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This paper studies the Mittag-Leffler stability and synchronization of fractional neural networks with multi-delays (FNNMs). First, by using the weak additivity E α (δ t 2 α) E α (δ t 1 α) ≤ E α (δ (t 1 + t 2) α) of the Mittag-Leffler function, the classical Halanay inequality is extended to a more comprehensive version c D 0 α r (t) ≤ κ (t) r (t) + χ (t) r (t - τ (t)) + ∫ t - ρ (t) t G (t , s) r (s) d s + c (t) , which is critical to the stability assessment research for fractional neural networks (FNNs). Based on the new inequality, a nonlinear matrix inequality (NMI) criterion for the Mittag-Leffler stability of FNNMs is derived. The nonlinear matrix criterion can be solved step by step with the linear matrix inequality (LMI) toolbox, thus avoiding the limitations of searching LMI criteria. In addition, a Mittag-Leffler synchronization criterion in terms of NMI is presented for master-salve FNNMs under an adaptive feedback controller. Finally, there are three numerical examples as a demonstration of the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this current study, novel fractional neural networks owning delay are formulated. Using Lipschitz condition, we demonstrate that the solution of the formulated fractional delayed neural networks exists and is unique. Applying a reasonable function, we handle the boundedness issue of solution to the formulated fractional delayed neural networks. Exploiting the stability criterion and bifurcation viewpoint of the fractional order delayed dynamical system, we explore the stability and bifurcation phenomenon of the established fractional delayed neural networks. Taking advantage of an adequate hybrid controller, we have efficaciously dominated the stability domain and the time of generation of bifurcation of the formulated fractional delayed neural networks. Ultimately, computer simulation graphs are provided to sustain our acquired outcomes. The acquired theoretical outcomes of this study possess considerable realistic meaning in regulating and controlling neural networks. [ABSTRACT FROM AUTHOR]

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly R L F σ , activation function in modeling the chaotic dynamics of Chua's circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including R L F σ , sigmoid, swish, and proportional Caputo derivative P C σ , and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed R L F σ consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of R L F σ . The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of R L F σ in real-world applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotical stability and synchronization of fractional neural networks. Multiple time-varying delays and distributed delays are taken into consideration simultaneously. First, by applying the Banach's fixed point theorem, the existence and uniqueness of fractional delayed neural networks are proposed. Then, to guarantee the asymptotical stability of the demonstrated system, two sufficient conditions are derived by integral-order Lyapunov direct method. Furthermore, two synchronization criteria are presented based on the adaptive controller. The above results significantly generalize the existed conclusions in the previous works. At last, numerical simulations are taken to check the validity and feasibility of the achieved methods. [ABSTRACT FROM AUTHOR]

In this paper, the robust asymptotic stability and projective synchronization of fractional-order time-varying delayed neural networks with uncertain parameters are studied. On account of the homeomorphism mapping theorem, free-weighting method and generalized Halanay inequality, several sufficient conditions of existence, uniqueness and asymptotic stability of the equilibrium point of the addressed models in the form of LMIs are established. In addition, some criteria ensuring the robust asymptotic projective synchronization between the master system and the slave system are deduced based on a suitable controller. Finally, two numerical simulations are designed to illustrate the effectiveness and rationality of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the synchronization of multi-order fractional neural networks (MoFNNs) with time-varying delays is investigated. Two kinds of controls, namely continuous control and quantized control, are introduced respectively to implement the synchronization. Moreover, by virtue of vector Lyapunov functions, sufficient criteria for realizing the synchronization of the MoFNNs with time-varying delays are deduced. The results of this paper cover the synchronization of traditional fractional neural networks with identical derivative order as a special case. Finally, a numerical example with two cases is given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

A recurrent neural network with fractional order dynamics is used for assessing the health of LFP rechargeable automotive batteries through incremental capacity analysis. The proposed algorithm learns a dynamical model of the battery voltage from samples of current and voltage taken on a vehicle. The output of this dynamical model is the sum of two terms: the over-potential and the open circuit voltage. The over-potential model includes an element with fractional order dynamics. The open circuit voltage model is designed to depend on the same parameters as the incremental capacity curve, so the expression of the learnt model can be directly related to the health of the battery. The performance of the new method is assessed in three different batteries with varying states of health. A usable estimation of the incremental capacity curve was attained for low to moderate discharge currents. The state of health estimation produced by the fractional order network was consistently better than statistical and fuzzy models, LSTM and Echo State Networks for all the batteries under study. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global Mittag-Leffler stability (GMLS) and global finite-time stability (GFTS) for fractional Hopfield neural networks (FHNNs) with Hölder neuron activation functions subject to nonlinear growth. Firstly, four functions possessing convexity are proposed, which can guarantee that four formulas with respect to the fractional derivative are established. Correspondingly, a novel principle of convergence in finite-time for FHNNs is developed based on the proposed formulas. In addition, by applying the Brouwer topological degree theory and inequality analysis techniques, the proof of the existence and uniqueness of equilibrium point is addressed. Subsequently, by means of the Lur'e-type Postnikov Lyapunov functional approach, and the presented principle of convergence in finite-time, the GMLS and GFTS conditions are achieved in terms of linear matrix inequalities (LMIs). Moreover, the upper bound of the setting time for the GFTS is calculated accurately. Finally, three numerical examples are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the global Mittag-Leffler stability of fractional Hopfield neural networks (FHNNs) with -inverse hölder neuron activation functions are considered. By applying the Brouwer topological degree theory and inequality analysis techniques, the proof of the existence and uniqueness of equilibrium point is addressed. By constructing the Lure's Postnikov-type Lyapunov functions, the global Mittag-Leffler stability conditions are achieved in terms of linear matrix inequalities (LMIs). Finally, three numerical examples are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, strength of fractional neural networks (FrNNs) is exploited to find the approximate solutions of nonlinear systems based on Riccati equations of arbitrary order. The feed-forward artificial FrNN are used to develop the energy function of the system by defining an error function in mean square sense. Design parameters for optimization of the energy function are adapted using viable local search with interior point methods (IPMs). The performance of design methodology in terms of accuracy and convergence is analyzed for two different variants of the nonlinear system. Comparison of the results with the exact solutions, as well as approximate numerical results, illustrates the correctness of the methodology. The worth of the scheme is established through statistical inferences based on a large number of simulation runs. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the globally asymptotic stability of the Riemann‐Liouville fractional‐order neural networks with time‐varying delays. The Lyapunov functional approach to stability analysis for nonlinear fractional‐order functional differential equations is discussed. By constructing an appropriate Lyapunov functional associated with the Riemann‐Liouville fractional integral and derivative, the asymptotic stability criteria of fractional‐order neural networks with time‐varying delays and constant delays are derived. The advantage of our proposed method is that one may directly calculate the first‐order derivative of the Lyapunov functional. Two numerical examples are also presented to illustrate the validity and feasibility of the theoretical results. With the increasing of the order of fractional derivatives, the state trajectories of neural networks show that the speeds of converging toward zero solution are faster and faster. [ABSTRACT FROM AUTHOR]

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented.

In this paper, a novel fractional-order sliding mode control strategy is introduced to synchronize two nonidentical fractional order neural networks (FNNs) in finite time. Firstly, by applying fractional-order calculation, the properties of the global asymptotical stability and convergence in finite time are developed for the fractional-order differential equation system. Secondly, based on the proposed principle of convergence in finite time, a new fractional-order sliding mode surface is presented and its global stability in finite time is analytically proved. In addition, an appropriate sliding mode controller is designed to drive the state trajectories of error system to the prescribed sliding surface in finite time and remain on it evermore. Meanwhile, by means of the fractional Lyapunov-like approach, the global projective synchronization condition is given, and the synchronization time is explicitly evaluated. As the special case, some sufficient conditions are given to guarantee global complete synchronization, anti-synchronization and stabilization of nonidentical FNNs. Finally, an example is given to demonstrate the validity of the proposed method. [ABSTRACT FROM AUTHOR]

Complex-valued neural networks (CVNNs) with integer-order have attracted much attention, and which have been well discussed. Fractional complex-valued neural networks (FCVNNs) are more suitable to describe the dynamical properties of neural networks, but have rarely been studied. It is the first time that the stability and bifurcation of a class of delayed FCVNN is investigated in this paper. The activation function can be expressed by separating into its real and imaginary parts. By using time delay as the bifurcation parameter, the dynamical behaviors that including local asymptotical stability and Hopf bifurcation are discussed, the conditions of emergence of bifurcation are obtained. Furthermore, it reveals that the onset of the bifurcation point can be delayed as the order increases. Finally, an illustrative example is provided to verify the correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the issue of stability and bifurcation for a delayed fractional neural network with three neurons by applying the sum of time delays as the bifurcation parameter. Based on fractional Laplace transform and the method of stability switches, some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. The analysis indicates that time delay can effectively enhance the stability of fractional neural networks. In addition, it is found that the stability interval can be varied by regulating the fractional order if all the parameters are fixed including time delay. Finally, numerical examples are presented to validate the derived theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper addresses the bipartite synchronization of coupled multi-order fractional neural networks (MFNNs) with time-varying delays. An effective event-triggered controller is proposed, and sufficient criteria for ensuring the bipartite synchronization are derived by using the Lyapunov function in vector form and the comparison principle for multi-order fractional differential equations. In addition, the preclusion of Zeno behavior is discussed. The results obtained in this paper cover the bipartite synchronization of both fractional neural networks with identical order and integer-order neural networks as special cases. A numerical example is given to verify the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]