On the Mittag-Leffler stability of mixed-order fractional homogeneous cooperative delay systems

TL;DR

This paper establishes Mittag-Leffler stability for multi-order fractional delay systems with cooperative and homogeneous vector fields, providing conditions for stability and convergence rates, applicable even with differing homogeneity degrees.

Contribution

It introduces a novel stability analysis method for multi-order fractional delay systems with structural assumptions, including cases with different homogeneity degrees.

Findings

Proves local and global Mittag-Leffler stability under certain conditions.

Derives convergence rates of solutions to equilibrium.

Validates results with two illustrative examples.

Abstract

In this paper, we study a class of multi-order fractional nonlinear delay systems. Our main contribution is to show the (local or global) Mittag-Leffler stability of systems when some structural assumptions are imposed on the "vector fields": cooperativeness, homogeneity, and order-preserving on the positive orthant of the phase space. In particular, our method is applicable to the case where the degrees of homogeneity of the non-lag and lag components of the vector field are different. In addition, we also investigate in detail the convergence rate of the solutions to the equilibrium point. Two specific examples are also provided to illustrate the validity of the proposed theoretical result.