分数阶网络系统稳定性分析新方法

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本文主要探讨了分数阶耦合系统稳定性分析的新方法，特别是针对网络上的一类非线性问题。作者是Hong-Li Li、Yao-Lin Jiang、Zuolei Wang、Xiaomei Feng和Zhidong Teng，他们的研究成果发表在2015年10月19日的《国际计算机数学杂志》(International Journal of Computer Mathematics)上，该刊的ISSN号为0020-7160（印刷版）和1029-0265（在线版）。文章的DOI为10.1080/00207160.2015.1105361，这意味着可以通过这个数字对象标识符访问到原文。文章的核心内容集中在分数阶微分方程耦合系统稳定性研究上，分数阶系统由于其在物理、工程和金融等领域中的广泛应用，使得这类系统的稳定性分析具有重要意义。Lyapunov函数是稳定性理论中的核心工具，作者通过构造合适的Lyapunov函数，提出了新的稳定性判据，这些判据能够有效地评估分数阶耦合系统是否稳定，这对于设计和控制复杂网络系统至关重要。 Lyapunov函数是一种用于确定动态系统稳定性的重要数学工具，它满足如果系统稳定，那么Lyapunov函数随着时间推移会单调减少。在本文中，作者可能利用了分数阶微分方程的特性，如 Caputo 导数或Riemann-Liouville导数，来构建适应分数阶系统特性的Lyapunov函数。这可能涉及到对系统动态的精细分析，以及对Lyapunov函数形式的特定选择，例如指数型、多项式型或者其他更复杂的函数结构。文章的贡献还包括提供了一套完整的使用和访问条件，可以在[Tandfonline](http://www.tandfonline.com/action/journalInformation?journalCode=gcom20)网站上找到。此外，作者在10月16日提交的接受稿版本于同年发布，这表明他们的研究成果已经过同行评审并准备发表。这篇论文为分数阶耦合系统稳定性分析提供了一个新颖且实用的方法，对于理解这类系统的长期行为和确保其安全性具有重要的实际意义。对于任何从事分数阶系统研究，特别是在网络科学、控制理论或信号处理领域工作的研究人员来说，这篇文章都是一份有价值的技术参考文献。

Accepted Manuscript

see [10-12] and references therein. Moreover, the method was extended to stochastic

CSIDENs, discrete-time CSIDENs, delayed CSIDENs, impulsive CSIDENs [13-16].

As we know, fractional diﬀerential equations are generalizations of integer-order diﬀer-

ential equations. These generalizations are not mere mathematical curiosities but rather

they have important applications in many areas of science and engineering such as elec-

trochemistry, control, porous media, electromagnetism, etc., see [17-22]. In [21], Wang et

al. analysed the synchronization conditions of the fractional order chaotic systems with

activation feedback method. Based on the above discussion, it is interesting to investigate

the dynamical behaviors for coupled systems of fractional diﬀerential equations on net-

works (CSFDENs). Wider applications of CSFDENs require the formulation of eﬀective

criteria for the stability of their solutions. However, to the best of our knowledge, there is

seldom work so far dedicated to the investigation of qualitative properties for CSFDENs.

In this paper, we study the stability, uniform stability and uniform asymptotic stability

of CSFDENs.

Compared with the existing results (e.g., [23-26] and other results), contributions of this

paper are as follows. Firstly, we provide a systematic method for constructing a Lyapunov

function for CSFDEN by combining graph theory and Lyapunov method. Secondly, some

new stability principles (stability principle, uniform stability principle, uniform asymptotic

stability principle), which have a close relation to the topology property of the network,

are given. Finally, by numerical simulations, we show the eﬀect of fraction order on the

solutions of the system.

This paper is organized as follows. In Section 2, we will give some preliminaries which

will be used in Section 3. In Section 3, we investigate the stability of CSFDEN, and some

suﬃcient conditions are given. In Section 4, some numerical simulations are presented to

verify the theoretical results. Finally, conclusions are given in Section 5.

2 Preliminaries

In this section, we introduce some basic deﬁnitions and lemmas. Since the system

considered in this paper is build on a directed graph, the necessary basic concepts and

theorems on graph are also reviewed in this section.

Let R

be the m-dimensional Euclidean space with norm k · k and let R

= [0, ∞)

and L = {1, 2, · · · , n}.

Deﬁnition 2.1. A function a(·) : R

→ R

belongs to class K if it is continuous, strictly

increasing and a(0) = 0.

Deﬁnition 2.2 [27]. For any t ≥ t

, the Caputo’s fractional derivative of order q, 0 <

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