Fractional-order Systems Research Articles

Exploring nonlinearity in quarter car models with an experimental approach to formulating fractional order form and its dynamic analysis.

This study explores the inherent nonlinearity of quarter car models by employing an experimental and numerical approach. The dynamics of vehicular suspension systems are pivotal for ensuring passenger comfort, vehicle stability, and overall ride quality. In this paper we assessed the impact of various parameters and components on suspension performance, enabled the optimization of ride comfort, stability, and handling characteristics. Firstly, experimental analysis allowed for the investigation of factors that are challenging to model theoretically, such as stiffness nonlinearity and damping characteristics, which may vary under different operating conditions. Time domain and frequency response diagram of the model has been obtained. Secondly, a quarter-car with single degree-of-freedom presented and investigated in fractional order form. Fractional order dynamics emphasize nonlinearities in quarter car models, capturing real-world dynamics effectively. The proposed fractional-order nonlinear quarter car model employed Caputo derivative. For numerical analysis of fractional order system, the Adam-Bashforth-Moulton method is used and the disturbance of road assumed to be stochastic. Results show that the dynamic response of the vehicle can be chaotic. Influence of road roughness amplitude and frequency on vehicle vibration is investigated.

Dynamic Analysis and Sliding Mode Synchronization Control of Chaotic Systems with Conditional Symmetric Fractional-Order Memristors

In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian Decomposition Method (ADM). Concurrently, the investigation probes into the existence of coexisting symmetric attractors, multiple coexisting bifurcation diagrams, and Lyapunov exponent spectra (LEs) utilizing system parameters as variables. Additionally, the system demonstrates an intriguing phenomenon known as offset boosting, where the embedding of an offset can adjust the position and size of the system’s attractors. To ensure the practical applicability of these findings, a fractional-order sliding mode synchronization control scheme, inspired by integer-order sliding mode theory, is designed. The rationality and feasibility of this scheme are validated through a theoretical analysis and numerical simulation.

Power-exponential and fixed-time consensus of conformable fractional-order quantum cellular neural networks via event-triggered control

Quantum neural networks (QNNs) are considered to be superior to classical ANNs in machine learning, memory capacity, information processing, and quantum system simulation. However, In a practical and complex system, the dynamic behavior of an open quantum system could not be accurately described by an integer-ordered Schrödinger equation. In this paper, the conformable time-fractional-order Schrödinger equation is proposed, and accordingly, the model of conformable fractional-order quantum cellular neural networks (CFOQCNNs) is established and derived from the as-proposed equation. The properties of the conformable fractional-order derivative are studied and several new inequalities regarding the power-exponential and fixed-time convergence of conformable fractional-order systems are obtained. To save the communication resource, we introduce the event-triggered mechanism to construct the controllers and then the power-exponential and fixed-time synchronizations of the master-slave systems derived from the above CFOQCNNs are studied. We also prove the absence of Zeno behaviors regarding the event-triggered strategies. According to the numerical simulation, the dynamic behavior of the CFOQCNNs is discussed and the dissipativity of the CFOQCNNs is briefly revealed. Then the synchronization behaviors of the master and slave CFOQCNNs under power-exponential and fixed-time event-triggered control are demonstrated, where the effectiveness of the event-triggered control strategy is verified. Control behaviors with different fractional orders are also presented. We also discuss the hybrid of power-exponential control and fixed-time control and illustrate the advantages of the hybrid strategy. In the last, we conclude our studies, analyze the drawbacks of this work, and briefly introduce our future research.

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems.

Chimera states in fractional-order coupled Rayleigh oscillators

Chimera states, characterized by the coexistence of synchronized and desynchronized oscillators, have been extensively described via integer differential equations over the past few decades. However, it is still unclear whether and under which conditions chimera states will appear in terms of the coupled systems involving fractional derivatives. In this work, we investigate the chimera states of Rayleigh oscillators in the fractional-order systems with attractive and repulsive couplings. Firstly, we introduce the repulsive control factor and show the various transition routes from the desynchronized state to the incoherent oscillation death by altering this factor and the fractional-order derivatives. Then, it is demonstrated that as the coupling strength varies, a vast repertoire of interesting collective behaviors will emerge, such as synchronization, solitary state, traveling wave, traveling chimera, oscillatory cluster, imperfect traveling chimera state, imperfect amplitude chimera state, and chimera death. Finally, we discuss the effect of attractive and repulsive couplings and find that decreasing the repulsive control factor can reduce the repulsive interaction, allowing the attractive coupling to dominate and promote the emergence of the coherent state.

Observer-based practical prescribed time control for fractional-order nonlinear systems with asymmetric state constraints

Observer-based practical prescribed time control for fractional-order nonlinear systems with asymmetric state constraints

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.

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

A powerful tool for dealing with high-dimensional fractional-order systems with applications to fractional Emden–Fowler systems

In this study, an approximation solution for a high-dimensional system in terms of the Caputo fractional derivative operator is obtained using the improved modified fractional Euler method, or IMFEM for short. To accomplish this aim, a result that can transform such a system into a double-equation, one-dimensional fractional-order system, is provided theoretically. Some physical applications, including fractional-order systems of equations of Emden–Fowler type, are discussed, and their graphs are plotted using MATLAB to demonstrate the IMFEM schema’s validity.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Image encryption based on a fractional-order hyperchaotic system and fast row-column-level joint permutation and diffusion

Image encryption based on a fractional-order hyperchaotic system and fast row-column-level joint permutation and diffusion

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

Dynamics of the Fractional-Order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation

Dynamics and digital circuit implementation of the fractional-order Lorenz system are investigated by employing Adomian decomposition method (ADM). Dynamics of the fractional-order Lorenz system with derivative order and parameter varying is analyzed by means of Lyapunov exponents (LEs), bifurcation diagram, chaos diagram and phase diagram. Results show that the fractional-order Lorenz system has rich dynamical behavior and it is a potential model for application. It is also found that the minimum order is affected by numerical algorithm and time step size. Finally, the fractional-order system is implemented on DSP digital circuit. Phase diagrams generated by the DSP are consistent with that generated by simulation.

Stability analysis of fractional difference equations with delay.

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ=1 and τ=2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays.

Explicit stability condition for delta fractional order systems with [formula omitted]

This paper delves into the stability of time-advance delta fractional order systems, with a specific emphasis on the order range (0,+∞) rather than the conventional range (0,1). The delta Laplace transform is used to investigate the stability of the suggested system, and a mapping relation ρ=ss+1 is introduced. The explicit stability condition is provided, articulated in relation to a specific distribution of eigenvalues of the system matrix. To validate this condition, the paper establishes equivalence between the delta difference and the nabla difference. Furthermore, both quantitative and qualitative analyses are conducted on the range of the unstable region. Finally, the correctness of the developed results is validated by three examples.

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

The analytical analysis of fractional differential system via different operators and normalization functions

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.

Mathematical analysis of a within-host dengue virus dynamics model with adaptive immunity using Caputo fractional-order derivatives

AbstractDengue fever poses a significant global health threat, with over 50 million annual infections spanning more than 100 countries. Given the absence of a specific treatment, medical intervention primarily targets symptom alleviation. The present study utilizes a Caputo-type fractional-order derivative operator to investigate and analyze the dynamics of dengue virus spread within a host with adaptive immune responses. The developed model describes and analyzes the dynamics of immune cells, free dengue particles, infected monocytes, and susceptible monocytes in the presence of cytotoxic T-Lymphocytes. A range of analytical methods is employed to probe the fractional-order within-host model. The application of the generalized mean value theorem aids in investigating the model’s solutions, employing positivity and boundedness theory. Furthermore, the Banach fixed-point approach is utilized to establish the existence and uniqueness of solutions. Employing the normalized forward sensitivity approach, the fractional-order system’s response to various model parameters is scrutinized. The study reveals that the dynamics of the viral model are significantly influenced by the transmission rate and parameters representing adaptive immune responses. Numerical simulations underscore the critical role of transmission rates and adaptive immune responses in the model. Additionally, the study examines the impact of memory on the density of susceptible monocytes, infected monocytes, free dengue particles, and immune cells to optimize immune responses. Through simulations, the study illustrates the influence of memory on immune dynamics.

Event-Triggered Adaptive Neural Network Control for State-Constrained Pure-Feedback Fractional-Order Nonlinear Systems with Input Delay and Saturation

In this research, the adaptive event-triggered neural network controller design problem is investigated for a class of state-constrained pure-feedback fractional-order nonlinear systems (FONSs) with external disturbances, unknown actuator saturation, and input delay. An auxiliary compensation function based on the integral function of the input signal is presented to handle input delay. The barrier Lyapunov function (BLF) is utilized to deal with state constraints, and the event-triggered strategy is applied to overcome the communication burden from the limited communication resources. By the utilization of a backstepping scheme and radial basis function neural network, an adaptive event-triggered neural state-feedback stabilization controller is constructed, in which the fractional-order dynamic surface filters are employed to reduce the computational burden from the recursive procedure. It is proven that with the fractional-order Lyapunov analysis, all the solutions of the closed-loop system are bounded, and the tracking error can converge to a small interval around the zero, while the state constraint is satisfied and the Zeno behavior can be strictly ruled out. Two examples are finally given to show the effectiveness of the proposed control strategy.