Research

16 pages, 6211 KiB

Open AccessArticle

Remaining Useful Life Prediction of a Planetary Gearbox Based on Meta Representation Learning and Adaptive Fractional Generalized Pareto Motion

by Hongqing Zheng, Wujin Deng, Wanqing Song, Wei Cheng, Piercarlo Cattani and Francesco Villecco

Fractal Fract. 2024, 8(1), 14; https://doi.org/10.3390/fractalfract8010014 - 22 Dec 2023

Viewed by 921

Abstract

The remaining useful life (RUL) prediction of wind turbine planetary gearboxes is crucial for the reliable operation of new energy power systems. However, the interpretability of the current RUL prediction models is not satisfactory. To this end, a multi-stage RUL prediction model is [...] Read more.

The remaining useful life (RUL) prediction of wind turbine planetary gearboxes is crucial for the reliable operation of new energy power systems. However, the interpretability of the current RUL prediction models is not satisfactory. To this end, a multi-stage RUL prediction model is proposed in this work, with an interpretable metric-based feature selection algorithm. In the proposed model, the advantages of neural networks and long-range-dependent stochastic processes are combined. In the offline training stage, a general representation of the degradation trend is learned with the meta-long short-term memory neural network (meta-LSTM) model. The inevitable measurement error in the sensor reading is modelled by white Gaussian noise. During the online RUL prediction stage, fractional generalized Pareto motion (fGPm) with an adaptive diffusion is employed to model the stochasticity of the planetary gearbox degradation. In the case study, real planetary gearbox degradation data are used for the model validation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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15 pages, 1153 KiB

Open AccessArticle

Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model

by Zhuang Cui, Yan Zhou and Ruimei Li

Fractal Fract. 2023, 7(7), 548; https://doi.org/10.3390/fractalfract7070548 - 16 Jul 2023

Cited by 2 | Viewed by 934

Abstract

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator [...] Read more.

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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18 pages, 1263 KiB

Open AccessArticle

Forecasting Cryptocurrency Prices Using LSTM, GRU, and Bi-Directional LSTM: A Deep Learning Approach

by Phumudzo Lloyd Seabe, Claude Rodrigue Bambe Moutsinga and Edson Pindza

Fractal Fract. 2023, 7(2), 203; https://doi.org/10.3390/fractalfract7020203 - 18 Feb 2023

Cited by 18 | Viewed by 10979

Abstract

Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies [...] Read more.

Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies have been conducted on cryptocurrency price prediction using different Deep Learning (DL) based algorithms. This study proposes three types of Recurrent Neural Networks (RNNs): namely, Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), and Bi-Directional LSTM (Bi-LSTM) for exchange rate predictions of three major cryptocurrencies in the world, as measured by their market capitalization—Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC). The experimental results on the three major cryptocurrencies using both Root Mean Squared Error (RMSE) and the Mean Absolute Percentage Error (MAPE) show that the Bi-LSTM performed better in prediction than LSTM and GRU. Therefore, it can be considered the best algorithm. Bi-LSTM presented the most accurate prediction compared to GRU and LSTM, with MAPE values of 0.036, 0.041, and 0.124 for BTC, LTC, and ETH, respectively. The paper suggests that the prediction models presented in it are accurate in predicting cryptocurrency prices and can be beneficial for investors and traders. Additionally, future research should focus on exploring other factors that may influence cryptocurrency prices, such as social media and trading volumes. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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11 pages, 310 KiB

Open AccessArticle

On Variable-Order Fractional Discrete Neural Networks: Existence, Uniqueness and Stability

by Othman Abdullah Almatroud, Amel Hioual, Adel Ouannas, Mohammed Mossa Sawalha, Saleh Alshammari and Mohammad Alshammari

Fractal Fract. 2023, 7(2), 118; https://doi.org/10.3390/fractalfract7020118 - 26 Jan 2023

Cited by 2 | Viewed by 1129

Abstract

Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete [...] Read more.

Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete nabla operator of variable-order. An adequate criterion for the existence of the solution in addition to its uniqueness for such systems is provided with the use of Banach fixed point technique. Moreover, the uniform stability is investigated. We provide at the end two numerical simulations illustrating the relevance of the aforementioned results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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18 pages, 9203 KiB

Open AccessArticle

Multistability and Phase Synchronization of Rulkov Neurons Coupled with a Locally Active Discrete Memristor

by Minglin Ma, Yaping Lu, Zhijun Li, Yichuang Sun and Chunhua Wang

Fractal Fract. 2023, 7(1), 82; https://doi.org/10.3390/fractalfract7010082 - 11 Jan 2023

Cited by 38 | Viewed by 1931

Abstract

In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the [...] Read more.

In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the LADM into two identical Rulkov neurons. Based on numerical simulation, the neural network manifested multistability and new firing behaviors under different system parameters and initial values. In addition, the phase synchronization between the neurons was explored. Additionally, it is worth mentioning that the Rulkov neurons showed synchronization transition behavior; that is, anti-phase synchronization changed to in-phase synchronization with the change in the coupling strength. In particular, the anti-phase synchronization of different firing patterns in the neural network was investigated. This can characterize the different firing behaviors of coupled homogeneous neurons in the different functional areas of the brain, which is helpful to understand the formation of functional areas. This paper has a potential research value and lays the foundation for biological neuron experiments and neuron-based engineering applications. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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24 pages, 1958 KiB

Open AccessArticle

Adaptive Quantized Synchronization of Fractional-Order Output-Coupling Multiplex Networks

by Yunzhan Bai, Juan Yu and Cheng Hu

Fractal Fract. 2023, 7(1), 22; https://doi.org/10.3390/fractalfract7010022 - 26 Dec 2022

Viewed by 970

Abstract

This paper is devoted to investigating the synchronization of fractional-order output-coupling multiplex networks (FOOCMNs). Firstly, a type of fractional-order multiplex network is introduced, where the intra-layer coupling and the inter-layer coupling are described separately, and nodes communicate with each other by their outputs, [...] Read more.

This paper is devoted to investigating the synchronization of fractional-order output-coupling multiplex networks (FOOCMNs). Firstly, a type of fractional-order multiplex network is introduced, where the intra-layer coupling and the inter-layer coupling are described separately, and nodes communicate with each other by their outputs, which is more realistic when the node states are unmeasured. By using the Lyapunov method and the fractional differential inequality, sufficient conditions are provided for achieving asymptotic synchronization based on the designed adaptive control, where the synchronized state of each layer is different. Furthermore, a quantized adaptive controller is developed to realize the synchronization of FOOCMNs, which effectively reduces signal transmission frequency and improves the effective utilization rate of network resources. Two numerical examples are given at last to support the theoretical analysis. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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23 pages, 26086 KiB

Open AccessArticle

Characteristic Analysis and Circuit Implementation of a Novel Fractional-Order Memristor-Based Clamping Voltage Drift

by Huaigu Tian, Jindong Liu, Zhen Wang, Fei Xie and Zelin Cao

Fractal Fract. 2023, 7(1), 2; https://doi.org/10.3390/fractalfract7010002 - 20 Dec 2022

Cited by 25 | Viewed by 1507

Abstract

The ideal magnetic flux-controlled memristor was introduced into a four-dimensional chaotic system and combined with fractional calculus theory, and a novel four-dimensional commensurate fractional-order system was proposed and solved using the Adomian decomposition method. The system orders, parameters, and initial values were studied [...] Read more.

The ideal magnetic flux-controlled memristor was introduced into a four-dimensional chaotic system and combined with fractional calculus theory, and a novel four-dimensional commensurate fractional-order system was proposed and solved using the Adomian decomposition method. The system orders, parameters, and initial values were studied as independent variables in the bifurcation diagram and Lyapunov exponents spectrum, and it was discovered that changing these variables can cause the system to exhibit more complex and rich dynamical behaviors. The system had an offset boosting, which was discovered by adding a constant term after the decoupled linear term. Finally, the results of the numerical simulation were verified through the use of analog circuits and FPGA designs, and a control scheme for the system circuit was also suggested. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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17 pages, 2900 KiB

Open AccessArticle

Multiscale Approach for Bounded Deformation Image Registration

by Yunfeng Du and Huan Han

Fractal Fract. 2022, 6(11), 681; https://doi.org/10.3390/fractalfract6110681 - 18 Nov 2022

Viewed by 1326

Abstract

Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last [...] Read more.

Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last decades, many smooth registration models (i.e., diffeomorphic registration) were proposed. However, image with strong noise may lead to discontinuous deformation, which cannot be modelled by smooth registration. To simulate this kind of deformation, some non-smooth registration models were also proposed. However, numerical algorithms for these models are easily trapped into a local minimum because of the nonconvexity of the object functional. To overcome the local minimum of the object functional, we propose a multiscale approach for a non-smooth registration model: the bounded deformation (BD) model. The convergence of the approach is shown, and numerical tests are also performed to show the good performance of the proposed multiscale approach. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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16 pages, 25100 KiB

Open AccessArticle

Dynamical Analysis of a Novel Fractional-Order Chaotic System Based on Memcapacitor and Meminductor

by Xingce Liu, Jun Mou, Jue Wang, Santo Banerjee and Peng Li

Fractal Fract. 2022, 6(11), 671; https://doi.org/10.3390/fractalfract6110671 - 13 Nov 2022

Cited by 22 | Viewed by 1223

Abstract

In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So [...] Read more.

In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So as to study the dynamic behavior in detail, not only the equilibrium stability of the system is analyzed, but also the dynamic characteristics are analyzed by means of a Bifurcation diagram and Lyapunov exponents (Les). By analyzing the dynamic behavior of the system, some special phenomena, such as the coexistence of attractor and state transition, are found in the system. In the end, the circuit implementation of the system is implemented on a Digital Signal Processing (DSP) platform. According to the numerical simulation results of the system, it is found that the system has abundant dynamical characteristics. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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19 pages, 8154 KiB

Open AccessArticle

Study on the Complex Dynamical Behavior of the Fractional-Order Hopfield Neural Network System and Its Implementation

by Tao Ma, Jun Mou, Bo Li, Santo Banerjee and Huizhen Yan

Fractal Fract. 2022, 6(11), 637; https://doi.org/10.3390/fractalfract6110637 - 01 Nov 2022

Cited by 28 | Viewed by 1952

Abstract

The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system [...] Read more.

The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system is solved. The dissipative characteristics of the system are discussed, as well as the equilibrium point is resolved. The characteristics of the dynamics through the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system are investigated. The circuit of the system was also designed, based on the Multisim simulation platform, and the simulation of the circuit was realized. The simulation results show that the proposed FOHNN system exhibits many interesting phenomena, which provides more basis for the study of complex brain working patterns, and more references for the design, as well as the hardware implementation of the realized fractional-order neural network circuit. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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