Integer-order Chaotic Systems Research Articles

Robust digital image watermarking scheme with a fractional-order discrete-time chaotic scheme and DWT-SVD transform

A robust digital image watermarking system with a fractional-order discrete-time chaotic system and discrete wavelet transform-singular value decomposition is presented. The inclusion method inserts an encrypted image into the dynamics of an integer-order discrete-time chaotic system and the resulting cipher serves as a host image watermark. A watermarking function (DWT-SVD transform) watermarks a host image before the watermark is extracted and decrypted at the receiver. As a contribution, our suggested approach introduces a new watermarking system based on discrete-time chaotic systems, as well as a hybridization of integer and fractional-order systems to ensure the watermarking scheme’s robustness. Our method achieved good results in terms of robustness, with normalized cross correlation values above 0.99 when subjected to a range of attacks. Results of the simulation underline practicality and robustness achieved by this approach. They also indicate that the proposed system resists various attacks.

A new 3D fractional-order chaotic system with complex dynamics

Compared to integer-order chaotic systems, fractional-order chaotic systems have more complex dynamical features due to the introduction of order. The application of fractional-order chaotic systems to chaotic cryptosystems makes the cryptosystems with higher security properties. In this paper, we developed a new 3D fractional-order chaotic system from a 3D integer-order chaotic system, and investigate the dynamical behaviors of this fractional-order system with different parameters and orders. Moreover, self-excited attractors appeared at lower orders through circuit simulations. Furthermore, the synchronization of the new fractional-order chaotic system in the presence of systematic uncertainties and perturbations was achieved using the sliding mode control technique, which sets the stage for the implementation of communication. Finally, offset boosting control was used to investigate the utility of the new chaotic system in engineering applications.

Fractional-order projection of a chaotic system with hidden attractors and its passivity-based synchronization

This paper presents the fractional-order projection of a chaotic system, which delivers a collection of self-excited and hidden chaotic attractors as a function of a single system parameter. Based on an integer-order chaotic system and the proposed transformation, the fractional-order chaotic system obtains when the divergence of integer and fractional vector fields flows in the same direction. Phase portraits, bifurcation diagrams, and Lyapunov exponents validate the chaos generation. Apart from these results, two passivity-based fractional control laws are designed effectively for the integer and fractional-order chaotic systems. In both cases, the synchronization schemes depend on suitable storage functions given by the fractional Lyapunov theory. Several numerical experiments confirm the proposed approach and agree well with the mathematical deductions.

A High-Security Probabilistic Constellation Shaping Transmission Scheme Based on Recurrent Neural Networks

In this paper, a high-security probabilistic constellation shaping transmission scheme based on recurrent neural networks (RNNs) is proposed, in which the constellation point probabilistic distribution is generated based on recurrent neural network training. A 4D biplane fractional-order chaotic system is introduced to ensure the security performance of the system. The performance of the proposed scheme is verified in a 2 km seven-core optical transmission system. The RNN-trained probabilistic shaping scheme achieves a transmission gain of 1.23 dB compared to the standard 16QAM signal, 0.39 dB compared to the standard Maxwell-Boltzmann (M-B) distribution signal, and a higher net bit rate. The proposed encryption scheme has higher randomness and security than the conventional integer-order chaotic system, with a key space of 10,163. This scheme will have a promising future fiber optic transmission scheme because it combines the efficient transmission and security of fiber optic transmission systems.

Encryption algorithm based on fractional order chaotic system combined with adaptive predefined time synchronization

An image encryption and decryption method of fractional order chaotic systems (FOCS) with predefined time synchronization is proposed in this article. Compared with the existing integer order chaotic systems (IOCS), fractional order chaotic systems has the advantage of increasing the complexity of the ciphertext. At the same time, by using the predefined synchronization time, the key space is expanded, the complexity of the key is increased, and the security of the algorithm is improved. To further improve the security of encryption and decryption process, this article uses a combination of DNA encoding, row/column cyclic shift and XOR diffusion, position scrambling and Arnold scrambling. The simulation tests of image encryption and decryption are carried out, and the effectiveness and advantages of the proposed encryption/decryption method are verified by histogram analysis, correlation analysis, entropy analysis, key sensitivity analysis and plaintext sensitivity analysis.

Special attractors and dynamic transport of the hybrid-order complex Lorenz system

Combining the advantages of both integer-order and fractional-order complex chaotic systems, we propose a hybrid-order complex Lorenz system. We demonstrate its abundant chaotic characteristics, including symmetry and dissipation, fixed points and their stability and Lyapunov exponents, with 0–1 test. Then we show that, as the initial value, parameters and the order are varying, the system exhibits diverse dynamical behaviors, with fixed points, limit cycles and chaotic attractors. We further show that the system has coexisting attractors and parametric attractors. In addition, we find that the system generates different chaotic attractors as the system hybrid order varies, referred to as order attractors. Finally, we examine the dynamic transport of the hybrid-order complex Lorenz system and design a piecewise continuous controller to realize offset boosting control. By varying the initial value, parameters or orders, we realize the dynamic transport of the system. Our simulation results confirm the dynamic transport of the hybrid-order complex Lorenz system.

Synchronization and Anti-Synchronization of a Novel Fractional Order Chaotic System with an exponential term

Today, chaotic systems have become one of the most important tools for encrypting and secure transmission of information. Other applications of these systems in economics, geography, sociology, and the like are not hidden from anyone. Despite the presentation of various chaotic systems, it is necessary to study and present new and more accurate chaotic systems. It is obvious that fractional models are more accurate and yield better results than integer order models. In this paper, the synchronization and anti-synchronization of an innovative fractional order chaotic system is investigated based on the nonlinear control method. In the proposed chaotic system, there is an exponential term that leads to behaviour very different from the integer order chaotic systems. Two different approaches have been proposed to achieve the synchronization and anti-synchronization goals between the proposed new fractional chaotic systems. A backstopping approach has been used to synchronize, and in addition to achieving this goal, it also ensures stability in Lyapunov's concept. Anti-synchronization between the two new fractional systems is also achieved by applying the active control method, and subsequently Lyapunov stability is shown under the proposed method. The simulation results in MATLAB environment show the synchronization and anti-synchronization effectiveness for the proposed innovative fractional order chaotic system.

Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System

In comparison with integer-order chaotic systems, fractional-order chaotic systems exhibit more complex dynamics. In recent years, research into fractional chaotic systems for the utilization of image cryptosystems has become increasingly highlighted. This paper describes the development, testing, numerical analysis, and electronic realization of a fractional-order memcapacitor. Then, a new four-dimensional (4D) fractional-order memcapacitive hyperchaotic system is suggested based on this memcapacitor. Analytically and numerically, the nonlinear dynamic properties of the hyperchaotic system have been explored, where various methods, including equilibrium points, phase portraits of chaotic attractors, bifurcation diagrams, and the Lyapunov exponent, are considered to demonstrate the chaos behaviour of this new hyperchaotic system. Consequently, an encryption cryptosystem algorithm is used for colour image encryption based on the chaotic behaviour of the memcapacitive model, where every pixel value of the original image is incorporated in the secret key to strengthen the encryption algorithm pirate anti-attack robustness. For generating the keyspace of that employed cryptosystem, the initial condition values, parameters, and fractional-order derivative value(s) (q) of the memcapacitive chaotic system are utilized. The common cryptanalysis metrics are verified in detail by histogram, keyspace, key sensitivity, correlation coefficient values, entropy, time efficiency, and comparisons with other recent related fieldwork in order to demonstrate the security level of the proposed cryptosystem approach. Finally, images of various sizes were encrypted and recovered to ensure that the utilized cryptosystem approach is capable of encrypting/decrypting images of various sizes. The obtained experimental results and security metrics analyses illustrate the excellent accuracy, high security, and perfect time efficiency of the utilized cryptosystem, which is highly resistant to various forms of pirate attacks.

Dynamic analysis of a fractional-order hyperchaotic system and its application in image encryption

Compared with integer order chaotic systems, fractional order chaotic systems can reflect natural phenomena more accurately, which are more suitable for chaotic cryptosystems. In order to explore the application of fractional order chaotic system in cryptography, a novel fractional order hyperchaotic system is constructed and implemented on DSP platform. More progressively, based on Adomian decomposition method, the dynamic behavior is studied by phase diagram, bifurcation diagram, Lyapunov exponent spectrum and spectral entropy (SE) complexity. It is found that each parameter and order have a large range of intervals that can keep the system in a hyperchaotic state. Therefore, the hyperchaotic sequences generated by the constructed fractional order hyperchaotic system have sufficient randomness and are well suited for applications in secure communications. In addition, a color image encryption scheme is designed based on the fractional order hyperchaotic system and DNA dynamic coding. Firstly, the improved Arnold algorithm is used to scramble the original image, then the column cyclic shift method is applied for secondary scrambling, and finally the pixel value is diffused by DNA sequence operation. The security analysis results indicate that the designed encryption algorithm can not only encrypt images effectively, but also has high security and can resist various common attacks.

Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems

In this article we studied a 3D autonomous system derived from Sprot B, C, Van der Schrier-Mass and Munmuangsaen Srisuchinwong chaotic systems for existence of zero Hopf bifurcation with the help of averaging theory of first order. Fractional order analysis of the derived system are discussed for stability of equilibrium points, chaotification condition, sensitivity dependence, Lyapunov exponents, Kaplan-Yorke dimension, chaotic time history and phase portraits. Novelty of the paper is investigation of integer and fractional order synchronization of derived system with famous Lorenz model by active control method under the same parametric values and initial conditions. By taking example of the model under consideration we generalized the synchronization for a class of integer and fractional order systems. We concluded that if a couple of integer order chaotic dynamical system is synchronized then its fractional order version will also be synchronized for same parametric values and initial conditions and vice versa. We also compared three different numerical techniques for synchronization. By calculating CPU timing for synchronization we determined that the integer order chaotic system was synchronized earlier than that of fractional order. The results so achieved show that it is sufficient to get synchronization of an integer order system if its fractional version also exists. This investigation contributes to minimize the cost of control for a class of dynamical systems when such control is made through synchronization. Numerical simulations are also provided to authenticate the analytical results.

High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point

Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

Complex compound-combination multi switching anti-synchronization of fractional-order complex chaotic systems and integer-order complex chaotic systems

Based on three fractional-order complex chaotic systems and two integer-order complex chaotic systems, we propose a novel synchronization scenario of complex compound-combination multi-switching anti-synchronization (CCCMSAS), which is the first of this kind. The CCCMSAS states are completed between three leader and two follower systems by adopting the nonlinear control method and choosing suitable Lyapunov function on the basis of the complex space. Furthermore, different switches and leader–follower systems of different dimensions are selected as the two examples to show the validity and maneuverability of the proposed scheme.

Fractional-order double-ring erbium-doped fiber laser chaotic system and its application on image encryption

Laser chaotic systems have been extensively studied and applied in the past 20 years. However, the fractional form of the chaotic system of erbium-doped fiber laser has not been studied yet. Therefore, this paper proposes a fractional-order double-ring erbium-doped fiber laser chaotic system. The dynamic characteristics of the system are studied by phase diagram, bifurcation diagram, Lyapunov exponent spectrum, Spectral Entropy (SE) complexity, and coexisting attractors. At the same time, the digital circuit of the new system is implemented on the digital signal processing (DSP) platform. Compared with integer-order double-ring erbium-doped fiber laser chaotic systems, this fractional-order chaotic system has more complex dynamics and higher sensitivity and randomness. Finally, the fractional-order double-ring erbium-doped fiber laser chaotic system is applied to the image encryption algorithm with the improved gravitation model. In this encryption algorithm, the diffusion part uses an improved universal gravitation model, and the Zigzag algorithm scrambles the pixels. The security of the algorithm is analyzed. The analyses of the experimental results show that the image encryption scheme proposed in this paper has a good encryption effect and good security performance. The research in this paper provides theoretical guidance and experimental basis for the research of fractional-order laser chaotic secure communication.

An experimental synthesis methodology of fractional-order chaotic attractors

A novel synthesis methodology of fractional-order chaotic systems, from the level of nonlinear systems until their experimental verification using microcontrollers, is presented. Firstly, the integer-order behavioral model of the Lorenz’s, Rossler’s, Chen’s, Liu’s, Saturated Nonlinear Function Series at one- and two-direction systems is briefly reviewed. Secondly, a first-order transfer function that approximates the behavior of a fractional-order integrator, based on continued fraction expansion method, is substituting the integer-order integrator inside the revised previously chaotic systems. Thirdly, the minimum phase Al-Alaoui’s transformation method is used for synthesizing all fractional-order chaotic systems in the discrete domain and are programmed in MATLAB and in the Arduino DUE development board. For a fair comparison, the minimum phase Al-Alaoui’s algorithm not only is used for solving the integer-order chaotic systems and it is also programmed in MATLAB and Arduino DUE board, but same initial conditions are also used for both interger- and fractional-order chaotic systems. Finally, experimental results of the integer- and fractional-order chaotic oscillators are shown. The results obtained not only allow a simple synthesis methodology of fractional-order chaotic attractors and their experimental evidence on reconfigurable hardware, but also demonstrate the viability of fractional attractors to be used in various applications such as secure communications, robot control, cryptography and so on.

Image Encryption Using Fractional Order Linear-Nonlinear Chaos

A fractional order chaos system is first designed. It is a linear-nonlinear hybrid using fractional order logistic map. The fractional order chaos system is closer to the real nature phenomena and has better cryptography features than the integer order chaotic systems in dynamics, such as larger range of key space and almost no periodic windows in bifurcation diagrams. Then, we propose an image encryption scheme employing the superior chaotic features of the fractional order chaos system. Which chaotic sequences generated are used for permutation and addition/subtraction operation of encryption depends on the plaintext image. Security analysis and test results indicate that our encryption scheme has a high efficiency and superior security.

Complexity Analysis and Synchronization Control of Fractional-Order Jafari-Sprott Chaotic System

Aiming at the complexity problem of fractional-order Jafari-Sprott chaotic system, in this paper, Adomian decomposition method is used to study its numerical analysis and a complexity analysis method of fractional-order Jafari-Sprott chaotic system based on fuzzy entropy algorithm, sample entropy algorithm and dispersion entropy algorithm is proposed. For the synchronization and control of fractional-order Jafari-Sprott chaotic system, sliding mode control is used to achieve synchronization of fractional-order Jafari-Sprott chaotic system and a control method of fractional-order Jafari-Sprott chaotic system is proposed based on frequency distribution model of fractional-order integral operator. The main results are as follows: (1) The complexity of the fractional-order Jafari-Sprott chaotic system is greater than the integer-order Jafari-Sprott chaotic system, and fractional-order chaotic system has better application prospects. (2) Moreover, it is concluded that the effect of the dispersion entropy algorithm on detecting complexity is the best, which provides theoretical and experimental basis for the practical engineering application of the fractional-order Jafari-Sprott chaotic system. (3) Synchronization and control of fractional-order Jafari-Sprott chaotic system is accomplished by sliding model control and frequency distribution model of fractional-order integral operator respectively. In particular, the control effect of each variable is accomplished by designing a control law based on the frequency distribution model of fractional integral operator.

The fractional-order unified chaotic system: A general cascade synchronization method and application

In this paper, we extend the cascade synchronization of integer-order chaotic systems to function cascade synchronization of fractional-order chaotic systems. The nice feature of our method is that a fractional-order chaotic system will synchronize to another system via a scaling function. The rich choices of the scaling function turn our method more general than some existing methods. As applications, we take a fractional-order unified chaotic system as an illustrative example to test the effectiveness.

An Efficient Image Encryption Scheme Based on S-Boxes and Fractional-Order Differential Logistic Map

In this work, an efficient image encryption based on S-boxes and fractional-order logistic map is proposed. The features of the fractional-order chaotic system in dynamical behaviors are exhibited. By simulation and comparison with the traditional logistic map, it is proved that the fractional-order logistic map contains larger key space and more parameters. Therefore, the fractional-order logistic system has better efficiency and security against cryptanalyst attacks. The S-boxes construction algorithm is proposed. By comparing with the S-boxes of the former schemes, the proposed S-boxes have good performance under Bits Independence Criterion (BIC), the Strict Avalanche Criterion (SAC) and the nonlinearity. Finally, the image encryption scheme is proposed for the verification. In the encryption process, the proposed S-boxes are used for scrambling and confusion. The simulation and experimental results indicate that the fractional-order method is a preferred approach to integer-order chaotic system.

Bifurcation analysis of a fractional order time-delay Chua’s circuit based on coupled memristors

In this paper, a fractional-order (and an integer-order) chaotic system, obtained from Chua’s circuit by substituting Chua’s diode with two active coupled memristors (MRs) characterized by quadratic nonlinearity, is introduced to probe the memristive coupling effect. Two MRs connected in parallel are coupled by the flux. For the integer-order memristive system, the dynamical characteristics depending on the coupling strength coefficient between MRs without changing the circuit parameters are illustrated theoretically and numerically by using phase portraits, time domain diagram, bifurcation diagram and the Lyapunov diagram. Then based on the Adams–Bashforth–Moulton algorithm, the study of dynamic behavior of the fractional-order memristive system containing the time-delay reveals that appropriately setting the coupling strength between MRs generates more interesting attractors that differ from its integer-order counterpart. Besides, the effects of the order and the time-delay on dynamics are discussed briefly. Finally, the simulation results verify the validity of the theoretical analysis.

Dynamics analysis of a fractional-order delayed SBT memristive chaotic system without equilibrium points

The SBT memristor is a physical memristor using a Sr0.95Ba0.05TiO3 nanometer film, which is presented with a flux-controlled mathematical model. The integer-order chaotic systems based on SBT memristor have attracted much attention and have been thoroughly discussed. Analysis shows that the fractional-order system is closer to real system. In this paper, a fractional-order delayed SBT memristive chaotic system is proposed. It is worth noting that this fractional-order delayed SBT memristive system can produce chaotic attractors although it has no equilibrium points. A key system parameter is investigated by theoretical analyses and numerical simulations using the modified Adams-Bashforth-Moulton method. The results indicate that the system parameter can significantly affect the dynamic behavior, which can be indicated by the bifurcation diagrams, the Max Lyapunov exponent (MLE) diagram, the time domain waveform, the phase portraits and the power spectrum of the oscillator. All research can lay a significant theoretical foundation for the physical realization of chaotic circuits.