Fractional-order Chaotic Systems Research Articles

A novel fractional-order 3-D chaotic system and its application to secure communication based on chaos synchronization

Abstract In this study, we introduce a new fractional-order chaotic system (FO-CS) that comprises six terms, setting it apart from classical chaotic models such as the Lorenz, Chen, and Lü systems. The proposed system, while having a different number of terms compared to the Lorenz and Chen systems, generates attractors that closely resemble those found in these conventional systems. The algebraic structure of the system is relatively simple, consisting of four linear terms and two quadratic terms. We conduct a comprehensive theoretical analysis and dynamic simulations of the system from both fractional and integer-order perspectives, exploring numerous dynamical characteristics, including Lyapunov exponent spectra, fractal dimensions, Poincaré maps, and bifurcation phenomena. Furthermore, we derive the Hamiltonian energy function for the proposed system through the application of Helmholtz’s theorem. To delve into synchronization within the system, we carry out numerical simulations alongside an active control method. The effective implementation of synchronization through this control strategy deepens our understanding of system dynamics and highlights its potential applications, particularly in secure communication. One significant application is the use of synchronization techniques for the secure transmission of real audio signals, showcasing the relevance of synchronization technique in enhancing communication security.

Chaos-based approaches to digital data security: Analysis of incommensurate fractional-order Arneodo chaotic system and engineering application on Nvidia Jetson AGX Orin

Chaos-based approaches to digital data security: Analysis of incommensurate fractional-order Arneodo chaotic system and engineering application on Nvidia Jetson AGX Orin

A novel multi remote sensing image encryption scheme exploiting modified zigzag transformation and S-Box

Abstract This paper proposed a novel multi-image remote sensing image encryption algorithm. The proposed algorithm leverages a novel fractional-order chaotic system, an enhanced Zigzag scanning technique, and a refined S-box for robust encryption. Initially, the plain remote sensing images are transformed into a one-dimensional sequence using an extended Zigzag transformation. Subsequently, chaotic sequence indices, generated by the advanced fractional-order chaotic system, are utilized for pixel position scrambling. During the diffusion phase, two differently ordered diffusions were performed to enhance the algorithm’s resistance to chosen-plaintext attacks after the S-box based encryption. To further augment the security of the proposed scheme, an XOR operation is executed on each color channel of the encrypted images. Additionally, to expand the key space and strengthen resistance to chosen-plaintext attacks, the initial values and parameters involved in the algorithm are intricately tied to the SHA3–512 hash value of the plaintext image. The experimental results show that the proposed algorithm not only meet the demand of efficiency, but also could resist commonly used security attacks.

A fractional-order chaotic Lorenz-based chemical system: Dynamic investigation, complexity analysis, chaos synchronization, and its application to secure communication

Abstract Synchronization of fractional-order chaotic systems is receiving significant attention in the literature due to its applications in a variety of fields, including cryptography, optics, and secure communications. In this paper, a three-dimensional fractional-order chaotic Lorenz model of chemical reactions is discussed. Some basic dynamical properties, such as stability of equilibria, Lyapunov exponents, bifurcation diagrams, Poincaré map, and sensitivity to initial conditions, are studied. By adopting the Adomian decomposition algorithm (ADM), the numerical solution of the fractional-order system is obtained. It is found that the lowest derivative order in which the proposed system exhibits chaos is q = 0.694 by applying ADM. The result has been validated by the existence of one positive Lyapunov exponent and by employing some phase diagrams. In addition, the richer dynamics of the system are confirmed by using powerful tools in nonlinear dynamic analysis, such as the 0–1 test and C 0 complexity. Moreover, modified projective synchronization has been implemented based on the stability theory of fractional-order systems. This paper presents the application of the modified projective synchronization in secure communication, where the information signal can be transmitted and recovered successfully through the channel. MATLAB simulations are provided to show the validity of the constructed secure communication scheme.

Adaptive arbitrary time synchronisation control for fractional order chaotic systems with external disturbances

In this paper, adaptive synchronisation of fractional order chaotic systems with external disturbances is studied. For this purpose, first, a new integer order system is introduced and its convergence to zero exponentially within an arbitrary time is proven. Using this system, a novel adaptive exponential arbitrary time sliding mode controller is established to synchronise the fractional order chaotic systems. Then, to estimate the disturbance terms of the fractional systems an exponential arbitrary time disturbance observer is developed. The use of the disturbance observer can lead to simple controller design, low chattering control input, and more accurate response. Finally, a disturbance observer based sliding mode control is presented, that by incorporating the disturbance estimation in controller design ensures synchronisation of chaotic systems. The exponential arbitrary time synchronisation and the stability of the closed loop control system are confirmed using the Lyapunov theory. Simulation results on synchronisation of different fractional order systems, validate the proposed control schemes. It is noteworthy that the introduced adaptive sliding mode controllers can be used for controlling a wide variety of fractional order systems.

Hidden complex multistable dynamical analysis and FPGA implementation of integer-fractional order memristive-memcapacitive chaotic system

Abstract A chaotic circuit based on a magnetic-controlled memristor and charge-controlled memcapacitor is proposed in this paper. The study reveals that it is a hyperchaotic system with hidden characteristics in integer-order. Furthermore, as the parameters change, the attractors exhibit rich evolutionary phenomena. Even after adjusting some parameters to very large values, the system still maintains hyperchaotic behavior. Interestingly, the basin of attraction shows the multistability of the system. Under initial value control, coexisting attractors are categorized into two types: those with initial offset-boosting behavior and nested attractors. When under parameter control, coexisting attractors are divided into two types: symmetric coexisting attractors and nested coexisting attractors. By analyzing the spectral entropy (SE) complexity of the system and using a complexity distribution diagram with two parameters and two initial values, the existence of multiple complex dynamic behaviors in the system has been verified. The fractional-order memristive-memcapacitive system based on the Grunwald-Letnikov algorithm and the five fractional-order values of q i (i = 1, 2, 3, 4, 5) are taken as different in the numerical simulation, it also displays multiple coexisting phenomena similar to those of the integer-order. Finally, Matlab/Simulink and DSP Builder software platform are used to design the fractional-order five-dimensional chaotic memristive-memcapacitive system, and then combined with VHDL and Verilog HDL hardware language, the proposed circuit system is verified on the EP4CE115F29C7 FPGA main chip of Cyclone IV E series. The consistency of hardware implementation and software simulation shows the correctness and feasibility of the design.

Novel Dynamic Behaviors in Fractional Chaotic Systems: Numerical Simulations with Caputo Derivatives

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain new types of complex and previously undiscussed dynamic behaviors.These phenomena, not recognized in prior numerical results or theoretical estimations, underscore the unique dynamics present in fractional systems. We also study the effects of the fractional parameters β1, β2, and β3 on the system’s behavior, comparing them to integer-order derivatives. It has been demonstrated via the findings that the suggested technique is consistent with conventional numerical methods for integer-order systems while simultaneously providing an even higher level of precision. It is possible to demonstrate the efficacy and precision of this technique through simulations, which demonstrates that this method is useful for the investigation of complicated chaotic models.

Does a Fractional-Order Recurrent Neural Network Improve the Identification of Chaotic Dynamics?

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, r2) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

Dynamics Analysis and Adaptive Synchronization of a Class of Fractional-Order Chaotic Financial Systems

It is of practical significance to realize a stable and controllable financial system by using chaotic synchronization theory. In this paper, the dynamics and synchronization are studied for a class of fractional-order chaotic financial systems. First, the stability and dynamics of the fractional-order chaotic financial system are analyzed by using the phase trajectory diagram, time series diagram, bifurcation diagram, and Lyapunov exponential diagram. Meanwhile, we obtain the range of each parameter that puts the system in a periodic state, and we also reveal the relationship of the derivative order and the chaotic behaviors. Then, the adaptive control strategy is designed to achieve synchronization of the chaotic financial system. Finally, the theoretical results and control method are verified by numerical simulations.

Synchronization of Bidirectionally Coupled Fractional-Order Chaotic Systems with Unknown Time-Varying Parameter Disturbance in Different Dimensions

In this article, the synchronization of bidirectionally coupled fractional-order chaotic systems with unknown time-varying parameter disturbance in different dimensions is investigated. The scale matrices are designed to address the problem of the synchronization for fractional-order chaotic systems across two different dimensions. Congelation of variables is used to deal with the unknown time-varying parameter disturbance. Based on Lyapunov’s stability theorem, the synchronization controllers in different dimensions are obtained. At the same time, adaptive laws of the unknown disturbance can be designed. Benefiting from the proposed methods, we verify all the synchronization errors can converge to zero as time approaches infinity, regardless of whether in n-D or m-D synchronization, simultaneously ensuring that both control and estimation signals are bounded. Finally, simulation studies based on fractional-order financial systems are carried out to validate the effectiveness of the proposed synchronization method.

Towards efficient identification of fractional-order systems

This study focuses on the efficient identification of fractional-order chaotic systems, particularly under challenging conditions where the time step, iteration time, orders, parameters, initial values, and system structure are all unknown. Unlike previous work, our approach emphasizes the complexities introduced by the unknown time step and iteration time. We introduce a novel differential evolution (DE) variant, termed global and local search using success-history based parameter adaptation for differential evolution (GL-SHADE), to tackle this problem in both noiseless and noisy environments. Comprehensive simulations show that GL-SHADE outperforms other comparative algorithms by delivering both higher precision and faster convergence under these uncertain conditions, highlighting the potential of advanced DE techniques in complex system identification.

A new four-dimensional chaotic system with rich transitional characteristics between dissipative and conservative.

The general form of the Hamiltonian function serves as the foundation for the creation of a new four-dimensional chaotic system in this study. We discover that the external excitation parameter d, the internal parameter a, and all initial values have a transforming influence on the system property. Additionally, the corresponding fractional-order chaotic system in accordance with the constructed four-dimensional chaotic system is proposed. It is found that as the order q rises, the system transforms gradually from a dissipative system to a conservative system. Multiple coexisting attraction flows based on the Hamiltonian energy magnitude are present in this dual-property chaotic system. The complexity analysis shows that the system has a high level of complexity. NIST test indicates that the chaotic sequences produced by this dual-property chaotic system exhibit good pseudo-randomness. Finally, a Digital Signal Processing-based hardware platform confirms the physical realizability of the system.

Application of Fractional-Order Multi-Wing Chaotic System to Weak Signal Detection

This work investigates a fractional-order multi-wing chaotic system for detecting weak signals. The influence of the order of fractional calculus on chaotic systems’ dynamical behavior is examined using phase diagrams, bifurcation diagrams, and SE complexity diagrams. Then, the principles and methods for determining the frequencies and amplitudes of weak signals are examined utilizing fractional-order multi-wing chaotic systems. The findings indicate that the lowest order at which this kind of fractional-order multi-wing chaotic system appears chaotic is 2.625 at a=4, b=8, and c=1, and that this value decreases as the driving force increases. The four-wing and double-wing change dynamics phenomenon will manifest in a fractional-order chaotic system when the order exceeds the lowest order. This phenomenon can be utilized to detect weak signal amplitudes and frequencies because the system parameters control it. A detection array is built to determine the amplitude using the noise-resistant properties of both four-wing and double-wing chaotic states. Deep learning images are then used to identify the change in the array’s wing count, which can be used to determine the test signal’s amplitude. When frequencies detection is required, the MUSIC method estimates the frequencies using chaotic synchronization to transform the weak signal’s frequencies to the synchronization error’s frequencies. This solution adds to the contact between fractional-order calculus and chaos theory. It offers suggestions for practically implementing the chaotic weak signal detection theory in conjunction with deep learning.

Fractional-order Sprott K chaotic system and its application to biometric iris image encryption

Fractional-order (FO) chaotic systems exhibit random sequences of significantly greater complexity when compared to integer-order systems. This feature makes FO chaotic systems more secure against various attacks in image cryptosystems. In this study, the dynamical characteristics of the FO Sprott K chaotic system are thoroughly investigated by phase planes, bifurcation diagrams, and Lyapunov exponential spectrums to be utilized in biometric iris image encryption. It is proven with the numerical studies the Sprott K system demonstrates chaotic behaviour when the order of the system is selected as 0.9. Afterward, the introduced FO Sprott K chaotic system-based biometric iris image encryption design is carried out in the study. According to the results of the statistical and attack analyses of the encryption design, the secure transmission of biometric iris images is successful using the proposed encryption design. Thus, the FO Sprott K chaotic system can be employed effectively in chaos-based encryption applications.

Fixed-time Synergetic Control for Synchronization of Fractional-order Chaotic Systems

In this paper, a fixed-time synergetic control is proposed for the synchronization of fractional-order chaotic systems. It is similar to the sliding mode approach but without chattering and achieves exact convergence of the macro-variable. Synergetic control is a robust approach that does not require linearization and is suitable for real-time implementation since it does not require a discontinuous term in its control law. The Lyapunov framework is used to ensure the convergence of the controlled system. Computer simulations are performed to illustrate the effectiveness of the proposed method.

Multistability Analysis of a Fractional-Order Multi-Wing Chaotic System and its Circuit Realization

Multistability Analysis of a Fractional-Order Multi-Wing Chaotic System and its Circuit Realization

Weak Signal Detection Application Based on Incommensurate Fractional-Order Duffing System

The Duffing Chaos System can detect weak signals that are obscured by Gaussian noise because it is sensitive to specific signal functions and can withstand noise. In this paper, we investigate the use of intermittent chaotic phenomena in fractional-order incommensurate Duffing chaotic systems for weak signal detection. This new intermittent chaotic state has not appeared in integer-order Duffing systems before, so this phenomenon reflects the superiority of fractional-order Duffing systems. We start by giving the incommensurate fractional-order Duffing system’s weak signal detection model. Then design a time series-based judgment method that successfully separates chaotic, intermittent chaotic, and limit cycle states. Finally, the intermittent chaotic of fractional-order detection system is used to determine the amplitude and frequency of the weak signals to calculate the detection performance. The results show that the weak signal can be detected at a maximum signal-to-noise ratio of -13.26 dB for single-detection oscillator amplitude detection. When detecting the frequency, a single-detection oscillator can detect the frequency range of 1050 rad/s, proving that the fractional-order chaos detection system is better than the integer-order chaos detection system.

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.

Solution of fractional order chaotic oscillator system

Solution of fractional order chaotic oscillator system

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