Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 30 April 2024 | Viewed by 1966

Special Issue Editors


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Guest Editor
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
Interests: fractional-order systems; distributed-order systems; fractional-order neural networks; impulsive systems; dynamics analysis and control of nonlinear systems

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Guest Editor
Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China
Interests: complex-valued neural networks; quaternion-valued neural networks; machine learning; convolutional neural network

Special Issue Information

Dear Colleagues,

Time-delay fractional-order systems are dynamical systems that exhibit both fractional-order differentiation and time-delay characteristics. They can be extensively applied in practical system modeling and control. Research into the theory, stability analysis, and applications of these systems is crucial for developing a profound understanding and for their utilization. In terms of theory, the study of time-delay fractional-order systems involves fractional calculus, the effects of time delay, and system modeling and analysis. Stability analysis holds great significance in the study of time-delay fractional-order systems as it explores stability conditions and boundaries to ascertain the system's stability characteristics. For time-delay fractional-order systems, stability analysis includes identifying stability regions, stability boundaries, and criteria. The goal of stability analysis is to ensure system stability and performance in the presence of time delays and fractional-order characteristics. Time-delay fractional-order systems find wide applications in various fields. For instance, in electrical engineering, time-delay fractional-order systems are used for stability analysis and control of power systems. In biomedical engineering, they are employed for neural signal processing and biological system modeling. In finance, time-delay fractional-order systems are useful for market forecasting and risk management. The application of time-delay fractional-order systems also extends to fields such as mechanical engineering, chemical engineering, and control systems. In conclusion, research on the theory, stability analysis, and applications of time-delay fractional-order systems in various domains is of significant importance for a comprehensive understanding and practical utilization of these systems.

The focus of this Special Issue is to continue to advance the research on topics relating to the theory, stability analysis, and applications of fractional-order systems with delay. Topics that are invited for submission include (but are not limited to):

  • Stability analysis of fractional-order systems;
  • Stability analysis and control design of fractional-order neural networks;
  • Nonlinear analysis of distributed-order systems and neural networks;
  • Fractional-order and distributed-order control systems and their implementation;
  • Effects of delay, impulse, and perturbance on fractional-order systems;
  • Dynamics analysis of stochastic fractional-order systems and neural networks.

Dr. Xujun Yang
Prof. Xiaofeng Chen
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional-order systems
  • distributed-order systems
  • fractional-order neural networks
  • delay, impulse, and perturbance
  • bifurcation, chaos, and synchronization
  • stability analysis
  • controller design

Published Papers (2 papers)

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Research

26 pages, 10413 KiB  
Article
Bifurcation Exploration and Controller Design in a Fractional Oxygen–Plankton Model with Delay
by Yunzhang Zhang and Changjin Xu
Fractal Fract. 2024, 8(4), 190; https://doi.org/10.3390/fractalfract8040190 - 27 Mar 2024
Viewed by 645
Abstract
Fractional-order differential equations have been proved to have great practical application value in characterizing the dynamical peculiarity in biology. In this article, relying on earlier work, we formulate a new fractional oxygen–plankton model with delay. First of all, the features of the solutions [...] Read more.
Fractional-order differential equations have been proved to have great practical application value in characterizing the dynamical peculiarity in biology. In this article, relying on earlier work, we formulate a new fractional oxygen–plankton model with delay. First of all, the features of the solutions of the fractional delayed oxygen–plankton model are explored. The judgment rules on non-negativeness, existence and uniqueness and the boundedness of the solution are established. Subsequently, the generation of bifurcation and stability of the model are dealt with. Delay-independent parameter criteria on bifurcation and stability are presented. Thirdly, a hybrid controller and an extended hybrid controller are designed to control the time of onset of bifurcation and stability domain of this model. The critical delay value is provided to display the bifurcation point. Last, software experiments are offered to support the acquired key outcomes. The established outcomes of this article are perfectly innovative and provide tremendous theoretical significance in balancing the oxygen density and the phytoplankton density in biology. Full article
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28 pages, 1633 KiB  
Article
Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays
by Călin-Adrian Popa
Fractal Fract. 2023, 7(11), 830; https://doi.org/10.3390/fractalfract7110830 - 20 Nov 2023
Viewed by 999
Abstract
Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They [...] Read more.
Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example. Full article
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