Fractional-order BAM Neural Networks Research Articles

Fractional order uncertain BAM neural networks with mixed time delays: An existence and Quasi-uniform stability analysis

This research investigates the presence of unique solutions and quasi-uniform stability for a class of fractional-order uncertain BAM neural networks utilizing the Banach fixed point concept, the contraction mapping principle, and analysis techniques. In order to guarantee the equilibrium point of fractional-order BAM neural networks with undetermined parameters, some new adequate criteria are devised, and both time delays result in quasi-uniform stability. The acquired results, which are simple to verify in practice, enhance and extend several earlier research works in some ways. Finally, two illustrative examples are provided to show the value of the suggested outcomes.

Further Research on the Problems of Synchronization for Fractional-Order BAM Neural Networks in Octonion-Valued Domain

Further Research on the Problems of Synchronization for Fractional-Order BAM Neural Networks in Octonion-Valued Domain

Insight into Hopf Bifurcation and Control Methods in Fractional Order BAM Neural Networks Incorporating Symmetric Structure and Delay

Insight into Hopf Bifurcation and Control Methods in Fractional Order BAM Neural Networks Incorporating Symmetric Structure and Delay

Synchronization Analysis of Fractional Order Delayed BAM Neural Networks via Multi-Delay-Boundary Inequality

This paper analyzes the synchronization of Caputo fractional-order BAM neural networks with multiple time-varying delays (FOBAMNNs-MTDs). The time-delay phenomenon is inevitable in the synchronization process. Thereby, it is crucial to fully consider the time-delay information on behalf of better studying the synchronization problem. In previously published studies of Caputo FOBAMNNs-MTDs, the time-delay information of bounded MTDs is implied in the delayed state term. Then, the boundary information of MTDs is not considered by applying the method mentioned above, which will be detrimental to the accuracy of the system dynamic characteristics analysis. This paper aims to propose a new time-delay information processing method to enhance the utilization of MTD information, thereby improving the synchronization analysis of FOBAMNNs-MTDs. Firstly, an inequality called multi-delay-boundary inequality (MDBI) containing all boundary information of time-varying delays and fractional-order information is proposed. That is, a new time-delay information processing method of MTDs with different boundary information is proposed, which makes full use of MTD information. Secondly, a less-conservatism synchronization criterion with the information of all delay boundary and fractional-order of Caputo FOBAMNNs-MTDs is established, which is more flexible than the delay-independent criterion. Finally, two numerical simulations are provided to verify the validity of the obtained results.

The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays

<abstract><p>The dynamic signal transmission process can be regarded as an anti-periodic process, and fractional-order inertial neural networks are widely used in signal processing and other fields, so anti-periodicity is also regarded as an important dynamic feature of inertial neural networks. This paper mainly studies the existence and Mittag-Leffler stability of anti-periodic solutions for a class of fractional-order inertial BAM neural networks with time-delays. By introducing variable substitution, the model with two different fractional-order derivatives is transformed into a model with only one fractional-order derivative of the same order. Using the properties of fractional-order calculus, the relationship between the fractional-order integral of the state function with and without time-delays is given. Firstly, the sufficient conditions for the boundedness and the Mittag-Leffler stability of the solutions for the system are derived. Secondly, by constructing the sequence solution of the function for the system and applying Ascoli-Arzela theorem, the sufficient conditions for the existence and Mittag-Leffler stability of the anti-periodic solution are given. Finally, the correctness of the conclusion is verified by a numerical example.</p></abstract>

Extended analysis on the global Mittag-Leffler synchronization problem for fractional-order octonion-valued BAM neural networks

In this paper, a new case of neural networks called fractional-order octonion-valued bidirectional associative memory neural networks (FOOVBAMNNs) is established. First, the higher dimensional models are formulated for FOOVBAMNNs with general activation functions and the special linear threshold ones, respectively. On one hand, employing Cayley–Dichson construction in octonion multiplication which is essentially neither commutative nor associative, the system of FOOVBAMNNs is divided into four fractional-order complex-valued ones. On the other hand, Caputo fractional derivative’s character and BAM’s interactive feature are also properly dealt with. Second, the general criteria are obtained by the new design of LKFs, the application of the related inequalities and the construction of the linear feedback controllers for the global Mittag-Leffler synchronization problem of FOOVBAMNNs. Finally, we present two numerical examples to show the realizability and progress of the derived results.

Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks

In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks.

Probing into bifurcation for fractional-order BAM neural networks concerning multiple time delays

This paper mainly examines the stability and the existence of Hopf bifurcation of fractional-order BAM neural networks with multiple delays. Based on Laplace transform, stability criterion and Hopf bifurcation theory of fractional-order differential equations, a new sufficient criterion to guarantee the stability and the existence of Hopf bifurcation for the involved fractional-order BAM neural networks with multiple delays is established. The investigation manifests that time delay plays an important role in maintaining network stability and controlling the appearance of Hopf bifurcation of fractional-order BAM neural networks. Numerical simulations are performed to check the rationality of the analytical conclusions. The obtained theoretical predictions of this paper have extremely vital guiding significance in designing and controlling networks.

New criteria on the finite-time stability of fractional-order BAM neural networks with time delay

New criteria on the finite-time stability of fractional-order BAM neural networks with time delay

New approach to finite-time stability for fractional-order BAM neural networks with discrete and distributed delays

• A novel generalized FODGI with two different orders is developed, which includes the one in [39, Theorem 3.2] as a special case and can be used to analyze the qualitative properties of various FODSs with two different orders. • As a straightforward generalization, a novel generalized FODGI with multiple different orders is also established, which includes the one in [40, Theorem 1.4] as a special case. • Two novel FTS criteria of FOBAMNNs with the orders 0 < ℘ < 1 and 1 < ℘< 2 are established by the obtained FODGI with two different orders. In this approach, the term w ( t ) and delay term w ( t − τ ) can be analyzed in a unified framework. This is to say, the delay term w ( t − τ ) needs not to be transformed to the term w ( t ). • Comparisons with the existing contributions: a) The obtained results in this article is less conservative than the ones in the existing works [30, 31]. b) There are delay terms in the obtained novel FODGI with multiple different orders. Therefore, the proposed inequality in this article is more general. The finite-time stability (FTS) for fractional-order bidirectional associative memory neural networks (FOBAMNNs) with discrete and distributed delays is investigated in this article. Firstly, a novel fractional-order delayed Gronwall inequality (FODGI) with two different orders is developed. As a straightforward generalization, a novel generalized FODGI with multiple different orders is also established, which can be applied to investigate the stability of fractional-order delayed systems (FODSs) with multiple different orders. Secondly, based on the former inequality, two novel FTS criteria of FOBAMNNs with the orders 0 < ℘ < 1 and 1 < ℘ < 2 are established. Finally, three examples are exhibited to illustrate the effectiveness and the less conservativeness of the proposed methods.

Finite-time stabilization of fractional-order fuzzy quaternion-valued BAM neural networks via direct quaternion approach

This paper investigates fractional-order fuzzy quaternion-valued BAM neural networks (FOFQBAMNNs) without decomposition. By virtue of a novel contraction mapping, the existence and uniqueness of the equilibrium point is yielded. Furthermore, according to some basic knowledge on fractional calculus, inequality techniques of fuzzy logic and reduction to absurdity, some criteria are yielded to guarantee finite-time stabilization of FOFQBAMNNs via original quaternion-valued controllers, and the settling times of corresponding finite-time stabilization are derived. Finally, the feasibility of our obtained theoretical results is illustrated by some numerical simulations.

Global synchronization of uncertain fractional-order BAM neural networks with time delay via improved fractional-order integral inequality

This paper is concerned with the problem of global synchronization for a class of uncertain fractional-order bidirectional associative memory (FOBAM) neural networks (NNs) with constant time delay. Some linear matrix inequality (LMI) based conditions have been established for ensuring the stability behavior of the governing augmented system derived from the considered fractional-order master system and the constructed slave system. In such LMI conditions, for utilizing the information on the time delay terms and the order of the fractional derivative, some fractional-order integral inequalities have been developed. Next, by defining a new Lyapunov–Krasovskii functional (LKF) and based on the improved fractional-order inequalities, the global synchronization criteria have been established in terms of a solvable set of LMIs for the considered class of uncertain FOBAM NNs without uncertain matrices. Further, the obtained results have been extended to the case of uncertain FOBAM NNs. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the proposed theoretical results.

Bifurcation Properties for Fractional Order Delayed BAM Neural Networks

In the past several decades, many papers involving the stability and Hopf bifurcation of delayed neural networks have been published. However, the results on the stability and Hopf bifurcation for fractional order neural networks with delays and fractional order neural networks with leakage delays are very rare. This paper is concerned with the stability and the existence of Hopf bifurcation of fractional order BAM neural networks with or without leakage delay. The Laplace transform, stability and bifurcation theory of fractional-order differential equations and Matlab software will be applied. The stability condition and the sufficient criterion of existence of Hopf bifurcation for fractional order BAM neural networks with delay (leakage delay) are established. It is found that when the sum of two delays (leakage delay) crosses a critical value, then a Hopf bifurcation will appear. The obtained results play an important role in designing neural networks. Also the derived results are new and enrich the bifurcation theory of fractional order delayed differential equations.

Improved approach to the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks based on new inequalities

This paper studies the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks (FOMVBAMNNs) with general activation functions (AFs). First, the unified model is established for the researched systems of FOMVBAMNNs which can be turned into the corresponding multidimension-valued systems as long as the state variables, the connection weights and the AFs of the neural networks are valued to be real, complex, or quaternion. Then, without any decomposition, the criteria in unified form are derived by constructing the new Lyapunov–Krasovskii functionals (LKFs) in vector form, combining two new inequalities and considering the easy controllers. It is worth mentioning that the obtained criteria have many advantages in higher flexibility, more diversity, smaller computation, and lower conservatism. Finally, a simulation example is provided to illustrate the availability and improvements of the acquired results.

A further study on bifurcation for fractional order BAM neural networks with multiple delays

In the present work, new fractional order BAM neural networks with multiple delays are formulated. Firstly, we study the existence and uniqueness of solution of the constructed neural networks by applying Lipschitz condition. Secondly, the boundedness of solution for the involved neural networks is analyzed by constructing an appropriate function. Thirdly, with the aid of the stability theorem and Hopf bifurcation theory of the fractional order delayed differential systems, the sufficient criterion to guarantee the stability and the existence of Hopf bifurcation for the addressed neural networks is set up. In the end, simulation results are displayed with computer software to support research findings. The established theoretical results of this work have significant theoretical guiding value in dominating and optimizing networks.

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

Controller design for finite-time and fixed-time stabilization of fractional-order memristive complex-valued BAM neural networks with uncertain parameters and time-varying delays

In this paper we investigate controller design problem for finite-time and fixed-time stabilization of fractional-order memristive complex-valued BAM neural networks (FMCVBAMNNs) with uncertain parameters and time-varying delays. By using the Lyapunov theory, differential inclusion theory, and fractional calculus theory, finite-time stabilization condition for fractional-order memristive complex-valued BAM neural networks and the upper bound of the settling time for stabilization are obtained. The nonlinear complex-valued activation functions are split into two (real and imaginary) components. Moreover, the settling time of fixed time stabilization, that does not depend upon the initial values, is merely calculated. A novel criterion for guaranteeing the fixed-time stabilization of FMCVBAMNNs is derived. Our control scheme achieves system stabilization within bounded time and has an advantage in convergence rate. Numerical simulations are furnished to demonstrate the effectiveness of the theoretical analysis.

Synchronization of multi-links impulsive fractional-order complex networks via feedback control based on discrete-time state observations

This paper addresses the issue of global Mittag-Leffler synchronization of multi-links impulsive fractional-order complex networks (MIFCNs). A class of feedback control based on discrete-time state observations is first adopted for synchronization of MIFCNs. By combining Lyapunov method with graph-theoretic approach, a synchronization criterion for MIFCNs is derived. In particular, this synchronization criterion is related to the order of fractional derivatives, the parameters of control and the topological structure of networks. Furthermore, as an application of our theoretical results, synchronization of multi-links impulsive fractional-order BAM neural networks is studied in detail and a synchronization criterion is also provided. Finally, a numerical example is presented to illustrate the validity and feasibility of theoretical results.

Comments on “Influence of Time Delay on Bifurcation in Fractional Order BAM Neural Networks With Four Delays”

Mathematical modeling is a tool that has a lot of applications in applied sciences. This has driven extensive investigations and the development of different methods in applied mathematics. Recently, neural networks have attracted a significant attention in various research areas owing to their promising application in signal and image processing, optimization solvers, intelligent control, quadratic optimization, and automatic control. In a recent paper <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , the authors discussed that the time delay has an important influence on the stability and Hopf bifurcation of the involved networks. The derived results in this paper are new and will play a key role in the future optimization of networks and the improvement of human life. However, the mathematical modeling discussed in <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> contains a wrong calculation in the early part of the paper related to the Hopf bifurcation, which is subsequently carried out throughout the manuscript. The purpose of this comment is to point out the calculation error in <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> and provide the correct calculation.

Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations

Abstract This manuscript concerns quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron activations (FBAMNNDDAs). Firstly, under the framework of Filippov solution and fractional-order differential inclusions analysis for the initial value problem of FBAMNNDDAs is presented. Secondly, two kinds of novel pinning controllers according to pinning control technique are designed. By means of fractional order Lyapunov method and designed pinning control strategy, the sufficient criteria is given first to ensure the quasi-synchronization for the dynamic behavior of FBAMNNDDAs. Furthermore, the error bound of pinning synchronization is explicitly evaluated. Thirdly, via Kakutani s fixed point theorem of set-valued map analysis, Razumikhin condition, and a nonlinear pinning controller, the existence and β-exponential stabilization of FBAMNNDDAs equilibrium point is obtained in the voice of linear matrix inequality (LMI) technique. Fourthly, based on as well as Mittag-Leffler function and growth condition, the global existence of a solution in the Filippov sense of such system is guaranteed with detailed proof. At last, a numerical example with computer simulations are performed to illustrate the effectiveness of proposed theoretical consequences.