Fractal and Fractional

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48 pages, 10712 KiB

Open AccessArticle

Bifurcation Analysis of Time-Delayed Non-Commensurate Caputo Fractional Bi-Directional Associative Memory Neural Networks Composed of Three Neurons

by Chengqiang Wang, Xiangqing Zhao, Qiuyue Mai and Zhiwei Lv

Fractal Fract. 2024, 8(2), 83; https://doi.org/10.3390/fractalfract8020083 - 26 Jan 2024

Viewed by 1187

Abstract

We are concerned in this paper with the stability and bifurcation problems for three-neuron-based bi-directional associative memory neural networks that are involved with time delays in transmission terms and possess Caputo fractional derivatives of non-commensurate orders. For the fractional bi-directional associative memory neural [...] Read more.

We are concerned in this paper with the stability and bifurcation problems for three-neuron-based bi-directional associative memory neural networks that are involved with time delays in transmission terms and possess Caputo fractional derivatives of non-commensurate orders. For the fractional bi-directional associative memory neural networks that are dealt with in this paper, we view the time delays as the bifurcation parameters. Via a standard contraction mapping argument, we establish the existence and uniqueness of the state trajectories of the investigated fractional bi-directional associative memory neural networks. By utilizing the idea and technique of linearization, we analyze the influence of time delays on the dynamical behavior of the investigated neural networks, as well as establish and prove several stability/bifurcation criteria for the neural networks dealt with in this paper. According to each of our established criteria, the equilibrium states of the investigated fractional bi-directional associative memory neural networks are asymptotically stable when some of the time delays are less than strictly specific positive constants, i.e., when the thresholds or the bifurcation points undergo Hopf bifurcation in the concerned networks at the aforementioned threshold constants. In the meantime, we provide several illustrative examples to numerically and visually validate our stability and bifurcation results. Our stability and bifurcation theoretical results in this paper yield some insights into the cause mechanism of the bifurcation phenomena for some other complex phenomena, and this is extremely helpful for the design of feedback control to attenuate or even to remove such complex phenomena in the dynamics of fractional bi-directional associative memory neural networks with time delays. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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14 pages, 349 KiB

Open AccessArticle

Finite Time Stability Results for Neural Networks Described by Variable-Order Fractional Difference Equations

by Tareq Hamadneh, Amel Hioual, Omar Alsayyed, Yazan Alaya Al-Khassawneh, Abdallah Al-Husban and Adel Ouannas

Fractal Fract. 2023, 7(8), 616; https://doi.org/10.3390/fractalfract7080616 - 10 Aug 2023

Cited by 4 | Viewed by 1226

Abstract

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. [...] Read more.

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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12 pages, 2884 KiB

Open AccessArticle

Designing Hyperbolic Tangent Sigmoid Function for Solving the Williamson Nanofluid Model

by Basma Souayeh and Zulqurnain Sabir

Fractal Fract. 2023, 7(5), 350; https://doi.org/10.3390/fractalfract7050350 - 25 Apr 2023

Cited by 6 | Viewed by 1381

Abstract

This study shows the design of the novel hyperbolic tangent sigmoid function for the numerical treatment of the Williamson nanofluid model (WNM), which is categorized as velocity, concentration, and temperature. A process of a deep neural network using fifteen and thirty neurons is [...] Read more.

This study shows the design of the novel hyperbolic tangent sigmoid function for the numerical treatment of the Williamson nanofluid model (WNM), which is categorized as velocity, concentration, and temperature. A process of a deep neural network using fifteen and thirty neurons is presented to solve the model. The hyperbolic tangent sigmoid transfer function is used in the process of both hidden layers. The optimization is performed through the Bayesian regularization approach (BRA) to solve the WNM. A targeted dataset through the Adam scheme is achieved that is further accomplished using the procedure of training, testing, and verification with ratios of 0.15, 0.13, and 0.72. The correctness of the deep neural network along with the BRA is performed through the overlapping of the solutions. The small calculated absolute error values also enhance the accurateness of the designed procedure. Moreover, the statistical observations are authenticated to reduce the mean square error for the nonlinear WNM. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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13 pages, 4520 KiB

Open AccessArticle

Chaotic Characteristic Analysis of Dynamic Gravity Model with Fractal Structures via an Improved Conical Volume-Delay Function

by Liumeng Yang, Ruichun He, Jie Wang, Wei Zhou, Hongxing Zhao and Huo Chai

Fractal Fract. 2023, 7(3), 278; https://doi.org/10.3390/fractalfract7030278 - 22 Mar 2023

Viewed by 1316

Abstract

Road traffic networks are chaotic and highly complex systems. In this paper, we introduce a dynamic gravity model that characterizes the behaviors of the O-D (origin-destination) traffic, such as equilibrium, period-doubling, chaos, and fractal in discrete time. In cases where the original cost [...] Read more.

Road traffic networks are chaotic and highly complex systems. In this paper, we introduce a dynamic gravity model that characterizes the behaviors of the O-D (origin-destination) traffic, such as equilibrium, period-doubling, chaos, and fractal in discrete time. In cases where the original cost function is used, the trip distribution model might degenerate into an all-or-nothing problem without the capacity constraints. To address this shortcoming, we propose substituting the original cost function with an improved conical volume-delay function. This new function retains some of the properties of the original cost function, and its parameters have the same meaning as those in the original function. Our analysis confirms that the double-constrained dynamic gravity model successfully characterizes complex traffic behavior because of the improved conical volume-delay function. Our analysis further shows that the three-parameter bifurcation diagram based on the period characteristics provides deep insight into the actual state of the road traffic networks. Investigating the properties of the model solutions, we further show that the new model is more effective in addressing the all-or-nothing problem. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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19 pages, 4623 KiB

Open AccessArticle

Stability and Hopf Bifurcation of a Class of Six-Neuron Fractional BAM Neural Networks with Multiple Delays

by Bingbing Li, Maoxin Liao, Changjin Xu, Huiwen Chen and Weinan Li

Fractal Fract. 2023, 7(2), 142; https://doi.org/10.3390/fractalfract7020142 - 2 Feb 2023

Cited by 6 | Viewed by 1352

Abstract

In this paper, we study the stability and Hopf bifurcation of a class of six-neuron fractional BAM neural networks with multiple delays. Firstly, the model is transformed into a fractional neural network model with two nonidentical delays by using variable substitution. Then, by [...] Read more.

In this paper, we study the stability and Hopf bifurcation of a class of six-neuron fractional BAM neural networks with multiple delays. Firstly, the model is transformed into a fractional neural network model with two nonidentical delays by using variable substitution. Then, by assigning a value to one of the time delays and selecting the remaining time delays as parameters, the critical value of Hopf bifurcation for different time delays is calculated. The study shows that when the time lag exceeds its critical value, the equilibrium point of the system will lose its stability and generate Hopf bifurcation. Finally, the correctness of theoretical analysis is verified by simulation. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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15 pages, 403 KiB

Open AccessArticle

Neutral-Type and Mixed Delays in Fractional-Order Neural Networks: Asymptotic Stability Analysis

by Călin-Adrian Popa

Fractal Fract. 2023, 7(1), 36; https://doi.org/10.3390/fractalfract7010036 - 29 Dec 2022

Cited by 4 | Viewed by 1781

Abstract

The lack of a conventional Lyapunov theory for fractional-order (FO) systems makes it difficult to study the dynamics of fractional-order neural networks (FONNs). Instead, the existing literature derives necessary conditions for various dynamic properties of FONNs using Halanay-type lemmas. However, when these lemmas [...] Read more.

The lack of a conventional Lyapunov theory for fractional-order (FO) systems makes it difficult to study the dynamics of fractional-order neural networks (FONNs). Instead, the existing literature derives necessary conditions for various dynamic properties of FONNs using Halanay-type lemmas. However, when these lemmas are used, the results are frequently more conservative than those produced for integer-order neural networks (NNs). In order to provide sufficient criteria that are less conservative than those found in other research, a novel application of the Halanay-type lemma is made within this study. Thus, for extremely general FONNs containing neutral-type, time-varying, and distributed delays, sufficient conditions presented by way of linear matrix inequalities (LMIs) and algebraic inequalities are achieved. For the FO scenario, a model this broad and including so many different kinds of delays is developed for the first time. Additionally, a novel form of Lyapunov-like function is built, which results in less stringent algebraic inequalities. One of the first times in the setting of FONNs, the free-weighting matrix method is also used to further lower the conservativeness of the obtained conditions. Based on different Lyapunov-type functions, three theorems are developed regarding the asymptotic stability of the proposed networks. Three numerical simulations are used to demonstrate the theoretical developments. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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13 pages, 440 KiB

Open AccessArticle

Switching-Jumps-Dependent Quasi-Synchronization Criteria for Fractional-Order Memrisive Neural Networks

by Yingjie Fan, Zhongliang Wei and Meixuan Li

Fractal Fract. 2023, 7(1), 12; https://doi.org/10.3390/fractalfract7010012 - 24 Dec 2022

Viewed by 1217

Abstract

This paper investigates the switching-jumps-dependent quasi-synchronization issue for fractional-order memristive neural networks (FMNNs). First, a simplied linear feedback controller is applied. Then, in terms of several fractional order differential inequalities and two kinds of Lyapunov functions, two quasi-synchronization criteria expressed by linear matrix [...] Read more.

This paper investigates the switching-jumps-dependent quasi-synchronization issue for fractional-order memristive neural networks (FMNNs). First, a simplied linear feedback controller is applied. Then, in terms of several fractional order differential inequalities and two kinds of Lyapunov functions, two quasi-synchronization criteria expressed by linear matrix inequality (LMI)-based form and algebraic form are established, respectively. Meanwhile, the co-designed scheme for error bound and control gain is established. Compared with the previous quasi-synchronization results, a strong assumption that the system states must be bounded is removed. Finally, some simulation examples are carried out to display the feasibility and validity of the proposed analysis methods. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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40 pages, 11166 KiB

Open AccessArticle

Bifurcation Phenomenon and Control Technique in Fractional BAM Neural Network Models Concerning Delays

by Peiluan Li, Yuejing Lu, Changjin Xu and Jing Ren

Fractal Fract. 2023, 7(1), 7; https://doi.org/10.3390/fractalfract7010007 - 22 Dec 2022

Cited by 5 | Viewed by 1548

Abstract

In this current study, we formulate a kind of new fractional BAM neural network model concerning five neurons and time delays. First, we explore the existence and uniqueness of the solution of the formulated fractional delay BAM neural network models via the Lipschitz [...] Read more.

In this current study, we formulate a kind of new fractional BAM neural network model concerning five neurons and time delays. First, we explore the existence and uniqueness of the solution of the formulated fractional delay BAM neural network models via the Lipschitz condition. Second, we study the boundedness of the solution to the formulated fractional delayed BAM neural network models using a proper function. Third, we set up a novel sufficient criterion on the onset of the Hopf bifurcation stability of the formulated fractional BAM neural network models by virtue of the stability criterion and bifurcation principle of fractional delayed dynamical systems. Fourth, a delayed feedback controller is applied to command the time of occurrence of the bifurcation and stability domain of the formulated fractional delayed BAM neural network models. Lastly, software simulation figures are provided to verify the key outcomes. The theoretical outcomes obtained through this exploration can play a vital role in controlling and devising networks. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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