp. 2473–2489, 2017.
View at: Publisher Site | Google Scholar
P. Muthukumar, P. Balasubramaniam, and K. Ratnavelu, “Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB),” Nonlinear Dynamics, vol. 80, no. 4, pp. 1883–1897, 2015.
View at: Publisher Site | Google Scholar
R. Li and H. Wu, “Secure communication on fractional-order chaotic systems via adaptive sliding mode control with teaching–learning–feedback-based optimization,” Nonlinear Dynamics, vol. 95, no. 2, pp. 1221–1243, 2019.
View at: Publisher Site | Google Scholar
A. Mohammadzadeh and S. Ghaemi, “Synchronization of uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network and its application to secure communication,” Nonlinear Dynamics, vol. 88, no. 1, pp. 1–19, 2017.
View at: Publisher Site | Google Scholar
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent-II,” The Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529–539, 1967.
View at: Publisher Site | Google Scholar
D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the IMACS IEEE-SMC, pp. 963–968, Lille, France, 1996.
View at: Google Scholar

Copyright

Copyright © 2019 Jiaxun 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1035

Downloads

1133

Citations