Yao Zichen, Yang Zhanwen, Fu Yongqiang
School of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China.
Chaos. 2024 Apr 1;34(4). doi: 10.1063/5.0188371.
We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.
我们介绍了一个关于具有Caputo阶数α∈(1,2)的非线性分数阶延迟微分方程（FDDEs）的线性化渐近稳定性的定理，该定理可直接用于分数阶延迟神经网络。它依赖于三个技术工具：对特征方程的详细根分析、对广义Mittag-Leffler函数的估计以及李雅普诺夫第一方法。我们通过对拉普拉斯变换得到的特征方程进行详细根分析，提出系数型准则以确保线性FDDEs的稳定性。此外，在这些准则下，我们给出了广义Mittag-Leffler函数的一个巧妙表达式，并证明了它们的多项式长时间衰减率。利用成熟的李雅普诺夫第一方法，我们证明，如果非线性Caputo FDDE在平衡点处的线性化系统是渐近稳定的，那么该平衡点就具有渐近稳定性。最后，作为我们结果的一个副产品，我们明确描述了分数阶延迟神经网络的渐近性质。为了说明我们理论结果的有效性，还给出了数值模拟。
