Fractional-order Cellular Neural Networks Research Articles

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.

Complex Dynamical Characteristics of the Fractional-Order Cellular Neural Network and Its DSP Implementation

A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption.

A fractional-order CNN hyperchaotic system for image encryption algorithm

This paper proposes a fractional-order cellular neural network (CNN) chaotic system for image encryption algorithm to explore the application of fractional-order CNN hyperchaotic system in chaotic secure communication. Firstly, a fractional-order CNN hyperchaotic system is defined based on CNN hyperchaotic system. The numerical solutions of the fractional-order CNN hyperchaotic system are calculated by Adomian decomposition algorithm. The dynamic characteristics of the of the fractional-order CNN hyperchaotic system are analyzed. Then to verify the image encryption application of the fractional-order CNN hyperchaotic system, we designed an image encryption scheme through fractional-order CNN hyperchaotic sequence, the principle of symmetry of main diagonal of matrix and XOR operation. Finally, the results illustrate that the fractional-order CNN hyperchaotic sequence has good randomness, which show that the fractional-order CNN hyperchaotic system more suitable for chaotic secure communication applications. The security performances of the algorithm show that the designed algorithm can effectively encrypt and decrypt image, and has better security performance.

Global stability analysis of S-asymptotically ω-periodic oscillation in fractional-order cellular neural networks with time variable delays

A delayed cellular neural networks with Caputo fractional-order derivative has been discussed in this paper. Firstly, the existence and uniqueness of S-asymptotically ω-periodic oscillation of the model are investigated by some important features of Mittag-Leffler functions and contraction mapping principle. Secondly, global asymptotical stability of the model is also studied by using Laplace transform, comparison principle and stability theorem of linear delayed Caputo fractional-order differential equations. Some better results are derived to improve and extend a few existing research findings. The research thoughts in this literature could be applied to research other fractional-order models in neural networks and physical areas.

Impulsive control strategy for the Mittag-Leffler synchronization of fractional-order neural networks with mixed bounded and unbounded delays

<abstract> <p>In this paper we apply an impulsive control method to keep the Mittag-Leffler stability properties for a class of Caputo fractional-order cellular neural networks with mixed bounded and unbounded delays. The impulsive controls are realized at fixed moments of time. Our results generalize some known criteria to the fractional-order case and provide a design method of impulsive control law for the impulse free fractional-order neural network model. Examples are presented to demonstrate the effectiveness of our results.</p> </abstract>

Uniform Euler approximation of solutions of fractional-order delayed cellular neural network on bounded intervals

In this paper, we study convergence characteristics of a class of continuous time fractional-order cellular neural network containing delay. Using the method of Lyapunov and Mittag-Leffer functions, we derive sufficient condition for global Mittag-Leffer stability, which further implies global asymptotic stability of the system equilibrium. Based on the theory of fractional calculus and the generalized Gronwall inequality, we approximate the solution of the corresponding neural network model using discretization method by piecewise constant argument and obtain some easily verifiable conditions, which ensures that the discrete-time analogues preserve the convergence dynamics of the continuous-time networks. In the end, we give appropriate examples to validate the proposed results, illustrating advantages of the discrete-time analogue over continuous-time neural network for numerical simulation.

Approximation of Solutions of Fractional-Order Delayed Cellular Neural Network on $$\varvec{[0,\infty )}$$ [ 0 , ∞ )

In this paper, we study a class of fractional-order cellular neural network containing delay. We prove the existence and uniqueness of the equilibrium solution followed by boundedness. Based on the theory of fractional calculus, we approximate the solution of the corresponding neural network model over the interval \([0,\infty )\) using discretization method with piecewise constant arguments and variation of constants formula for fractional differential equations. Furthermore, we conclude that the solution of the fractional-delayed system can be approximated for large t by the solution of the equation with piecewise constant arguments, if the corresponding linear system is exponentially stable. At the end, we give two numerical examples to validate our theoretical findings.

Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays

Some comparison principles for fractional-order linear systems with multiple time delays are established, after Mittag–Leffler functions are showed to be positive. Then by the stability theory of fractional linear delayed systems, the comparison system with multiple time delays is showed to be asymptotically stable under some conditions. Based on the comparison results, the asymptotical stability of the original systems follows from that of the comparison system. Then the obtained results are applied to investigate the asymptotical stability of nonlinear fractional-order cellular neural networks with multiple time delays. In terms of the inequality satisfied by the fractional derivative of Lyapunov function, some criteria ensuring asymptotical stability of fractional neural models are derived. Numerical simulations are presented to demonstrate the validity and feasibility of the proposed stability criteria.

Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays

In this paper we consider a class of impulsive Caputo fractional-order cellular neural networks with time-varying delays. Applying the fractional Lyapunov method and Mittag-Leffler functions, we give sufficient conditions for global Mittag-Leffler stability which implies global asymptotic stability of the network equilibrium. Our results provide a design method of impulsive control law which globally asymptotically stabilizes the impulse free fractional-order neural network time-delay model. The synchronization of fractional chaotic networks via non-impulsive linear controller is also considered. Illustrative examples are given to demonstrate the effectiveness of the obtained results.

Chaos and hyperchaos in fractional-order cellular neural networks

In this paper, a fractional-order four-cell cellular neural network is proposed and the complex dynamical behaviors of such a network are investigated by means of numerical simulations. Several varieties of interesting dynamical behaviors, such as periodic, chaotic and hyperchaotic motions, are displayed. In addition, it can be found that the network does exhibit hyperchaotic phenomena over a wide range of values of some specified parameter. The existence of chaotic and hyperchaotic attractors is verified with the related Lyapunov exponent spectrum, bifurcation diagram and phase portraits. Meanwhile, the Lyapunov exponents and Poincaré sections are calculated for some typical parameters, respectively.