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Stability for coupled systems on networks with Caputo-Hadamard fractional derivative. (English) Zbl 1488.34022

Summary: This paper discusses stability and uniform asymptotic stability of the trivial solution of the following coupled systems of fractional differential equations on networks \[ \left\{\begin{array}{lll} ^{cH}D^{\alpha}x_i=f_i(t,x_i)+\sum\limits_{j=1}^ng_{ij}(t,x_i,x_j),\quad t>t_0,\\ x_i(t_0)=x_{i0}, \end{array}\right. \] where \(^{cH}D^{\alpha}\) denotes the Caputo-Hadamard fractional derivative of order \(\alpha,1<\alpha\leq 2\), \(i=1,2,\dots,n\), and \(f_i:\mathbb{R}_+\times\mathbb{R}^{m_i}\to\mathbb{R}^{m_i}\), \(g_{ij}:\mathbb{R}_+\times\mathbb{R}^{m_i}\times\mathbb{R}^{m_j}\to\mathbb{R}^{m_i}\) are given functions. Based on graph theory and the classical Lyapunov technique, we prove stability and uniform asymptotic stability under suitable sufficient conditions. We also provide an example to illustrate the obtained results.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34D20 Stability of solutions to ordinary differential equations
05C90 Applications of graph theory
92B20 Neural networks for/in biological studies, artificial life and related topics
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