Fractional-order Chaotic Systems Research Articles

Enhancing the trustworthiness of chaos and synchronization of chaotic satellite model: a practice of discrete fractional-order approaches

Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research.

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features

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 novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0–1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

Modeling and optimal control of nonlinear fractional order chaotic system of factors affecting money laundering: genetic algorithms and particle swarm optimization

ABSTRACT The purpose of this article is to model and optimally control the non-linear chaotic system of the fractional order of factors affecting money laundering. The model of factors affecting money laundering was expressed as a system of fractional order differential equation. Due to the presence of chaos in this model, optimal control was provided for it. For optimal control of the chaos in the proposed model, particle swarm optimization algorithm and genetic algorithm were used. The implementation and simulation of this research was done by coding in MATLAB software. The results of the research show that the optimal control applied to the model can control the factors affecting money laundering. When the controller is applied from scratch, the results of the genetic algorithm method are excellent. All the results obtained for the particle swarm optimization method show that this method is also very successful and the results are very close to the genetic algorithm method. Due to the necessity of conducting this research, it can be mentioned that money laundering as an economic crime has a significant negative impact on the economic growth and development of countries.

A fractional-order hyperchaotic system that is period in integer-order case and its application in a novel high-quality color image encryption algorithm

A new fractional-order 5D hyperchaotic system based on memristor is constructed in this paper, with the speciality that the system exists chaotic and hyperchaotic states in the fractional-order case, while in periodic state in the integer-order. In addition, it has a variety of special phenomena at fractional-order such as infinite initial value range, parameter-dependent offset-boosting and amplitude control, attractor coexistence, and fractional order complexity greater than integer order. The correctness and feasibility of the system is verified by analog circuit simulation and hardware circuit implementation. Combining this system with image encryption algorithms, two new scrambling algorithms and a diffusion algorithm are proposed. And a high-quality encryption scheme that can be applied to a wide range of color images is proposed. The scheme is found to have excellent security after verification by various security analyses and comparison with other literatures. This paper provides a basis for the superiority of fractional-order chaotic systems and provides new methods in the field of image encryption.

Command Filtered Adaptive Backstepping Fuzzy Synchronization Control of Uncertain Fractional Order Chaotic Systems with External Disturbance

Command Filtered Adaptive Backstepping Fuzzy Synchronization Control of Uncertain Fractional Order Chaotic Systems with External Disturbance

Adaptive Terminal Sliding-Mode Synchronization Control with Chattering Elimination for a Fractional-Order Chaotic System

In this paper, an adaptive terminal sliding-mode control (ATSMC) method is proposed for the synchronization of uncertain fractional-order chaotic systems with disturbances. According to the sliding-mode control theory, a non-singular sliding surface is constructed. To overcome the chattering problem of ATSMC, a smooth term is used in the controller. In order to reduce the estimation error of an uncertain parameter, adaptive laws are designed to adjust the amplitude of the continuous function. Based on the Lyapunov stability theory, a stability analysis of the error system is performed to ensure that the tracking error eventually converges to the origin. The effectiveness and applicability of the proposed control strategy are verified using the simulation results.

Circuit realization and FPGA-based implementation of a fractional-order chaotic system for cancellable face recognition

Circuit realization and FPGA-based implementation of a fractional-order chaotic system for cancellable face recognition

Adaptive T-S fuzzy synchronization for uncertain fractional-order chaotic systems with input saturation and disturbance

This paper proposes an adaptive Takagi-Sugeno (T-S) fuzzy control scheme to synchronize two uncertain fractional-order (FO) chaotic systems with input saturations and external disturbances. Under the FO Lyapunov criterion, a novel adaptive T-S FO controller with a simple architecture is developed. System constraints are handled by modeling a saturator as a fraction of unknown functions. Due to the presence of system uncertainties, FO adaptive laws are established to adjust controller parameters. The asymptotic convergence of tracking errors and the boundedness of all closed-loop signals can be ensured. Especially, the feedback control is achieved in T-S FO chaotic systems and the posed method avoids solving linear matrix inequalities, which relieves the calculation burden. The feasibility of the presented method is confirmed through numerical simulations.

Finite-Time Synchronization of Fractional-Order Nonlinear Systems with State-Dependent Delayed Impulse Control

This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.

A new synchronization result for fractional-order discrete-time chaotic systems via bandlimited channels

Recently, digital chaotic communication with limited bandwidth has seized the attention of several researchers. This interest stems from the fact that all physical communication channels are limited in bandwidth for data transmission. In furtherance of this, researchers and investigators in this area have put forward various new communication strategies based on synchronizing and controlling coupled chaotic systems evolving in discrete time. In this article, we present a new robust communication scheme based on the synchronization of two fractional-order discrete-time chaotic systems connected by bandlimited channels. The major contribution is to give a sufficient condition in term of the existing and updated parameters of the incorporated fractional-order system on one hand and the inserted filter coefficients, on the other hand. Once applied for data communications, the currently drawn up synchronization scheme pledges to enhance the transmission robustness. Indeed, the new considered parameters act like novel secret keys which widen the cipher key space. Primary and pertinent simulation curves and results performed on Matlab software portray and illustrate the effectiveness of our approach.

Stability analysis of two fractional-order Duffing-Holmes chaotic systems using a generalized predictive controller

Stability analysis of two fractional-order Duffing-Holmes chaotic systems using a generalized predictive controller

Adaptive fuzzy echo state network optimal synchronization control of hybrid–order chaotic systems via reinforcement learning

In this paper, a novel optimal synchronization control scheme for fractional–integer hybrid–order chaotic systems is formulated. To deal with the fractional–order (FO) constraint, a transformation programme is developed and then the master system considered as an FO chaotic system is transformed into an integer–order one. A fuzzy echo state network (FESN) with the advantages of both fuzzy logic system and echo state network is introduced to approximate system uncertainty. Simultaneously, to alleviate the resource pressure, an optimal synchronization control is proposed in the light of the reinforcement learning mechanism, in which critic–actor update laws are constructed through the negative gradient of a positive function with regard to Bellman residual. Compared to typical fuzzy synchronization control, the devised FESN synchronization scheme with the same computational complexity possess superior approximation ability and synchronization performance. Ultimately, three simulation cases are exhibited to check the validity of the proposed approach.

Color image encryption algorithm based on Mackey–Glass time-delay chaotic system and quantum random walk

To ensure the confidentiality and integrity of image data and prevent unauthorized data tampering and privacy leaks. This study proposes a new color image encryption scheme based on the Mackey–Glass time-delay chaotic system and quantum random walk. This approach fully leverages the unpredictability of quantum random walks to generate random values. It combines the differences in Hamming distance between the three RGB channels of color images to create a highly complex and random key. The overall image and the three independent RGB channels are arranged in ascending order using Logistic-tent chaotic mapping and the Mackey–Glass time-delay chaotic system to obfuscate the image data. The deformed fractional-order Lorenz chaotic system is introduced, integrated with DNA encoding and decoding technology, and XOR operations are performed to achieve encryption at the spatial and pixel levels, thereby increasing the complexity of decryption. Through extensive experimental research, this solution has demonstrated excellent results in tests such as adjacent pixel correlation, information entropy, and key sensitivity. It has an excellent ability to protect the privacy of images and provides a reliable guarantee for the security of image data.

Compounding and Synchronization of Fractional Order Chaotic Systems With Prescribed Performance for Secure Communication

Compounding and Synchronization of Fractional Order Chaotic Systems With Prescribed Performance for Secure Communication

Neural network control of fractional-order chaotic systems with unknown control direction

In this paper, neural network control of fractional order chaotic systems (FOCSs) with input saturation and unknown sign of the controller gain is addressed by employing the Nussbaum function, where neural networks are utilized to model system uncertainties. To get rid of the limitation that reaching phase should be active before sliding motion in the traditional sliding mode control, a stable sliding surface is constructed. Then, by using the integer order Nussbaum gain control method, a novel controller with neural network sliding mode variable structure is designed. Finally, the practicality of the designed method is confirmed by a simulation experiment.

Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication

In this paper, we use two Fractional-Order Chaotic Systems (FOCS)—one for the sender and one for the receiver—to determine the optimal synchronisation for a secure communication technique. With the help of the Step-By-Step Sliding-Mode Observer (SBS-SMO), this synchronisation is accomplished. An innovative optimisation method, known as the artificial Harris hawks optimisation (HHO), was employed to enhance the observer’s performance. This method eliminates the random parameter selection process and instead selects the optimal values for the observer. In a short amount of time, the secret message that could have been in the receiver portion (signal, voice, etc.) was successfully recovered using the proposed scheme. The experimental validation of the numerical results was carried out with the assistance of an Arduino microcontroller and several electronic components. In addition, the findings of the experiments were compared with the theoretical calculations, revealing a satisfactory level of agreement.

Seagull Optimization Algorithm with Share Creation with an Image Encryption Scheme for Secure Vehicular Ad Hoc Networks

Α Vehicular Ad hoc Network (VANET) allows transmission, amid moving or stationary vehicles via wireless technology. Amongst several problems, safe transmission is the most important one in smart VANETs in 5G networks. Smart vehicles require integration with advanced road systems encompassing smart payment and traffic control systems. Numerous security mechanisms are used in VANETs to ensure safe communication. One such mechanism is cryptographic digital signatures based on encryption. This study introduces the new seagull optimization algorithm involving share creation with an image encryption scheme (SGOA-SCIES) for secure VANET transmissions. The goal of the SGOA-SCIES technique is to create a considerable number of shares and encrypt them to accomplish security. In the SGOA-SCIES technique, a Multiple Share Creation (MSC) scheme is employed to generate numerous share sets. For the share encryption process, the SGOA-SCIES technique engages the Fractional-Order Chaotic System (FOCS) approach to encrypt the generated shares. The optimal keys of the FOCS method can be chosen by the SGOA usage, which ameliorates the security level. The performance evaluation of the SGOA-SCIES method is examined on benchmark data. The simulations demonstrate the enhanced SGOA-SCIES methodology outcome and compare it with the ones of other existing systems and under the implementation of various measures.

Combination-combination synchronization of chaotic fractional order systems

The quest for simple and practically implementable synchronization control functions is the fundamental motive for this research work because most of the control functions for synchronization are bulky thereby leading to complexity in implementation and high cost. This research paper examines the combination-combination synchronization of chaotic fractional order systems of identical order evolving from different initial conditions via the backstepping nonlinear control technique with the aim of reducing the control function’s complexity and cost. So, based on the stability theory of fractional order systems, stable synchronization are designed via a backstepping technique using chaotic fractional order Duffing and Aneodo as the paradigm. Numerical simulations are presented to confirm the feasibility of the analytical technique. The number of control functions has been sufficiently reduced compared to previous work hence reduction in control functions complexity and reduced cost of implementation. The outcome of this work may be useful to give a better understanding of the underlying dynamical behaviour among several interacting particles particularly in particle physics.